Nonlinear Response Of Strongly Intera ting Quantum Systems in
Nonequilibrium
A Thesis
submitted to the Fa ulty of the
Graduate S hool of Arts and S ien es
of Georgetown University
in partial ful�llment of the requirements for the
degree of
Do tor of Philosophy
in Physi s
By
Khadijeh Naja�, M.S .
Washington, DC
April 5, 2018
Copyright
© 2018 by Khadijeh Naja�
All Rights Reserved
ii
Nonlinear Response Of Strongly Intera ting Quantum Systems
in Nonequilibrium
Khadijeh Naja�, M.S .
Thesis Advisor: James K. Freeri ks, Ph.D. in Physi s
Abstra t
The re ent developments in experimental te hniques su h as pump-probe spe -
tros opy, new te hnologies in building two dimensional materials, and the advent of
highly tunable systems in the form of ultra old opti al latti es, trapped ions and
array of atoms have opened a new path to investigate the dynami s of strongly or-
related system out of equilibrium, whi h would not be possible with the onventional
methods. These new developments not only provided a powerful tool to study some of
the most fundamental questions in the nonequilibrium regime of quantum systems but
also made it possible to explore new phenomena whi h has not been observed before.
Motivated by some of these experiments, in this thesis, we study a wide range of
strongly orrelated systems out of equilibrium. We will use a ombination of theoret-
i al and omputational methods su h as sum rules, nonequilibrium Green's fun tion,
nonequilibrium dynami al mean-�eld theory (DMFT), and an exa t solution of the
non-intera ting model to address a wide spe trum of systems out of equilibrium. First,
we have generalized a formalism for the nth derivative of a time-dependent operator
in the Heisenberg representation and employ it to the spe tral sum rules in whi h we
obtain a qualitative understanding of the pump e�e t on ele tron-phonon oupling in
high Tc super ondu tors. Then, by using the nonequilibrium DMFT for the Fali ov-
Kimball model, we obtain the urrent-voltage pro�le of a multilayer devi e whi h
onsists of a single barrier region (usually insulator plane) onne ted to a number
of metalli leads in both sides. To improve the onservation of urrent and �lling
iii
a ross the barrier region, we further develop an optimization method. From a dif-
ferent aspe t, we use the DMFT solution of the Fali ov-Kimball model to enhan e
the riti al temperature of quantum ordering, whi h is a hallenging problem as the
riti al temperature lies below urrently a essible temperatures. We further propose
a few mixtures su h as Yb− Cs and Sr− Cs as a possible andidate for dete ting the
riti al temperature enhan ement e�e t. Finally, in the last hapter, we study some
of the most interesting dynami al quantities su h as the probability of revivals, the
light one velo ity, formation probabilities and Shannon information in the XY hain.
Although XY hain is a free fermioni system, it has been onsidered as one of the
most interesting models from both theoreti al and experimental views. Be ause this
model, not only manifests a quantum phase transition but it has been the subje t of
the array of trapped ions and neutral atoms whi h are the most promising andidates
for a quantum simulator. We show that the formation probabilities, revival probabili-
ties, and observed propagation velo ity are a tually state-dependent and non-trivially
predi table in the XY hain.
Index words: Nonequilbrium dynami al mean-�eld theory, Fali ov-Kimball
model, ele tron-phonon intera tion, high Tc super ondu tor,
Sum rules, Multilayer devi e, Current-voltage pro�le,
Antiferromagneti ordering, quantum ordering enhan ement,
Quantum spin hain, Formation probabilities, Shannon
information, Light- one velo ities, thesis
iv
Dedi ation
To my parents, Akbar and Batool.
v
A knowledgments
First and foremost I would like to thank my advisor, Professor. Jim Freeri ks,
who has supported me throughout my Ph.D. with his patien e and guidan e at key
moments while allowing me to work independently. It has been a true honor to work
with him. In parti ular, I appre iate all of his e�orts, en ouragements, and time to
make my Ph.D. experien e very produ tive and rewarding. Thanks to him, I had a
spe ta ular opportunity to work on multiple subje ts whi h have helped me tremen-
dously to deepen my intuition and knowledge in the �eld of strongly orrelated sys-
tems out of equilibrium.
Next, I would like to o�er my sin erest gratitude to my olleague and the best
ollaborator I ould ask for, Mohammad Ali Rajabpour, whom not only has been a
great person but also has thauht me how to be ome a persistent resear her. I am
truly grateful for all of his endless support in my professional areer and personal life
sin e 2010.
I would like also to take the opportunity to thank my dissertation ommittee:
Prof. Amy Liu, Edward Van Kueren and Hans Engler for their time and e�ort for
helping me through my dissertation. In parti ular, Prof. Amy Liu, who has been a
great support throughout my Ph.D. and an ex ellent example of a su essful female
physi ist.
I am also grateful to my other ollaborators: Prof. Ma iej Maska, Paul. S. Julienne,
Thomas Devereaux, Alexander F. Kemper, and Ja opo Viti who I had the honor to
work with on di�erent proje ts. Additionally, I would like to o�er my spe ial thanks
vi
to Woonki Chung, the systems administrator who has helped me with o asional
omputational issues.
During my graduate study, I have met many post-do s whose advi e and friendship
I have bene�ted from in my professional areer. I am espe ially grateful to Ehsan
Khatami, Oleg Matveev, Herbert Fotso, Karlis Mikelsons, Juan Carrasquilla, Rubem
Mondiani, Mohammad Maghrebi, Miles Stoudenmire, Andreas Dirks, Greg Boyd,
Louk Rademaker, and Valentine Stanev.
In my daily work, I have been blessed with a friendly and supportive group of
fellow students and post-do s. In parti ular, I would like to thank Jesús Cruz-Rojas,
Bry e Yoshimura, Oliver Albertini, Je� Cohn, Claudia Dessi, Mona Kaltho�, and
Manuel Weber, with whom I have enjoyed to share many s ienti� dis ussions and
unforgettable memories.
Prof. Freeri ks, provided me a fantasti opportunity to mentor multiple under-
graduate students. Parti ularly, I would like to thank Kahlil Dixon, Forest Yang, and
Alex Ja obi who ontributed to my Ph.D. resear h with their talent and enthusiasm.
I also would like to express my gratitude to Physi s department sta�; Amy Hi ks,
Thomas Lewis, Jennifer Liang, Jannet Gibson, and Mary Rashid who have been a
great support in administrative work.
Beyond physi s, my time at Georgetown was made memorable due to the many
friends that be ame a part of my life. More spe i� ally, I am grateful to the friend-
ship of Eri Urano, Maryam Shojaie, Azadeh Ghayomi, Afsaneh Ranghiani, Miriam
Bolanes, Azadeh Bagheri, Neghar Ghahremani, Samyeh Mahmoodian, Sara Shahabi,
and Behnaz Bagheri from whom I have bene�ted tremendously of their emotional
support.
Last but not least, I would like to express my sin ere gratitude to my parents
and family members for their endless support and love throughout my life. Thanks
vii
to them and espe ially my oldest brother Dariush Naja�, who made it possible for
me to pursue my dream of be oming a physi ist.
viii
Table of Contents
1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Nonequilibrium dynami al mean �eld theory . . . . . . . . . . . . . . . 7
2.1 Nonequilibrium Green's fun tion formalism . . . . . . . . . . . . 8
2.2 Nonequilibrium dynami al mean �eld theory for Fali ov-Kimball
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Numeri al implementation of nonequilibrium DMFT . . . . . . . 25
3 Nonequilibrium spe tral moment sum rules for systems with ele tron-
phonon intera tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Spe tral sum rules for the retarded Green's fun tion . . . . . . 36
3.2 Nonequilibrium sum rules for the Holstein model . . . . . . . . . 44
3.3 Nonequilibrium sum rules for the Hubbard-Holstein model . . . 54
4 Current-voltage pro�le of a strongly orrelated materials heterostru ture
using non-equilibrium dynami al mean �eld theory . . . . . . . . . . . 73
4.1 Non equilibrium DMFT formalism for a multilayer devi e . . . . 75
4.2 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Fixing the boundary ondition . . . . . . . . . . . . . . . . . . . 81
4.4 Impurity solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5 Cal ulating the urrent in multilayer devi e . . . . . . . . . . . . 88
4.6 Optimizing of the DMFT-Zip algorithm . . . . . . . . . . . . . . 99
4.7 Con lusion: Current-Voltage (I-V) pro�le . . . . . . . . . . . . . 102
5 Designing mixtures of ultra old atoms to boost Tc by using dynami al
mean �eld theory solution . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1 Intera tion of light and matter . . . . . . . . . . . . . . . . . . . 107
5.2 DMFT solution for mixture of heavy and light parti les in opti al
latti e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 Enhan ing quantum order with fermions by in reasing spe ies
degenera y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Nonequilbrium dynami s of XY hain . . . . . . . . . . . . . . . . . . . 124
6.1 XY spin hain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2 Dynami s of observables in XY spin hain . . . . . . . . . . . . 130
ix
6.3 On the possibility of omplete revivals after quantum quen hes
to a riti al point . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.4 Light- one velo ities after a global quen h in a non-intera ting
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.5 Formation probabilities and Shannon information and their time
evolution after quantum quen h in the transverse-�eld XY hain 173
7 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8 Publi ation List of Khadijeh Naja� . . . . . . . . . . . . . . . . . . . . 212
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
x
List of �gures
2.1 Kadano�-Baym-Keldysh ontour . . . . . . . . . . . . . . . . . . . . 12
2.2 Mapping a latti e model into a single impurity problem . . . . . . . 18
2.3 S hemati of dis retization on the Keldysh-Baym ontour . . . . . . . 26
2.4 Boundary ondition on the Keldysh-Baym ontour . . . . . . . . . . . 28
3.1 The s hemati pi ture of tr-Arpes experiment . . . . . . . . . . . . . 33
3.2 Time-resolved spe tra of high Tc uprate super ondu tor . . . . . . . 35
4.1 S hemati pi ture of a multilayer devi e . . . . . . . . . . . . . . . . 75
4.2 Retarded Green's fun tion in bulk and barrier . . . . . . . . . . . . . 91
4.3 Retarded Green's fun tion at the �rst plane . . . . . . . . . . . . . . 92
4.4 Retarded Green's fun tion at one and two adja ent planes to the barrier 93
4.5 Lo al lesser Green's fun tion in the bulk at low temperature T = 0.01 94
4.6 Current through the multilayer devi e. . . . . . . . . . . . . . . . . . 95
4.7 Current of multilayer devi e in the presen e of ele tri �eld A = π/20 97
4.8 Current and �lling through the multilayer devi e for A = 6π/20 . . . 98
4.9 Current and �lling through the multilayer devi e in the presen e of
ele tri �eld for A = 2π/20 . . . . . . . . . . . . . . . . . . . . . . . 98
4.10 Optimization of urrent for di�erent values of ve tor potential with
single layer nonzero ele tri �eld . . . . . . . . . . . . . . . . . . . . . 101
4.11 Comparing the urrent obtained from the optimization algorithm . . 102
4.12 Current-Voltage pro�le for a multilayer devi e . . . . . . . . . . . . . 103
5.1 Enhan ement of quantum ordering riti al temperature . . . . . . . . 105
5.2 Comparison of the 2D DMFT and MC . . . . . . . . . . . . . . . . . 118
5.3 The maximal riti al temperature plotted as a fun tion of mobile
fermion degenera y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1 Di�erent riti al regions in the quantum XY hain . . . . . . . . . . 128
6.2 S hemati set up of quen h of trapped ion experiment . . . . . . . . . 132
6.3 Distribution of domain size of a one dimensional spin hain with 53 spins133
6.4 Di�erent regions in the phase diagram of the quantum XY hain . . 137
6.5 Logarithmi �delity for the periodi riti al Ising hain starting . . . 140
6.6 Logarithmi �delity for the open riti al Ising hain . . . . . . . . . . 141
6.7 Group velo ity vφ with respe t to φ . . . . . . . . . . . . . . . . . . . 142
6.8 Maximum group velo ity . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.9 Los hmidt e ho on the Ising line for periodi and open hains . . . . 144
6.10 Los hmidt e ho on the XY riti al line for periodi and open hains . 144
6.11 Los hmidt e ho on the line h = 0.8 for periodi and open boundary
ondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
xi
6.12 ∆ln(t) for the XX hain with di�erent initial states . . . . . . . . . . 157
6.13 ∆ln(t) for the XY hain with various γ and h parameters . . . . . . . 159
6.14 |Cln|2, |Fln|2, and ∆ln(t) for the XY hain . . . . . . . . . . . . . . . 160
6.15 The ontinuous blue urve is the fun tion −Ai2(−X) . . . . . . . . . 163
6.16 The evolution of the entanglement entropy in the XY hain . . . . . . 166
6.17 ve�max(2) and ve�
max(1) for di�erent values of a and h . . . . . . . . . . . 172
6.18 Π(l, L)− αl for periodi system with total length L = 2000 . . . . . . 187
6.19 Π(l, L)− αl for open system with total length L = 2000 . . . . . . . . 188
6.20 The oe� ient of the logarithmi term for two on�gurations . . . . . 190
6.21 Π(l, L)−αl for periodi system with total length L = 300 with respe t
to l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.22 Values of f(x), αmin and αmax with respe t to x . . . . . . . . . . . . 196
6.23 The ontributions of di�erent ranks k in the Shannon information for
two sizes l = 14 and 26 . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.24 The error E(xm, l) in the evaluation of Shannon information oming
from the trun ation at the rank k = xml . . . . . . . . . . . . . . . . 198
6.25 The ontributions of di�erent ranks k in the Shannon information . . 199
6.26 The evolution of logarithmi formation probability of di�erent on�g-
urations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.27 The evolution of Shannon information of a subsystem with di�erent sizes201
6.28 The mutual information between a pair of dimers . . . . . . . . . . . 202
6.29 The evolution of mutual information between two adja ent subsystems
in two di�erent ases . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
xii
List of tables
6.1 Properties of the four L × L blo ks of the matrix T for a quadrati
Hamiltonian (6.37). The notation is obvious and time dependen e is
omitted here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2 Fitting parameters for the logarithmi formation probabilities . . . . 187
6.3 Fitting parameters for the logarithmi formation probability of anti-
ferromagneti on�gurations . . . . . . . . . . . . . . . . . . . . . . . 191
6.4 Fitting parameters for di�erent on�gurations . . . . . . . . . . . . . 192
A1 Shannon information al ulated for sizes l = 1, 2, ..., 39 . . . . . . . . 207
xiii
1
Introdu tion
� More often seems like it should be more of the same, but more is di�erent.�
� P.W. Anderson
The �eld of nonequilibrium phenomena of strongly orrelated systems has be ome
one of the most a tive and hallenging parts of ondensed matter physi s. The strong
intera tion between ele tron, phonon, and spin degrees of freedom in strongly or-
related systems has reated a variety of interesting phases of matter su h as Mott
insulators (metal to insulator transition), super ondu tivity, harge density waves,
and antiferromagneti ordering to name a few. The number of exoti phenomena
with emerging phases of the matter even in reases as we drive the system out of equi-
librium [1℄. In general, theoreti al study of these systems be omes hallenging be ause
the strongly orrelated systems have a non-perturbative nature, and most of the usual
te hniques whi h are based on a perturbative approa h annot be applied. Moreover,
in the nonequilibrium ase, one needs to study the time evolution of the many-body
system whi h be omes problemati as the Hilbert spa e grows exponentially. The
exponential growth of the Hilbert spa e also remains a problem even in the equilib-
rium ase. However, there has been a number of omputational methods developed
for studying strongly orrelated systems with di�erent hara teristi s, ea h one with
its own advantages and limitations. In addition, various developments in experiments
with higher ontrol and a ura y have provided a great opportunity to study the
1
dynami s of su h systems. Here, we provide a list of su h experimental developments
on whi h we will be fo using in this thesis:
1. Developments in ultrafast pump-probe spe tros opy whi h have provided a
powerful tool to study the ex itation and relaxation dynami s of orrelated ele troni
and phononi degrees of freedom.
2. Designing and building two-dimensional materials that manifest di�erent ele -
troni properties from their bulk omponents, whi h leads to new emerging phe-
nomena.
3. Development of opti al latti es and trapped ion hain that provides a highly
tunable and ontrollable system.
The nonlinear behavior of a wide range of strongly orrelated systems whi h are
driven into nonequilibrium states will be the subje t of this thesis. For this purpose,
we have used various theoreti al and omputational methods of whi h we will provide
a brief des ription as follows. As we know, one of the simplest models to des ribe the
strongly orrelated systems is the Hubbard model whi h is de�ned as,
HHub = −∑
ijσ
tij(c†iσcjσ + c†jσciσ) + U
∑
i
ni↑ni↓ (1.1)
where throughout this thesis, c†i and cj indi ate the fermioni reation and annihilator
operator on site i and j, respe tively, and U is the Coulomb repulsion. However, ea h
hapter of this thesis will be devoted to slightly di�erent model in order to investigate
the various intera tions between harge, spin, and phononi degrees of freedom. Below,
�rst, we will des ribe some of the related experiments as a motivation to our work,
and subsequently, des ribe the models and methods that we have used to study ea h
parti ular system in nonequilibrium regime.
In hapter 2, we will fo us on the simpli�ed version of the Hubbard model, the
so- alled Fali ov-Kimbal model where only one of the spe ies of ele trons hop on the
2
latti e ( ele tron) and the other spe ies is lo alized (f ele tron) [2℄:
HFK = −∑
ij
tij(c†icj + c†jci) + Ucf
∑
i
c†icif†i fi + Ef
∑
i
f †i fi, (1.2)
where, Ucf is the intera tion between itinerant and lo alized ele trons, and Ef is the
energy of the lo alized ele trons. The advantages of the Fali ov-Kimball model is that
the dynami al mean �eld theory(DMFT) an be solved exa tly as nf = f †i fi ommutes
with Hamiltonian and onsequently, one an obtain the Green's fun tion with respe t
to c ele tron [3℄. The DMFT is onsidered as one of the most powerful methods to
study strongly orrelated systems out of equilibrium. In hapter 2, �rst, we provide
an introdu tion to nonequilibrium Green's fun tion whi h is essential to des ribe the
DMFT. Then, we explain the nonequilibriumDMFT whi h was su essfully developed
by Freeri ks, Turkowski, and Zlati¢ (2006) where they employed the Kadano�-Baym
formalism to des ribe the real-time evolution of a latti e system deriven by an external
ele tri �eld [4℄. Then, we provide a brief des ription of how one an implement
the nonequilibrium DMFT whi h has been proposed by Freeri ks et al in Ref.[5℄.
Furthermore, we will use the DMFT algorithm in hapters 4 and 5.
As we mentioned, the pump-probe spe tros opy provides a powerful tool to ex ite
individual ele tron or phonon degrees of freedom [6, 7℄. Subsequently, in hapter 3,
we will fo us on a re ent study where the role of the pump on the weakening of the
ele tron-phonon oupling in a high Tc Cuprate based super ondu tor[8, 9℄. In order
to investigate the ele tron-phonon intera tion and its interplay with the ele tron-
ele tron intera tion, we will onsider the Hubbard-Holstein model whi h is des ribed
3
as
HHH(t) = −∑
ijσ
tijc†iσcjσ +
∑
i
Uini↑ni↓
+∑
iσ
[gixi − µi]niσ
+∑
i
1
2Mip2i +
∑
i
1
2κix
2i
, (1.3)
where in addition to the hopping and on-site Coulomb intera tion, we have to add
the Hamiltonian of the harmoni os illator and it's oupling with harges. In the
above Hamiltonian, p and x indi ate the momentum and position of the phonon
degrees of freedom, and µ indi ates the hemi al potential. Furthermore, g is the
oupling strength between ele tron and phonon degrees of freedom. First, we derive
a general pro edure for evaluating the nth derivative of a time-dependent operator
in the Heisenberg representation and employ this approa h to al ulate the spe tral
fun tion of retarded Green's fun tion. The sum rules are an exa t formalism whi h an
be applied both in equilibrium and nonequilibrium to study the nonlinear behavior of
strongly orrelated systems. We will losely follow the generalization of the sum rule
in nonequilibrium whi h has been previously obtained by Freeri ks et al for various
models[10, 11, 12, 13℄ and report the results of the spe tral moment sum rules both for
the Holstein model whi h has been published in Ref. [14℄ and, the Hubbard-Holstein
model whi h is near submission.
In addition, from a di�erent aspe t, strongly orrelated materials entered a new
era with the development of thin-�lm growth te hnology to reate two-dimensional
materials whi h have revolutionized our daily life. One ategory of su h materials is
transition metal oxides (TMO) whi h is an ideal andidate for the study of ele tron
orrelation as ele trons in transition metals lie in narrow d orbitals whi h ause strong
4
intera tions. This be omes even more fas inating as new phenomena an emerge in
the interfa e of TMOs [15℄. For example, in a re ent study, two layers of intrinsi
insulators LaAlO3 and SrTiO3 grown on top of ea h other forms a highly ondu tive
layer at the interfa e [16℄. In fa t. one of the most important hara teristi s of these
systems is the urrent-voltage pro�le. In hapter 4, �rst, we introdu e a multilayer
devi e onsisting of an array of metal and insulator planes, and then, we provide
a summary of urrent biasing DMFT method for the Fali ov-Kimball model whi h
has been developed by Freeri ks [17℄. In order to maintain the urrent and �lling
onservation, we develop an optimization method. We report the results before and
after the optimization in whi h manifest an improvement in both urrent and �lling
results. Finally, we present a urrent-voltage pro�le for the di�erent regime of barrier
planes.
Furthermore, the DMFT solution of Fali ov-Kimball model provides a path to
enhan e the quantum ordering in the mixture of heavy and light mixture [18℄. In
hapter 5, we fo us on the hallenges of dete ting a quantum ordering at low temper-
ature. Although, there has been an intensive e�ort to dete t the antiferromagneti
phase of Hubbard model, a hieving low temperature below the Neel ordering tem-
perature still remains hallenging[19, 20℄. Instead, we use the results of al ulation of
spin and harge sus eptibility of Fali ov-Kimball model obtained by Freeri ks et al in
Ref. [18℄ whi h suggest that in reasing the degenera y of light(fermioni ) spe ies will
lead to an in rease in the riti al temperature of quantum ordering. First, we brie�y
explain the intera tion of light and matter whi h plays a ru ial role both in trapping
and ooling of atoms in ultra old atoms. Then, we losely follow the derivation of
harge sus eptibility of the Fali ov-Kimball model from Ref. [18℄ for a pedagogi al
reason. Finally, we report the published results where we have used the DMFT to al-
ulate the enhan ement of riti al temperature of the light-heavy mixture for di�erent
5
degenera ies of light spe ies and proposed a few di�erent mixtures whi h satisfy the
required riteria for su h enhan ement whi h an be onsidered as a good andidates
to manifest su h e�e ts[21℄.
Finally, we have devoted the last hapter of this thesis to study the exa t dynami s
of the free fermioni system. There are multiple reasons to study su h a system.
First, it turns out that one an map the free fermioni system into the XY quantum
hain[22℄ whi h provides a solid laboratory to study the quantum phase transition[23℄
and has importan e from a fundamental point of view. Furthermore, theXY quantum
hain plays an important role in quantum simulators su h as trapped ion experi-
ment [25, 26, 27, 28, 29℄ and old atoms [30℄. In this hapter, we �rst, introdu e
the XY model and we explain how one an map it onto the fermioni degrees of
freedom. Then, we report our published results about the dynami al properties of
the XY hain su h as time evolution of orrelation and entanglement, revival proba-
bility and �nally, the formation probability, and Shannon information and their time
evolution [31, 32, 33℄. Furthermore, we have al ulated the post-measurement entan-
glement entropy and full ounting statisti s of the quantum spin hain, whi h are
beyond the s ope of this thesis and we refer the interested reader to Refs.[34, 35℄.
6
2
Nonequilibrium dynami al mean field theory
�The simpli ities of natural laws arise through the omplexities of the lan-
guage we use for their expression.�
� Eugene Wigner
There exist a variety of methods to study systems in equilibrium su h as mean
�eld theory and the renormalization group. These methods allow us to understand
the experimental data of a omplex systems at low temperature by des ribing their
intera tions in terms of simple e�e tive models with relatively few degrees of freedom.
The problem be omes less lear for systems driven into the nonequilibrium regime,
as it is not obvious that the dynami s of a omplex system an be explained by the
dynami al des ription of simpli�ed models. The e�ort to �nd an analyti al approa h
to understand the time evolution of generi many body quantum systems started in
early 1960's, when S hwinger, Kadano�, Baym and, Keldysh developed the nonequi-
librium Green's fun tion formalism [36, 37, 38℄. This formalism is an extension of
the imaginary time equilibrium formulation [39℄. Later, the quantum kineti approa h
was developed by Rammer [40℄, whi h is an alternative method based on the nonequi-
librium Green's fun tion. The kineti approa h des ribes the evolution of a system
towards its thermal state on a long time s ale and it an not apture the dynami s of
the system at short times. Other methods su h as the density matrix renormalization
group (DMRG) [41℄, are dire tly involved in the evolution of wavefun tion and are
7
more suitable for one-dimensional systems [42, 43℄. In ontrast, the nonequilibrium
dynami al mean-�eld theory (DMFT) works e�e tively in the thermodynami limit
and it an be applied to both short and long times ales. Although the main restri tion
of DMFT lies in the lo al approximation of the self-energy, whi h may not be appro-
priate for systems with spatial nonlo al orrelations; these orrelations an be taken
into a ount by additional diagrammati and luster expansions. In this hapter, we
will explain the nonequilibrium dynami al mean-�eld theory. First, in se tion 2.1,
we will explain the nonequilibrium Green's fun tion formalism and will des ribe the
Kadano�-Baym formalism for the ontour ordered Green's fun tion and the Keldysh
formalism for the nonequilibrium steady state. In se tion 2.2, we will explain the
nonequilibrium dynami al mean-�eld theory and in parti ular, we will dis uss an
exa t formalism for the the Fali ov-Kimbal model in whi h it was su essfully gen-
eralized by Freeri ks and et al in Ref. [4℄. Then, in se tion 2.3 we will explain how
one an imply the omputational for the DMFT formalism in nonequilbrium whi h
is proposed by Freeri ks et al in Ref. [5℄.
2.1 Nonequilibrium Green's fun tion formalism
The main goal of many body theory is to al ulate orrelation fun tions. In this se -
tion, we will des ribe the nonequilibrium Green's fun tion approa h whi h an be
appli able to arbitrary time evolution of orrelated systems and ontains informa-
tion regarding the orrelation of the strongly orrelated system. First, we will review
the Heisenberg representation as we will use it frequently in this thesis. Then, we
explain the Kadano�-Baym formalism as a general formulation of a Green's fun tion
in nonequilbrium, whi h an immediately be des ribed by the Heisenberg representa-
tion of the operators. The nonequilibrium Green's fun tion is a general formalism in
8
whi h does not involve any assumption about the distribution of strongly orrelated
system out of equilibrium as the time evolution of the distribution fun tion an be
obtained from the initial distribution. So in this sense the nonequilibrium Green's
fun tion formalism is an initial value problem while in a di�erent formalism needed
for open system whi h is based on boundary ondition problem. For open systems
with driving for es and dissipation, the system is expe ted to rea h a nonequilibrium
steady state, and onsequently, the Green's fun tion does not hange with time and
they an be solved by the boundary ondition �xed by the bath. This spe i� ase,
is known as the Keldysh formalism, in whi h we will explain afterwards. Finally, in
the last part of this se tion, we will review the results for the equilibrium ase in the
eigenstate basis whi h is known as Lehmann representation.
2.1.1 Heisenberg representation
Before explaining the nonequilibrium formalism for the Green's fun tion, we �rst
review the Heisenberg representation as a powerful tool to study the time evolution
of physi al system in nonequilibrium. For simpli ity, we start with a general time-
dependent impurity Hamiltonian denoted by Hs(t) in the S hrödinger representation.
We know that, in the Heisenberg representation, the time dependen e is en oded in
the operator AH(t), whi h is related to the S hrödinger representation operator As(t)
by
AH(t) = U †(t, t0) As(t) U(t, t0), (2.1)
where t0 is the referen e time (whi h in our ase is equal to −∞) and U(t, t0) is the
evolution operator whi h is de�ned as
U(t, t0) = Te−i
∫ tt0
dtHs(t), (2.2)
9
and
U †(t, t0) = T ei∫ tt0
dtHs(t), (2.3)
Here, T is the time ordering operator whi h orders the operator with later time to
the left. A ordingly T is the anti time ordering operator whi h a ts in the opposite
way. Moreover, the evolution operator satis�es the equation of motion
i∂tU(t, t0) = Hs(t) U(t, t0). (2.4)
Now, let us derive a useful identity for the derivative of c(t). Starting from the de�-
nition of the time dependent operator
c(t) = T ei∫ tt0
dtHs(t) c Te−i
∫ tt0
dtHs(t), (2.5)
we see immediately that
dc(t)
dt= i e
i∫ tt0
dtHs(t)Hs(t) c e−i
∫ tt0
dtHs(t)
− i e−i
∫ tt0
dtHs(t) c Hs(t) e−i
∫ tt0
dtHs(t), (2.6)
whi h be omes
dc(t)
dt= iU †(t, t0) [Hs(t), c] U(t, t0). (2.7)
Now we onsider the following time-independent Hamiltonian
H0s (t) = H − µN = −µc†c, (2.8)
where H0s is the full Hamiltonian in the S hrödinger representation and, for simpli ity,
we absorb the hemi al potential into the de�nition of H0s (t). For this Hamiltonian,
we have
[H0s (t), c] = −µ[c†c, c] = µc, (2.9)
10
so that
dc(t)
dt= iµc(t), (2.10)
whi h is solved by
c(t) = eiµtc. (2.11)
In a similar way, using the ommutation relation [H0s , c
†] = −µ[c†c, c†] = −µc† we
�nd
c†(t′) = e−iµt′c†. (2.12)
2.1.2 Kadanoff-Baym-Keldysh ontour
As we mentioned, orrelation fun tions are one of the important quantities to des ribe
strongly orrelated systems. In this se tion, we fo us on single parti le Green's fun -
tions whi h provide information about the single ex itation and distribution of par-
ti les. We start with the ontour-ordered Green's fun tion whi h is de�ned as:
GCij(t, t
′) = − i〈TC ci(t)c†j(t
′)〉
= − iθC(t, t′)Tre−βH(−∞)ci(t)c
†j(t
′)/Z
+ iθC(t′, t)Tre−βH(−∞)c†j(t
′)ci(t)/Z. (2.13)
whi h is de�ned on the ontour C that runs from tmin to tmax, and ba k from tmax
to tmin and then, goes to tmin − iβ. In addition, β = 1/kBT is de�ned as the inverse
temperature and Z = Tre−βH(−∞). In this approa h, one onsiders the initial orrela-
tion by introdu ing it via time evolution on the imaginary axis, see �gure 2.1. The Tc
is time ordering on the ontour with ordering along tmin → tmax → tmin → tmin − iβ.
One an determine di�erent types of Green's fun tion by hoosing the time variables
11
Figure 2.1: Kadano�-Baym-Keldysh ontour with thermal state at t = tmin.
to lie on di�erent bran hes. For example, the time-ordered Green's fun tion is when
t and t′ both lie on the upper bran h,
GTij(t, t
′) = −iTr TT e−βH(−∞)ci(t)c
†j(t
′)/Z, (2.14)
while, the anti-time-ordered Green's fun tion is for the ase when both times lies on
the lower bran h,
GTij(t, t
′) = −iTr TT e−βH(−∞)ci(t)c
†j(t
′)/Z, (2.15)
where TT denotes anti-time ordering and a ts in opposite way of time ordering. Sim-
ilarly, if t lies on the upper bran h and t′ lies on the lower bran h, we get the lesser
Green's fun tion de�ned as
G<ij(t, t
′) = iTr e−βH(−∞)c†j(t′)ci(t)/Z, (2.16)
and greater Green's fun tion where t and t′ lies on the lower and upper bran h respe -
tively,
G>ij(t, t
′) = −iTr e−βH(−∞)ci(t)c†j(t
′)/Z. (2.17)
Subsequently, there is a ase where both terms lie on the imaginary axis,
GMij (τ, τ
′) = −〈T ci(τ)c†j(τ ′)〉. (2.18)
12
this ase has spe ial properties whi h we will explain in the end of this se tion. In
addition, there are Green's fun tions whi h are an be obtained from the mix of real
and imaginary bran hes whi h normally are useful when one studies the transient
behavior of the systems. In general, the Green's fun tion an be expressed as 3 × 3
matrix, where, for simpli ity, we drop the position dependen e,
G =
G++ G+− G+I
G−+ G−− G−I
GI+ GI− GII
, (2.19)
where the indi es +,−, and I orresponds to the time lying on the upper, lower and
imaginary bran hes, respe tively. In fa t, it turns out that in addition to the above
Green's fun tion that an be dire tly obtained from di�erent ordering on the ontour,
one may de�ne a few other important Green's fun tion known as retarded, advan ed,
and Keldysh Green's fun tion,
GRij(t, t
′) = −iθ(t− t′)〈{ci(t), c†j(t′)}〉, (2.20)
GAij(t, t
′) = iθ(t′ − t)〈{ci(t), c†j(t′)}〉, (2.21)
GKij (t, t
′) = −i〈[ci(t), c†j(t′)]〉, (2.22)
where the simple and urely bra ket denotes the ommuting and anti ommuting rela-
tion between fermioni operators, respe tively. In fa t, the lesser and greater Green's
fun tion an be written in terms of retarded, advan ed and Keldysh Green's fun tion:
G<ij(t, t
′) =1
2[GK
ij (t, t′)−GR
ij(t, t′) +GA
ij(t, t′)]
G>(t, t′) =1
2[GK
ij (t, t′) +GR
ij(t, t′)−GA
ij(t, t′)], (2.23)
13
in whi h one may get GKij (t, t
′) = G<ij(t, t
′) + G>(t, t′). From the de�nition of the
Green's fun tion it is straightforward to �nd a relation between the Green's fun tion
and their Hermitian onjugates:
GRij(t, t
′)∗ = GAij(t
′, t),
G<,>,Kij (t, t′)∗ = −G<,>,K
ij (t′, t), (2.24)
Now, we fo us on the Matsubara Green's fun tion as we mentioned that has a few
advantages: it is always translationally invariant GMij (τ, τ
′) = GMij (τ − τ ′), it is Her-
mitian GMij (τ)
∗ = GMij (τ), and it is anti-periodi for fermions GM
ij (τ) = −GMij (τ + β).
Furthermore, one an perform a Fourier transform into the frequen y domain by
GMij (τ) = T
∑
n
e−iωn(τ)GMij (iωn), (2.25)
where
GMij (iωn) =
∫ β
0
dτeiωn(τ)GMij (τ), (2.26)
where ωn = (2n+1)πT for fermioni ase. Finally, in the Green's fun tion formalism,
the intera tion an be taken into a ount by introdu ing the self-energy, whi h is
de�ned on the ontour and it satis�es the same symmetry and boundary ondition
as the Green's fun tion. The self-energy may be introdu ed by the Dyson equation :
G(t, t′) = G0(t, t′) +
∫
Cdt
∫
Cdt′G0(t, t)Σ(t, t′)G(t′, t
′). (2.27)
In the next se tion, we will explain how one may employ the dynami al mean-�eld
theory to obtain the self-energy of systems with lo al intera tions.
2.1.3 Keldysh formalism for nonequilibrium steady state
It turns out that, it is possible to simplify the nonequilibrium formalism for open
systems. This is possible, be ause in this ase, the energy pumped into system by
14
external driving an dissipate through the onne tion with a heat bath, and the
system an rea h a steady state. Furthermore, one may also ignore the imaginary
bran h, as there is no orrelation between the initial state and time evolving states.
The reasoning for ignoring the initial orrelation arises from the dissipation in the
system and onne tion to the heat bath whi h provides large number of degrees of
freedom that in�uen es the long time dynami of the system by stripping out the
information from the initial states. So, this makes it possible to use the Keldysh
formalism. By this methodology, the ontour ordered Green's fun tion be omes a
2× 2 matrix,
G =
G++ G+−
G−+ G−−
. (2.28)
One an rewrite the above matrix with respe t to retarded, advan ed and, Keldysh
Green's fun tions by performing the Larkin-Ov hinkov transformation [45℄
G ≡ Lτ3GL† =
GR GK
0 GA
, (2.29)
where L and τ3 are de�ned as
L =1√2
1 −1
1 1
, (2.30)
and
τ3 =
1 0
0 −1
. (2.31)
We will return to the Keldysh formalism in hapter 4 and provide more information
of how it an be imply to obtain the urrent-voltage pro�le of a multilayer devi e.
15
2.1.4 Lehmann representation
Before ending this se tion, we also brie�y omment on the so alled Lehmann repre-
sentation in whi h one uses a set of eigenstates {|n〉} of the full Hamiltonian. In equi-
librium, the Hamiltonian is independent of time, and the Green's fun tion be omes a
fun tion of time di�eren e, trel = t′ − t. Now, if the energy eigenstates of the Hamil-
tonian are de�ned as, H|n〉 = En|n〉, then by onsidering the diagonal form of the
greater Green's fun tion in equation 2.17, we have
G>(k, t− t′) = −i 1Z 〈ck(t)c†k(t′)〉 = −i 1Z∑
n
〈n|e−βHck(t)c†k(t
′)|n〉
= −i 1Z∑
nn′
e−βEn〈n|ck|n′〉〈n′|c†k|n〉ei(En−En′ )(t−t′). (2.32)
where in the se ond line, we have inserted 1 =∑
n |n〉〈n| and we have absorbed
the hemi al potential inside the Hamiltonian. In addition, by rewriting the ck(t)
in Heisenberg representation, we have used the following identity: 〈n|ck(t)|n′〉 =
〈n|eiHt ck e−iHt|n′〉 = ei(En−En′)(t−t′)〈n|ck|n′〉. By performing the Fourier transform
with respe t to trel = t− t′ into the frequen y domain, we get:
G>(k, ω) =−2πi
Z∑
nn′
e−βEn〈n|ck|n′〉〈n′|c†k|n〉δ(En −En′ + ω), (2.33)
similarly, for the lesser Green's fun tion we get:
G<(k, ω) =2πi
Z∑
nn′
e−βEn〈n|c†k|n′〉〈n′|ck|n〉δ(En − En′ − ω)
=2πi
Z∑
nn′
e−βEn′ 〈n′|c†k|n〉〈n|ck|n′〉δ(En′ − En − ω)
. =2πi
Z∑
nn′
e−β(En+ω)〈n′|c†k|n〉〈n|ck|n′〉δ(En′ −En − ω)
= −G>(k, ω)e−βω. (2.34)
where we have ex hanged the En and En′on the se ond line. In the ase of retarded
Green's fun tion, when we take the Fourier transform,we have to remind ourself about
16
the in�nitesimal onvergen e fa tor iη → i0+
GR(k, ω) = − i
Z
∫ ∞
0
dt
ei(ω+iη)t∑
nn′
e−βEn
(
〈n|ck|n′〉〈n′|c†k|n〉ei(En−En′ )t + 〈n|c†k|n′〉〈n′|ck|n〉e−i(En−En′ )t)
=1
Z∑
nn′
e−βEn
( 〈n|ck|n′〉〈n′|c†k|n〉ω + En − En′ + iη
+〈n|c†k|n′〉〈n′|ck|n〉ω −En + En′ + iη
)
=1
Z∑
nn′
〈n|ck|n′〉〈n′|c†k|n〉ω + En − En′ + iη
(e−βEn + e−βEn′ ). (2.35)
The imaginary part of of the retarded Green's fun tion is de�ned as the single parti le
spe tral fun tion,
A(k, ω) =−1
πIm[GR(k, ω)] =
1
Z∑
nn′
(e−βEn + e−βEn′ )|〈n|ci|n′〉|2δ(ω + En −En′)
=−1
πIm[GR(k, ω)] =
1
Z∑
nn′
e−βEn(1 + e−βω)|〈n|ci|n′〉|2δ(ω + En −En′)
= −i(1 + e−βω)G>(k, ω). (2.36)
where we have used the Dira identity
1ω+iη
= P 1ω− iπδ(ω). Consequently, we may
derive the following relations known as �u tuation-dissipation relations:
G<ij(ω) = 2πiAij(ω)f(ω),
G>ij(ω) = −2πiAij(ω)[1− f(ω)], (2.37)
where f(ω) = 11+eβω is the Fermi-Dira distribution.
2.2 Nonequilibrium dynami al mean field theory for Fali ov-Kimball
model
A stati mean �eld theory, su h as a Weiss mean-�eld theory (whi h maps a spin
system onto an e�e tive single site in the average �eld) has been known for a long time.
A similar stati mean-�eld theory for intera ting ele trons is the Hartree approa h
17
whi h takes an average, time-independent potential as an approximation for the
Coulombi intera tion. However, the orrelation of ele trons are time dependent,
as they move in the latti e and one needs to onsider their dynami features to
apture the physi al behavior of strongly orrelated system. The DMFT approa h
approximates a latti e problem with many degrees of freedom by means of a single
site e�e tive problem with fewer degrees of freedom. The underlying physi al idea is
to map the many-body latti e problem into a many-body lo al problem (an impu-
rity problem), intera ting with an e�e tive bath reated by all the other degrees of
freedom of other sites. The ru ial di�eren e between stati and dynami mean-�eld
arises from the fa t that the DMFT has been built upon the e�e tive theory for the
lo al frequen y-dependent Green's fun tion and the impurity site is onne ted with
the bath via time dependent oupling λ(t, t′) whi h mimi s the dynami s of the hop-
ping of ele trons in the latti e and it is determined in a self- onsistent formalism, see
�gure 2.2.
λ(t, t′)
Lattice model Single impurity
Figure 2.2: Mapping a latti e model into a single impurity problem with a time
dependent hybridization fun tion λ(t, t′). The urved arrow in the �gure demonstrates
the hopping of an ele tron from a site onto a nearest neighbor site in a latti e model
and hopping from a single impurity into the bath in the impurity model.
In this se tion, we will explain the DMFT for the Fali ov-Kimball model. As we
mentioned in introdu tion, the Fali ov-Kimball model des ribes itinerant ele trons
hopping in the ba kground of lo alized ele trons, whi h are denoted by c and f
18
ele trons, respe tively. For a reminder, we rewrite the Hamiltonian in Eq. 2.38:
HFK(t) = −∑
ij
tij(c†icj + c†jci) + Ucf (t)
∑
i
c†icif†i fi + Ef
∑
i
f †i fi, (2.38)
Noti e that, in this ase, the Hamiltonian is time-dependent. The DMFT approa h
has been su essful in des ribing the so alled Mott-Hubbard transition and it pro-
vides insight about the dynami s of the quantum system. For the half-�lled ase
(µ = Ucf (t → −∞)/2), half of the sites are �xed as they are o upied with lo al-
ized f ele trons. For large enough intera tions, the repulsion between lo alized and
ondu tion ele trons prevents the double o upan y of sites, and, sin e there are no
other empty sites, the system will be frozen and be ome an insulator. In addition,
the Fali ov-Kimball model has an advantage sin e the impurity model in DMFT an
be solved exa tly [3℄. In this se tion, we will demonstrate how to use DMFT to study
the time-dependent Fali ov-Kimball model. First, we will explain how one an �nd
the impurity solver for the Fali ov-Kimball model, whi h is normally the hardest part
of the DMFT algorithm and then, we will des ribe the DMFT algorithm.
2.2.1 Impurity problem solver
As we explained earlier, in the DMFT approa h, one maps the many body latti e
problem onto an impurity problem, that is the lo al many-body problem in the pres-
en e of an external dynami al mean-�eld. The idea in DMFT is to onsider the evolu-
tion operator with an additional time dependent �eld, the so- alled dynami al mean
�eld that mimi s the dynami s of the latti e many body problem. In general, any
time dependen e of the many-body problem, su h as hopping of ele trons on latti e
sites an be onsidered inside the dynami al mean-�eld parameter. In this se tion, we
will des ribe the impurity problem solver losely following the derivation proposed
by Freeri ks in Ref. [17℄. For simpli ity, we will solve the impurity problem in the
19
presen e of an evolution operator for H0s (t) and H1
s (t) separately, and we will use
those results to �nd the impurity Green's fun tion for the generalized Hamiltonian
HFKs (t) = −µc†c+ Eff
†f + Ucf(t)c†cf †f .
First, we onsider the impurity Hamiltonian in equation (2.8). Now, in the presen e
of the �eld, the time ordered Green's fun tion is de�ned as
GC(t, t′) = − i〈TC S(λC) c(t)c†(t′)〉
= − iθC(t, t′)Tr{e−βH0
s (−∞)S(λC)c(t)c†(t′)}/Z0
+ iθC(t′, t)Tr{e−βH0
s (−∞)S(λC)c†(t′)c(t)}/Z0, (2.39)
where S(λC) is the evolution operator on the ontour whi h satis�es
S(λC) = TC exp[−i∫
Cdt
∫
Cdt′c†(t′)λC(τ, τ
′)c(t)], (2.40)
and the operators are in the Heisenberg representation. Moreover, the impurity par-
tition fun tion is de�ned as Z0(λ, µ) = TrC{TCe−βH0s (−∞)S(λC)}. Due to the presen e
of the evolution operator it is not easy to al ulate the tra e, so we use a fun tional
derivative to obtain the hange of the partition fun tion due to a small hange of λc
δZ0(λC, µ) = TrC{TCe−βH0s (−∞)δS(λC)}. (2.41)
Using the al ulus of variations, we obtain
δS(λC) = −iTC [S(λC)∫
Cdt
∫
Cdt′δλC(t, t
′)c†(t)c(t′)], (2.42)
Re alling the de�nition of the Green's fun tion in Eq 2.13, we obtain
δZ0(λc, µ) = −Z0(λ, µ)
∫
Cdt
∫
Cdt′δλC(t, t
′)GC,0(t′, t), (2.43)
whi h an be solved by
GC,0(t, t′) = −δ lnZ0(λC, µ)
δλC(t′, t). (2.44)
20
The next step is to solve the EOM to �nd the impurity Green's fun tion. Due
to the presen e of the time ordered produ t with respe t to two times in the double
integral, al ulating the derivative of S(λc) is ompli ated. Fortunately, one may split
the time evolution operator into two parts to simplify the time derivative
TC[S(λC) c(t)c†(t′)] = TC[S1(λC)]c(t)[S2(λc)]c
†(t′), (2.45)
with
S1(λc) = exp[−i∫ ∞
t
dt′′∫
Cdt′′′λc(t
′′, t′′′)c†(t′′)c(t′′′)], (2.46)
and
S2(λc) = exp[−i∫ t
−∞dt′′∫
Cdt′′′λc(t
′′, t′′′)c†(t′′)c(t′′′)]. (2.47)
Evaluating the time derivative of the Green's fun tion in equation(2.13), we obtain
∂GC,0(t, t′)
∂t′= −iδC(t, t′) + iµG0
C(t, t′)− i
∫
Cdt′′λC(t, t
′′)GC,0(t′′, t′), (2.48)
where the �rst term −iδC(t, t′) omes from the derivative of the step fun tion along
the ontour, while the se ond term is the expli it derivative of c(t) and the last term
omes from the time derivative of the evolution operator. It is straightforward to show
that the above equation an be solved by
GC,0(t, t′) = [(i∂ + µ)δC(t, t′)− λC(t, t
′)]−1. (2.49)
Using the same pro edure, we al ulate the impurity Green's fun tion for the following
Hamiltonian.
H1s (t) = −µc†c+ Ucf(t)c
†c, (2.50)
where the Green's fun tion is de�ned as
GC,1(t, t′) = − i〈TC S(λC) c(t)c†(t′)〉
= − iθC(t, t′)Tr{e−βH1
s (−∞)S(λC)c(t)c†(t′)}/Z1
+ iθC(t′, t)Tr{e−βH1
s (−∞)S(λC)c†(t′)c(t)}/Z1. (2.51)
21
The impurity partition fun tion be omes Z1(λ, µ−Ucf(−∞)) = TrC{TCe−βH1s (−∞)S(λC)},
while S(λC) is de�ned by equation (2.40) and it is straightforward to show that the
Green's fun tion satis�es a similar equation to equation (2.44),
GC,1(t, t′) = −δ lnZ1(λc, µ)
δλc(t′, t). (2.52)
To �nd the impurity Green's fun tion, we have to take the time derivative of the
Green's fun tion; the only hange is the derivative of c(t), whi h involves the om-
mutation [H1s (t), c(t)], and �nally we obtain
GC,1(t, t′) = [(i∂ + µ− Ucf(t))δC(t, t′)− λC(t, t
′)]−1. (2.53)
Now, we may use the impurity Green's fun tion that we obtained for H0s and H1
s
to al ulate the impurity Green's fun tion for the time-dependent Fali ov-Kimball
model:
HFKs (t) = −µc†c+ Eff
†f + Ucf(t)c†cf †f, (2.54)
where wi = f †f is the number of lo alized ele trons, and the se ond term is the
energy of the lo alized ele trons on the latti e, and the last term denotes the repul-
sive intera tion between ondu tion and lo alized ele trons, whi h is expli itly time
dependent. The fa t that wi = f †f ommutes with the Hamiltonian and the evolution
operator, separates the Hilbert spa e into two subspa es with wi = 0 and wi = 1,
making the al ulation of the tra e simpler. The partition fun tion an be written as
ZFK = Z0(λ, µ) + e−βEfZ0(λ, µ− Ucf(t)), (2.55)
and the Green's fun tion be omes
GC,FK(t, t′) = − i〈TC S(λC) c(t)c†(t′)〉
= − iθC(t, t′)Tr{e−βHFK
s (−∞)S(λC)c(t)c†(t′)}/ZFK
+ iθC(t′, t)Tr{e−βHFK
s (−∞)S(λC)c†(t′)c(t)}/ZFK. (2.56)
22
Due to the presen e of the time-ordered produ t and the tra e operator, whi h a ts
over ondu tion and lo alized ele trons, �nding the equation of motion is ompli ated.
However, employing the al ulus of variations, it is straightforward to show that the
Green's fun tion satis�es
GFKC (t, t′) = −δ lnZ
FK(λC, µ)
δλC(t′, t), (2.57)
whi h simpli�es to
GC,FK(t, t′) = − Z0(λC, µ)
ZFK
δ lnZ0(λC, µ)
δC(t′, t)
+e−βEfZ1(λC, µ)
ZFK
δ lnZ0(λC, µ− Ucf(−∞))
δλC(t′, t). (2.58)
Using equation(2.44) and equation(2.52), we an rewrite it as
GC,FK(t, t′) = w0GC,0(t, t′) + w1G
C,1(t, t′), (2.59)
where w0 =Z0(λC ,µ)ZFK , w1 =
e−βEfZ0(λC ,µ−Ucf (−∞))
ZFK and it is straightforward to show that
w0 + w1 = 1. Finally, using equations 2.49 and 2.53, we obtain
GC,FK(t, t′) = (1− w1)[(i∂t + µ)δC(t, t′)− λC(t, t
′)]−1
+ w1[(i∂t + µ− Ucf (t))δC(t, t′)− λC(t, t
′)]−1. (2.60)
2.2.2 Self- onsistent Dynami al mean field theory loop
In order to solve the many-body problem, we need to determine the ele troni Green's
fun tion. In large spatial dimensions d → ∞, the ele tron self-energy be omes lo al
and makes the many-body problem simpler. This approximation is alled dynam-
i al mean-�eld theory [3℄. We start with the non-intera ting ontour-ordered Green's
fun tion in momentum spa e, whi h is de�ned as
GC(eq,non)(t, t′) = − iT r TC e−βHeq,nonc
k
(t)c†k
(t′)/Zeq,non, (2.61)
23
where ck
(t) and c†k
(t′) are reation and annihilator operator in momentum spa e:
ck
=1√L
∑
j
e−ik.Rjcj
c†k
=1√L
∑
j
eik.Rjc†j . (2.62)
Furthermore, Heq,non is the equilibrium non-intera ting Hamiltonian de�ned as
Heq,non =∑
k
(ǫk
− µ)c†k
ck
, (2.63)
with ǫR
= limd→∞− t∗∑di=1 cos(ki)/
√d to be the band stru ture for hyper- ubi lat-
ti e. Metzner and Vollhardt showed that hoosing the nearest-neighbor hopping s ales
as t = t∗
2√don a simple hyper- ubi latti e leads to �nite average kineti energy. [46℄
. In this ase, the nonintera ting density of states be ome Gaussian with an in�nite
bandwidth de�ned as
ρ(ǫ) =1√πe−ǫ2 , (2.64)
noti e that in our al ulation we have used t∗ as our energy unit and the latti e
onstant is set equal to 1.
Here, we explain the DMFT algorithm for nonequilibrium in whi h losely follows
the algorithm des ribed in Ref. [17℄. First, we need to generalize the Hilbert transform
for the nonequilibrium ase. Primarily, we onstru t the lo al ordered Green's fun tion
as GCloc =
∑
kGCk(t, t
′). Starting with lo al self-energy
∑C(t, t′) (whi h is usually
hosen as equilibrium self-energy) and by using Dyson's equation, we an rewrite the
lo al Green's fun tion as
GCloc(t, t
′) =
∫
dǫρ(ǫ)[(I −GC,nonΣC)−1GC,non](t, t′), (2.65)
24
noti e that within the nonequilibrium formalism, all the Green's fun tions and the
self-energies are ontinuous matrix operators with two time variables running over
the ontour. The next step is to extra t an e�e tive dynami al mean-�eld λC(t, t′).
First, we need to �nd the e�e tive medium, using the Dyson equation, we have
GC0(t, t
′) = [(GCloc)
−1 + ΣC ]−1(t, t′), (2.66)
then the dynami al mean �eld be omes
λC(t, t′) = (i∂Ct + µ)δC(t, t′)− (GC
0)−1(t, t′)
= (i∂Ct + µ)δC(t, t′)− (GC
loc)−1(t, t′) + ΣC(t, t′). (2.67)
On e the dynami al mean �eld has been determined, we need to �nd the impurity
Green's fun tion whi h evolves in the presen e of the dynami al mean �eld. We have
already explained this step in great detail for the Fali ov-Kimball model when the
intera tion parameter is time dependent, see equation(2.60). To lose the DMFT loop,
we use the Dyson equation to extra t the impurity self-energy from the impurity
Green's fun tion and the e�e tive medium. Then we need to iterate the loop until it
onverges.
2.3 Numeri al implementation of nonequilibrium DMFT
To implement the algorithm numeri ally, we need to dis retize the ontour, as there is
no way to al ulate ontinuous matrix operators. We will de�ne the ontinuous matrix
operator as the limit where the dis retization size goes to zero using an extrapolation
pro edure. For pedagalogi al reason, in this se tion, we explain the dis retization
pro edure whi h has been proposed in Ref. [5℄ by Freeri ks et al. First, we will split
the upper bran h ontour with Nt points starting from tmin to tmax −∆t, Nt points
25
in the lower bran h starting from tmax to tmin +∆t and Nτ points on the imaginary
axis starting from tmin to tmin − iβ +∆τ , see �gure 2.3
ti = tmin + (i− 1)∆t 1 ≤ i ≤ Nt
= tmax − (i−Nt − 1)∆t Nt + 1 ≤ i ≤ 2Nt
= tmin − (i− 2Nt − 1)∆τ 2Nt + 1 ≤ i ≤ 2Nt + 100, (2.68)
where ∆t = tmax−tmin
Ntand we �xed the Nτ = 100 points on the imaginary axis.
−tmin
−tmin − iβ
tmax
∆t
∆τ Upper branch
Lower branch
Imaginary branch
Figure 2.3: S hemati of dis retization on the Keldysh-Baym ontour.
The ontour ordered Green's fun tion satis�es the boundary ondition similar to
the antiperiodi ity property of thermal Green's fun tion
GCii(tmin, t
′) = −GCii(tmin − iβ, t′). (2.69)
Our numeri al dis retization varies from ∆t = 0.1 to 0.02 on the real axis and it
is �xed to be ∆τ = 0.1i on the imaginary axis. We will use the so- alled Wigner
oordinates to transform to new oordinates de�ned as [47℄
trel = t− t′ , T =t+ t′
2. (2.70)
Now we explain the te hni al details for performing the DMFT algorithm. Starting
from equation(2.65), one performs the Hilbert transform to �nd the lo al Green's
26
fun tion; it is ne essary to perform the matrix multipli ation and matrix inverse. The
matrix multipli ation in a dis ritized form is de�ned as
∫
Cdt′′A(t, t′′)B(t′′, t′) =
∑
k
A(ti, tk)wkB(tk, tj). (2.71)
Using the identity
∫
C dt′′A(t, t′′)A−1(t′′, t′) = δC(t, t′), we an de�ne the inverse for the
ontinuous matrix
∑
k
A(ti, tk)wkA−1(tk, tj) =
1
wiδij . (2.72)
Hen e, to �nd the inverse of any matrix, we need to multiply the rows and olumns
by quadrature weights. The standard linear algebra pa kage (LAPACK and BLAS)
is used to manipulate the general, omplex operators. Additionally, we will use the
left point re tangular integration rule, de�ned as
∫
Cdtf(t) =
2Nt+Nτ∑
i=1
wif(ti), (2.73)
where
wi = ∆t 1 ≤ i ≤ Nt
= −∆t Nt + 1 ≤ i ≤ 2Nt
= −0.1i 2Nt + 1 ≤ i ≤ 2Nt + 100. (2.74)
However, in our ase, the omputations will be less intensive, sin e the Hilbert trans-
form is redu ed to integration over one band stru ture ǫ, simplifying the equation
and making possible the exploration of longer times in the nonequilibrium formalism.
We will dis uss the details in next se tion, when we explain about the ben hmarking
of nonequilibrium formalism. In this ase, the numeri s works better by using the
so alled point splitting for delta fun tion, we will losely follow the omputational
27
pro edure as des ribed in Ref.[5℄. Re alling that the delta fun tion is the derivative
of the theta fun tion, the dis retized delta fun tion an be de�ned as
δC(ti, tj) =1
wiδij+1 for integration over j
=1
wi−1
δij+1 for integration over i (2.75)
where wi are quadrature weight as de�ned in Eq. 2.74. Noti e that the above formula
works only for i 6= 1, when i = 1 the only nonzero elements is 1, 2Nt +Nτ , whi h has
a sign hange due to a boundary ondition, see �gure 2.4 For al ulating the impurity
i = 1
i = 2Nt + 100
i i+ 1
Nt
Figure 2.4: Boundary ondition on the Keldysh-Baym ontour. The points tmin(i =1) and tmin − iβ are identi al, ex ept the sign hange due to boundary ondition.
problem, one needs to evaluate (i∂t +µ)δC(t, t′) and (i∂t +µ−Ucf(t))δC(t, t′). We are
looking for proper matrix Mjk as following
[i∂t + µ]δC(tj, tk) = i1
wjMjk
1
wk. (2.76)
28
Using the de�nition of derivative in dis retized form, we an write the �rst operator
as
[i∂t + µ]δC(tj , tk) =[
i(δjk − δjk+1
∆t) + µδjk+1
] 1
wj. (2.77)
It is obvious that the matrix will have many zero elements and will have nonzero
elements on diagonal and sub diagonal,
[
i(δjk − δjk+1
∆t) + µδjk+1
] 1
wj
= i1
wj
1
wk
k = j
= [−1− iµ∆t]1
wj
1
wk
k = j + 1. (2.78)
Comparing equations(2.76), and (2.78), the matrix Mjk be omes
Mjk =
1 0 0 · · · 1 + i∆τµ
−1 − i∆tµ 1 0 · · · 0
0 −1 − i∆tµ 1 0
.
.
.
0 −1 + i∆tµ 1 0
0 −1 + i∆tµ 1
.
.
.
−1 − ∆τµ 1
−1 − ∆τµ 1
,
where the top blo k orresponds to upper bran h, the middle blo k orresponds to
lower bran h and the right bottom blo k orresponds to imaginary bran h. The upper
right hand matrix element omes from the orre t boundary ondition. One ould
dire tly al ulate the Green's fun tion of a spinless Green's fun tion with a hemi al
potential, and the inverse should equal to Mjk. In our ase, the determinant of the
29
inverse of the matrix should be equal the partition fun tion.
det Mjk = 1 +(−1)2Nt+Nτ−1(1 + i∆tµ)(−1 − i∆tµ)Nt−1
× (−1 + i∆tµ)Nt(−1− i∆tµ)Nτ
≃ 1 + (1 + ∆τµ)Nτ +O(∆t2). (2.79)
Re alling that ∆τ = βNτ(sin e the imaginary bran h has length β), we lead to 1 +
exp(βµ) in the limit ∆t,∆τ → 0. This shows the importan e of point splitting and
veri�es that our al ulation is orre t. In similar way
[i∂t + µ− Ucf(t)]δC(tj , tk) = i1
wj
Ljk1
wk
, (2.80)
where
Ljk =
1 0 0 · · · 1 + i∆tu1
−1 − i∆tu2 1 0 · · · 0
0 −1 − i∆tu3 1 0
.
.
.
0 −1 + i∆tuNt+1 1 0
0 −1 + i∆tuNt+2 1
.
.
.
−1 − ∆τu2Nt+1 1
.
.
.
−1 − ∆τu2Nt+100 1
and the ui = µ − Ucf(ti) oe� ients are time-dependent and evaluated at ea h dis-
ritized point using the left point rule, as we des ribed in equation(2.68). For example,
the oe� ient hanges from u1 = µ−Ucf (t1) to uNt = µ−Ucf(tmax−∆t) in the upper
bran h, and from uNt+1 = µ − Ucf(tmax) to u2Nt = µ − Ucf (tmin + ∆t) in the lower
bran h, and from u2Nt+1 = µ−Ucf(tmin) to u2Nt+100 = µ−Ucf(tmin− iβ+0.1i) in the
imaginary bran h. The time-dependent parameter is hanging from some initial value
30
at tmin to rea h the �nal value at tmax in upper bran h, and then needs to evolve ba k
to it's initial value toward tmin in the lower bran h. This way of time evolution, on
the time ordered ontour, is responsible for orresponden e in time dependent oe�-
ients; for example the oe� ients uNt+1,...,u2Nt in the lower bran h be ome identi al
to uNt,...,u1 in upper bran h. Moreover, the imaginary oe� ient are the same and
we will denote them by uβ = µ− Ucf(−∞). So the matrix Lij simpli�es to
Ljk =
1 0 0 · · · 1 + i∆tu1
−1 − i∆tu2 1 0 · · · 0
0 −1 − i∆tu3 1 0
.
.
.
0 −1 + i∆tuNt1 0
0 −1 + i∆tuNt−1 1
.
.
.
.
.
.
−1 − ∆τuβ 1
. Consequently, the determinant be omes
det Ljk = 1 + (−1)2Nt+Nτ−1 (1 + i∆tu1)(−1 − i∆tu2) · · · (−1 − i∆tuNt)
× (−1 + i∆tuNt)(−1 + i∆tuNt−1) · · · (−1 + i∆tu1)
× (−1 −∆τuβ)(−1−∆τuβ) · · · (−1 −∆τuβ) ,(2.81)
simplifying the above equation we obtain
det Ljk = 1 + [1 + ∆t2u21] · · · [1 + ∆t2u2Nt]× (1 + ∆τuβ)
Nτ , (2.82)
and the determinant equals the partition fun tion when ∆t,∆τ → 0:
det Ljk = 1 + exp[β(µ− Ucf(−∞))]. (2.83)
31
3
Nonequilibrium spe tral moment sum rules for systems with
ele tron-phonon intera tion
�S ien e is spe tral analysis. Art is light synthesis.�
� Karl Kraus
Most of the interesting strongly orrelated systems su h as strongly orrelated
oxide multilayers, harge- and spin-density wave materials and high Tc uprates
exhibit strong ele tron-ele tron or ele tron-phonon ouplings while understanding
the non equilibrium dynami s of ele tron-phonon intera tion still remains as one of
the most intriguing topi s both from experimental and theoreti al points of view.
Re ent development in pump-probe spe tros opy has provided a powerful tool to
study the non-equilibrium properties of a large variety of strongly orrelated sys-
tems with oupled ele tron, phonon and spin degrees of freedom within the relevant
time-s ale for the ele tron-phonon dynami s. The original angle-resolved photoemis-
sion spe tros opy (Arpes) is used to obtain the energy and momentum of ele troni
band stru ture with high resolution, but an not provide information regarding the
band stru ture in ex ited states. Instead, the time and angle-resolved photoemission
spe tros opy (trARPES) adds the femtose ond time-resolution to the onventional
ARPES whi h provides a powerful tool to study the elementary s attering pro ess
in the ele troni stru tures. The set up for the pump-probe spe tros opy is shown in
�gure 3.1. The pump is a strong ultrashort laser pulse whi h an be used to ex ite
32
either the ele troni or phononi state generating a non-equilibrium state, followed
by the UV pulse whi h probes the pump-indu ed hanges in the system after a time
delay∆t. Moreover, in a re ent experiment [8℄, the ultrafast response of the self-energy
Figure 3.1: The s hemati pi ture of tr-Arpes experiment. The probe whi h is a
femtose ond infrared pulse reates an ex itation in sample, while the se ond pulse
whi h is a UV pulse, is used to probe the transient behavior of ele troni stru ture
after the time ∆t. Pi ture taken from Shen Laboratory, Stanford.
(a fundamental quantity des ribing many-body intera tions) of a high-temperature
super ondu tor has been investigated both in the normal and super ondu ting state.
The most dire t eviden e of ele tron-phonon oupling in uprate high Tc super on-
du tors is known as a universal ele tron self-energy renormalization whi h manifests
itself as a kink in photoemission spe tra [9℄, although whether it is related to super-
ondu tivity still remains un lear. In �gure 3.2, we show the results of the ele troni
dispersion of a high Tc uprate super ondu tor [8℄. The data is taken at 17K < Tc
at equilibrium, t = −1ps and in nonequlibrium at t = 1ps and t = 10ps. In part
(a), we an observe the kink around 70mev at equilibrium, while after applying the
pump, there is a signi� ant loss in the spe tral weight, whi h is happening at t = 1ps
33
and �nally, at a later time of t = 10ps the transient spe tra re overs the equilibrium
state. In part (b), the ele troni dispersion is shown before and after applying the
laser pump. By omparing the momentum dispersion urve (MDC) after applying the
pump shown in red and the equilibrium dispersion urves shown in bla k, we observe
that the signi� ant hanges in the MDC are happening around the kink energy. Fur-
ther results from the same experiment [8℄, show that the pump indu ed hanges in
the self-energy in the super ondu ting state are di�erent from the normal state as it
is more on�ned in the vi inity of the kink and an not be explained by a temper-
ature broadening e�e t whi h hanges over a broad range of energies. These results
raise interesting questions regarding the origin of su h behavior and it an provide
an insight to understand the role of ele tron-phonon ouplings in high Tc uprate
super ondu tors.
In this hapter, we will explain how one an use sum rules in a form of exa t
analyti al results to investigate the response of strongly orrelated system out of
equilibrium. In addition to the fa t that sum rules have been used to develop a
di�erent approximation to investigate the feature of systems, they provide powerful
tools to be used in the ben hmarking of omputational work to he k the pre ision
of numeri al solutions. This approa h has been developed by White by applying the
exa t sum rules for the zeroth and the �rst two moments of the spe tral fun tion
to estimate the a ura y of Monte-Carlo solutions of the Hubbard model in two-
dimensions [54℄. Moreover, higher moments an reveal useful information about the
spontaneous magneti order in orrelated systems [55℄. The appli ation of the sum
rules for the self-energy has attra ted mu h interest from angle-resolved experiments
on the high-temperature super ondu tors [56, 57, 58℄. However, in nonequilibrium the
Green's fun tion and self-energies depend on two times, nevertheless, the exa t sum
rules have been developed by Freeri ks et al [10, 11, 12, 13℄. In the next se tion, we
34
Figure 3.2: Time-resolved spe tra of high Tc uprate super ondu tor,
Bi2Sr2Ca2Cu2O8+δ. a) Photoemission intensity as a fun tion of energy and momentum
measured at equilibrium (t = −1ps) and nonequilibrium (t = 1ps and t = 10ps) afterapplying the ultra fast pump. The intensity shown in false olor, indi ates a lear
loss of spe tral weight after applying the pump at t = 1ps. The arrow indi ates
the position of kink in the momentum distribution urve whi h happens at 70mev.b) Comparing the momentum distribution urve (MDC) at di�erent time shown by
bla k for t = −1ps, red for t = 1ps and gray for t = 10ps. In the insets, omparison
of (MDC) is done at di�erent binding energies [8℄.
35
will show how one an al ulate the nth moment sum rules for the spe tral fun tion
of a generi time-dependent Hamiltonian. Then, in se tion 2, we will fo us on the
Holstein model as a basi model to des ribe the ele tron-phonon intera tion. We will
derive the moments of the retarded Green's fun tion and self-energy up to se ond and
zeroth order respe tively. These results are reported from a paper where I al ulated
the atomi limit of the Holstein model [14℄. In se tion 3, we go one step further and
we add the ele tron-ele tron intera tion. We have al ulated the sum rules for the
retarded Green's fun tion up to third order whi h allows us to al ulate the moment
of the self-energy to �rst order.
3.1 Spe tral sum rules for the retarded Green's fun tion
The nonequilibrium retarded Greens fun tion an be de�ned as,
GRij(t1, t2) = −iθ(t1, t2)〈{ci(t1), c†j(t2)}〉, (3.1)
where 〈O〉 = Tr [exp(−βHeq)O]/Z. Moreover the fermioni operators are written in
the Heisenberg representation ci(t) = U †(t, t′) ci U(t, t′), where the evolution operator
satis�es the S hrödinger equation, idU(t,t′)dt
= H(t)U(t, t′). The two times lie on the
Kadano�-Baym-Keldysh ontour, where one starts from tmin and runs in the positive
dire tion until tmax and ba k to tmin in opposite dire tion and �nally goes to tmin− iβ
parallel to the imaginary axis with β = 1/T . It is onvenient to onvert to so alled
Wigner oordinates: the average time T = t+t′
2and the relative time t = t1 − t2. By
using a Fourier transform with respe t to relative time, we an �nd the frequen y
dependent retarded Green's fun tion for ea h average time
GRij(T, ω) =
∫ ∞
0
dt eiωtGRij(T +
t
2, T − t
2). (3.2)
36
The nth spe tral moment in real spa e is de�ned as
µRnij (T ) = −1
π
∫ ∞
−∞dω ωnImGR
ij(T, ω). (3.3)
It is straightforward to show that one an rewrite the moments as derivatives in time
µRnij (T ) = Im 〈in+1 d
n
dtn{ci(T +
t
2), c†j(T − t
2)} |t=o+〉. (3.4)
So pra ti ally the problem of �nding the nth moment of the spe tral fun tion redu es
to al ulating the nth derivative of {ci(T + t2), c†j(T − t
2)} with respe t to t. Below
we show how one an al ulate this quantity in the Heisenberg representation. Let
us onsider a physi al system with a generi time-dependent Hamiltonian denoted by
Hs(t) in the S hrödinger representation. We know that in the Heisenberg represen-
tation the time dependen e is en oded in the operator AH(t) whi h is related to the
S hrödinger representation operator As(t) by
AH(t) = U †(t, t0) As(t) U(t, t0), (3.5)
where t0 is the referen e time and U(t, t0) is the evolution operator whi h is de�ned
as
U(t, t0) = T e−i∫ tt0
dtHs(t), (3.6)
and the onjugate operator is de�ned as
U †(t, t0) = T ei∫ tt0
dtHs(t), (3.7)
here T is the time ordering operator whi h orders the operator with later time to the
left. A ordingly T is the anti time ordering operator and a ts in the opposite way.
Moreover, the evolution operator satis�es the equation of motion
i∂tU(t, t0) = Hs(t) U(t, t0). (3.8)
37
Then it is easy to show that the Heisenberg equation of motion implies,
dAHdt
= iU †(t, t0)[Hs(t), A]U(t, t0) + U †(t, t0)∂A
∂tU(t, t0) (3.9)
for simpli ity hereafter we will drop the index for the Hamiltonian and we get
dAHdt
= i[H(t), AH] +∂AH∂t
, (3.10)
where the Hamiltonian appearing in the equation is in the Heisenberg represen-
tation and the symbol AH = U †(t, t0)AU(t, t0). Using the de�nition LnAH =
[...[[AH, H(t)], H(t)]...H(t)] and DnAH =∂An
H
∂tn, we an rewrite above equation as,
idAHdt
= L1AH + iD1AH, (3.11)
and onsequently we an al ulate the higher order derivatives as follows,
i2dA2
Hdt2
= L1L1AH + iD1L1AH + iL1D1AH + i2D1D1AH
= L2AH + iD1L1AH + iL1D1AH + i2D2AH, (3.12)
i3dA3
Hdt3
= L3AH + iL1D1L1AH + iL2D1AH + i2L1D2AH +
iD1L2AH + i2D2L1AH + i2D1L1D1AH + i3D3AH. (3.13)
Noti e that we have used a ontra tion rule, where we have ombined the two oper-
ators in the ase that identi al operators lie beside ea h other, for example L1L1 an
be written as L2. In general, when the order of the derivative is smaller than the
number of the nested ommutator, one has to onsider all possible ommutators, we
will show it with a further example later. As we an see there are 2n distin t terms
for the nth derivative. So n = 4 has 16 di�erent terms and the number of terms grows
exponentially. However most of the time, the operator has no expli it time depen-
den e in the S hrödinger representation and the derivative terms with respe t to the
38
operator itself vanishes, here we use a tilde notation to indi ate that the operator
does not have expli t time dependen e. So we lead to,
idAHdt
= L1AH, (3.14)
i2dA2
Hdt2
= L2AH + iD1L1AH, (3.15)
i3dA3
Hdt3
= L3AH + iL1D1L1AH + iD1L2AH + i2D2L1AH. (3.16)
One ould see that for higher derivatives the equation involves a di�erent order of
derivative of the Hamiltonian and keeping tra k of all terms be omes a hard task.
Here we derive a simple identity that helps to tra k all possible terms.
indnAHdtn
=2n∑
sequence=1
(i)m Tuple[{D1, L1}, n]AH, (3.17)
where Tuple[list, n] is n-tuple, whi h is de�ned as a sequen e of elements with length
n. The sum runs over all the sequen es of n-tuple and as we mentioned before, we
have 2n possible sequen es as the list only ontains two elements, L1 and D1. The
index m indi ates the order of derivative of ea h sequen e and it an be obtained by
summing over the number of times that operator D1 appears in that sequen e.
To larify the notation, let us al ulate the derivative for n = 1 and n = 2 by using
the de�nition of n-tuple. For n = 1, the 1-tuple list ontains only two sequen es with
one element, Tuple[{D1, L1}, 1] = {{L1}, {D1}}, while for n = 2, we have 4 sequen es
with two elements, Tuple[{D1, L1}, 2] = {{L1, L1}, {L1, D1}, {D1, L1}, {D1, D1}}.
Now using the identity (3.17), we an al ulate the derivative of the operator AH as
follows,
39
idAHdt
= L1AH + iD1AH, (3.18)
and
i2dA2
Hdt2
= L1L1AH + iL1D1AH + iD1L1AH + i2D1D1AH, (3.19)
Noti e that we an use similar ontra tion rule to ombine the index of identi al
operators that lie beside ea h other to rea h equations (3.11) and (3.12). Again for
operators whi h do not have expli t time dependen e, half of the terms be ome zero.
Now that we �nd a generalized way to al ulate the nth derivative of the operator
in the Heisenberg representation, we ome ba k to equation (3.4). Here we expli itly
derive the derivatives of {ci(T + t2), c†j(T − t
2)} up to third order. The zeroth order
is trivial sin e it simply be omes the anti ommutator of {ci, c†j} whi h is equal to δij
for the fermioni operators. For simpli ity we drop the time arguments of operators,
but we keep in mind that ea h time we take the derivative with respe t to ci(T + t2)
and c†j(T − t2), we get a fa tor of 1/2 and −1/2 respe tively. Now we show how one
an al ulate the nth moment independent of the lower order derivatives. Sin e both
operators ci(T + t2) and c†j(T − t
2) are time dependent, �rst we use Leibniz rule to
al ulate the proper nth derivative,
indn
dtn{ci(T +
t
2), c†j(T − t
2)} =
1
2n
n∑
k=0
(−1k)
(n
k
) {
[d
dt]n−kci(T +
t
2) , [
d
dt]kc†j(T − t
2)}
,(3.20)
where the derivative with respe t to t will bring down a 1/2n fa tor for ci and (−1/2)n
for c†j. After generating the orre t orders of derivatives and using equation 3.17 one
an generate the proper terms for individual derivatives. For n = 0, we simply get
40
the anti ommutator {ci, c†j}. Below we show the results from n = 1 to n = 3,
id
dt{ci, c†j} =
1
2
({ d
dtci , c
†j
}
−{
ci ,d
dtc†j
})
=1
2
(
{L1ci, c†j} − {ci, L1c
†j})
. (3.21)
i2d2
dt2{ci, c†j} =
1
4
({ d2
dt2ci , c
†j
}
− 2{ d
dtci ,
d
dtc†j
}
+{
ci ,d2
dt2c†j
})
=1
4
({
(L2 + iD1L1)ci , c†j
}
− 2{
L1ci , L1c†j
}
+{
ci , (L2 + iD1L1)c†j
})
,
separating the di�erent order of derivatives we get,
i2d2
dt2{ci, c†j} =
1
4
(
{L2ci, c†j} − 2{L1ci, L1c
†j}+ {ci, L2c
†j})
+i
4
(
{D1L1ci, c†j}+ {ci, D1L1c
†j})
. (3.22)
Similarly, for n = 3 we get,
i3d3
dt3{ci, c†j} =
1
8
({ d3
dt3ci , c
†j
}
− 3{ d2
dt2ci ,
d
dtc†j
}
+ 3{ d
dtci ,
d2
dt2c†j
}
−{
ci ,d3
dt3c†j
})
=1
8
(
{L3ci, c†j} − 3{L2ci, L1c
†j}+ 3{L1ci, L2c
†j} − {ci, L3c
†j})
+i
8
(
{L1D1L1ci, c†j}+ 2{D1L2ci, c
†j} − 3{D1L1ci, L1c
†j}
+ 3{L1ci, D1L1c†j} − {ci, L1D1L1c
†j} − 2{ci, D1L2c
†j})
− 1
8
(
{D2L1ci, c†j} − {ci, D2L1c
†j})
, (3.23)
Now we write the results for the spe tral moments,
µR0ij = Re {ci, c†j}, (3.24)
41
µR1ij = Re
1
2
({
[ci, H ] , c†j
}
−{
ci , [c†j , H ]})
, (3.25)
µR2ij = Re
1
4
({
[[ci, H ], H ] , c†j
}
− 2{
[ci, H ] , [c†j , H ]}
+{
ci , [[c†j, H ], H ]
})
+ Rei
4
({
[ci,∂H
∂t] , c†j
}
+{
ci , [c†j ,∂H
∂t]})
, (3.26)
µR3ij = Re
1
8
({
[[[ci, H ], H ], H ] , c†j
}
− 3{
[[ci, H ], H ] , [c†j, H ]}
+ 3{
[ci, H ] , [c†j, H ], H ]}
−{
ci , [[[c†j , H ], H ], H ]
})
+ Rei
8
(
2{
[[ci,∂H
∂t], H ] , c†j
}
+{
[[ci, H ],∂H
∂t] , c†j
}
− 3{
[ci,∂H
∂t, [c†j , H ]
}
+ 3{
[ci, H ] , [c†j,∂H
∂t]}
− 2{
ci , [[c†j ,∂H
∂t], H ]
}
−{
ci , [[c†j , H ],
∂H
∂t]})
− Re1
8
({
[ci,∂2H
∂t2] , c†j
}
−{
ci , [c†j,∂2H
∂t2]})
, (3.27)
As n in reases, the number of terms in reases, but we go one step further and we
show that we an sum up all terms with m = 0 whi h are the zero derivative. Let us
start with {L0ci, Lnc†j}, whi h an be written as
{L0ci, Lnc†j} = {L0ci, L1Ln−1c
†j}} = {ci, [[...[c†j , H ]...], H ]}. (3.28)
where, the urly bra kets indi ate the anti- ommutation, so we an use the following
Ja obian identity [X, {Y, Z}] − {Z, [X, Y ]} + {Y, [Z,X ]} ≡ 0. Now, we ompare the
42
�rst and the last term of nth derivative with zero derivative order,
{ci, [[...[c†j , H ]...], H ]} = −{[ci, H ], [[...[c†j , H ]...]]} − [H, {ci, [[...[c†j , H ]...]}], (3.29)
and
{[[...[ci, H ]...], H ], c†j} = −{[[[...[ci, H ]...]], [c†j, H ]} − [H, {[[...[ci, H ]...], c†j}], (3.30)
the se ond term is zero sin e the internal anti ommutator always is a number and
ommutes with Hamiltonian. So we end up with
{L0ci, Lnc†j} = −{L1ci, Ln−1c
†j}. (3.31)
We repeat the same pro edure one more time,
{L1ci, Ln−1c†j} = −{L2ci, Ln−2c
†j} (3.32)
ombining Eqs. 3.31 and 3.32, we get
{L0ci, Lnc†j} = {L2ci, Ln−2c
†j} (3.33)
repeating the pro edure s times we observe that
{L0ci, Lnc†j} = (−1)s{Lsci, Ln−sc
†j} (3.34)
where the sign of term will be positive or negative depending on whether the degree
of L shifts even or odd times respe tively. Similarly, using the Ja obi identity the
se ond term
{
ci , [c†j ,∂H∂t]}
an be obtained by swapping
∂H∂t
over the ommutator
from the �rst term
{
[ci,∂H∂t] , c†j
}
. However, as we des ribed above, there is a (−1)s
oe� ient whi h makes this two term to an el ea h other. One has to be areful
for the derivatives in higher moments as there are terms that are not identi al due
to the fa t that H and
∂H∂t
do not ommute. However, one an simplify some of the
43
terms from the Ja obi identity. Below, we rewrite the �nal answers for the spe tral
moments after the simpli� ation,
µR0ij = Re {ci, c†j}, (3.35)
µR1ij = Re
{
[ci, H ] , c†j
}
, (3.36)
µR2ij = Re
{
[[ci, H ], H ] , c†j
}
(3.37)
µR3ij = Re
{
[[[ci, H ], H ], H ] , c†j
}
+ Rei
2
({
[[ci,∂H
∂t], H ] , c†j
}
−{
[[ci, H ],∂H
∂t] , c†j
}
− Re1
8
({
[ci,∂2H
∂t2] , c†j
}
−{
ci , [c†j,∂2H
∂t2]})
, (3.38)
3.2 Nonequilibrium sum rules for the Holstein model
In re ent years, we have seen signi� ant advan es in time-resolved experiments on
systems that have strong ele tron-phonon intera tions [59, 60℄. These experiments
study how energy is transferred between the ele troni and phononi parts of the
system. One of the interesting e�e ts that have been seen in these experiments is
the so- alled phonon-window e�e t [61℄, where ele trons with energies farther than
the phonon frequen y from the Fermi level relax qui kly ba k to equilibrium after
the pulsed �eld is applied, but those lose to the Fermi level relax on a mu h longer
time s ale, be ause their relaxation involves multiparti le pro esses due to a restri ted
phase spa e. It is lear that this experimental and theoreti al work is just starting
This hapter is reprinted from J. K. Freeri ks, K. Naja�, A. F. Kemper, and T. P. Dev-
ereaux, FEIS 2013, Copyright(2013) Key West, FL, USA, 2013O.
44
to analyze ele tron-phonon intera ting systems in the time domain. Hen e, any exa t
results that an be brought to bear on this problem will be important.
In this work, we derive sum rules for the zeroth and �rst two moments of the
retarded ele troni Green's fun tion and for the zeroth moment of the retarded self-
energy. The moment sum rules have already been derived in equilibrium [56, 58℄, but
they a tually hold true, un hanged, in nonequilibrium as well [10, 11, 12, 13℄. With
these sum rules, one an understand how the ele tron-phonon intera tion responds
to nonequilibrium driving, and how di�erent response fun tions will behave.
We start with the so- alled Holstein model [62, 63℄, given by the following Hamil-
tonian in the S hroedinger representation:
HH(t) = −∑
ijσ
tij(t)c†iσcjσ +
∑
iσ
[g(t)xi − µ]c†iσciσ +∑
i
p2i2m
+1
2κ∑
i
x2i (3.39)
where c†iσ (ciσ) are the fermioni reation (annihilation) operators for an ele tron at
latti e site i with spin σ (with anti ommutator {ciσ, c†jσ′}+ = δijδσσ′), and xi and
pi are the phonon oordinate and momentum (with ommutator [xi, pj]− = i~δij),
respe tively. The hopping −tij(t) between latti e sites i and j an be time dependent
[for example, an applied ele tri �eld orresponds to the Peierls' substitution [64℄, µ
is the hemi al potential for the ele trons, g(t) is the time-dependent ele tron-phonon
intera tion, m is the mass of the opti al (Einstein) phonon and κ is the orresponding
spring onstant. The frequen y of the phonon is ω =√
κ/m. It is often onvenient
to also express the phonon degree of freedom in terms of the raising and lowering
operators a†i and ai (with ommutator [ai, a†j]− = δij) with xi = (a†i + ai)
√
~/(2mω)
and pi = (−a†i + ai)√
~mω/2/i. This Hamiltonian involves ele trons that an hop
between di�erent sites on a latti e and intera t with harmoni Einstein phonons that
have the same phonon frequen y for every latti e site. The hopping and the ele tron-
45
phonon oupling are taken to be time dependent for the nonequilibrium ase. We set
~ = 1 and kB = 1 for the remainder of this work.
3.2.1 The ele troni sum rules
These moments an now be evaluated straightforwardly, although the higher the
moment is the more work it takes. We �nd the well-known result
µR0ijσ(T ) = δij (3.40)
for the zeroth moment. The �rst moment satis�es
µR1ijσ(T ) = −tij(T )− µδij + g(T )〈xi(T )〉δij (3.41)
and the se ond moment be omes
µR2ijσ(T ) =
∑
k
tik(T )tkj(T ) + 2µtij(T ) + µ2δij (3.42)
− tij(T )g(T )〈xi(T ) + xj(T )〉 − 2µg(T )〈xi(T )〉δij
+ g2(T )〈x2i (T )〉δij.
Unlike in the ase of the Hubbard or Fali ov-Kimball model, where the sum rules
relate to onstants or simple expe tation values [11, 12, 13℄, one an see here that
one needs to know things like the average phonon oordinate and its �u tuations in
order to �nd the moments. We will dis uss this further below.
Our next step is to al ulate the self-energy moments in the normal state, whi h
are de�ned via
CRnijσ(T ) = −1
π
∫
dω ωnImΣRijσ(T, ω). (3.43)
Note that the self-energy is de�ned via the Dyson equation
GRijσ(t, t
′) = GR0ijσ(t, t
′) +∑
kl
∫
dt
∫
dt′GR0ikσ(t, t)Σ
Rklσ(t, t
′)GRljσ(t
′, t′), (3.44)
46
where GR0is the nonintera ting Green's fun tion and the time integrals run from −∞
to ∞. The strategy for evaluating the self-energy moments is rather simple. First,
one writes the Green's fun tion and self-energy in terms of the respe tive spe tral
fun tions
GRijσ(T, ω) = −1
π
∫ImGR
ijσ(T, ω′)
ω − ω′ + i0+dω′
(3.45)
and
ΣRijσ(T, ω) = ΣR
ijσ(T,∞)− 1
π
∫ImΣR
ijσ(T, ω′)
ω − ω′ + i0+dω′. (3.46)
Next, one substitutes those spe tral representations into the Dyson equation that
relates the Green's fun tion and self-energy to the nonintera ting Green's fun tion.
By expanding all fun tions in a series in 1/ω for large ω, one �nds the spe tral formulas
involve summations over the moments. By employing the exa t values for the Green's
fun tion moments, one an extra t the moments for the self-energy. Details for the
formulas appear elsewhere [12℄. The end result is
ΣRijσ(T,∞) = g(T )〈xi(T )〉δij (3.47)
and
CR0ijσ(T ) = g2(T )[〈x2i (T )〉 − 〈xi(T )〉2]. (3.48)
So, the total strength (integrated weight) of the self-energy depends on the �u tua-
tions of the phonon �eld.
3.2.2 Formalism for the phononi sum rules
The retarded phonon Green's fun tion is de�ned in a similar way, via
DRij(t, t
′) = −iθ(t − t′)Tre−βH(tmin)[xi(t), xj(t′)]−/Z, (3.49)
with the operators in the Heisenberg representation. The moments are de�ned in the
same way as before. First one onverts to the average and relative time oordinates
47
and Fourier transforms with respe t to the relative oordinate
DRij(T, ω) =
∫
dtreleiωtrelDR
ij(T +1
2trel, T − 1
2trel), (3.50)
and then one omputes the moments via
mRnij (T ) = −1
π
∫
dω ωnImDRij(T, ω). (3.51)
The zeroth moment vanishes be ause xi ommutes with itself at equal times. For the
higher moments, we also derive a formula similar to what was used for the ele troni
Green's fun tions. In parti ular, we have
mR1ij (T ) = −1
2Im{
〈[x′i(T ), xj(T )]−〉 − 〈[xi(T ), x′j(T )]−〉}
(3.52)
for the �rst moment. But x′i(T ) = −i[xi(ttave),HH(T )]− = pi(T )/m, so we �nd
mR1ij (T ) =
1
mδij . (3.53)
Similarly,
mR2ij (T ) = −1
4Imi{
〈[x′′i (T ), xj(T )]−〉 − 2〈[x′j(T ), x′j(T )]−〉
+ 〈[xi(T ), x′′j (T )]−〉}
. (3.54)
Using the fa t that x′′i (T ) = −i[pi(T ),HH(T )]− = −g(T )(ni↑(T ) + ni↓(T ))− κxi(T ),
then we an show that mR2ij (T ) = 0, sin e all ommutators vanish. We don't analyze
the phonon self-energy here. Unlike the ele troni moments, the phononi moments
are mu h simpler, and do not require any expe tation values to evaluate them.
We end this se tion by showing that the imaginary part of the retarded phonon
Green's fun tion is an odd fun tion of ω, whi h explains why all the even moments
vanish. If one evaluates the omplex onjugate of the retarded phonon Green's fun -
tion, one �nds
DRij(t, t
′)∗ = iθ(t− t′)Tr[xj(t′), xi(t)]−e
−βH(tmin)/Z = DRij(t, t
′) (3.55)
48
where the last identity follows by swit hing the order of the operators in the ommu-
tator and the invarian e of the tra e under a y li permutation. Hen e, the phonon
propagator in the time representation is real. Evaluating the frequen y-dependent
propagator, then it shows that DR∗ij (T, ω) = DR
ij(T,−ω) by taking the omplex onju-
gate of equation (3.50). Hen e the real part of the retarded phonon propagator in the
frequen y representation is an even fun tion of frequen y while the imaginary part is
an odd fun tion of frequen y, and therefore all even moments vanish.
3.2.3 Atomi limit of the Holstein model
To get an idea of the phonon expe tation values and the �u tuations, we solve expli -
itly for the expe tation values for the Holstein model in the atomi limit, where
tij(t) = 0 and we an drop the site index from all operators. In this limit, one an
exa tly determine the Heisenberg representation operator x(t) by solving the equation
of motion for the Heisenberg representation operators a(t) and a†(t). This yields
x(t) =ae−iωt + a†eiωt√
2mω− 2Re
{
ie−ωt
∫ t
0
dt′eiωt′
g(t′)
}n↑ + n↓2mω
(3.56)
where the ele troni number operators ommute with H now, so they have no time
dependen e. Sin e the atomi sites are de oupled from one another, we an fo us on
just a single site. The partition fun tion for a single site an be evaluated dire tly by
employing standard raising and lowering operator identities. To begin, we note that
the Hilbert spa e is omposed of a dire t produ t of the harmoni os illator states
|n〉 = 1√n!
(a†)n |0〉 (3.57)
and the fermioni states
|0〉, | ↑〉 = c†↑|0〉, | ↓〉 = c†↓|0〉, | ↑↓〉 = c†↑c†↓|0〉. (3.58)
49
The partition fun tion satis�es
Zat =∑
0,↑,↓,↑↓
∞∑
nb=0
〈nb, nf | exp[−β{(gx− µ)(nf↑ + nf
↓) + ω(nb +1
2)}]|nb, nf〉, (3.59)
where nfdenotes the Fermi number operator and nb
the Boson number operator
(we will drop the exp[βω/2] term whi h provides just a onstant). Sin e the produ t
states are not eigenstates ofH, we annot immediately evaluate the partition fun tion.
Instead, we need to �rst go to the intera tion representation with respe t to the
bosoni Hamiltonian in imaginary time (and we drop the onstant term from the
Hamiltonian), to �nd that
Zat = TrfTrbe−βωnbTτ exp
[
−∫ β
0
dτ ′{e−ωτ ′a+ eωτ
′a†√
2mωg(tmin)− µ
}
nf
]
,
= TrfTrbeβωnb
UI(β) (3.60)
where the time-ordering operator is with respe t to imaginary time and the time-
ordered produ t is the evolution operator in the intera tion representation and
denoted by UI(τ). Be ause the only operators that don't ommute in the evolution
operator are a and a†, and their ommutator is a -number, one an get an exa t
representation for the evolution operator via the Magnus expansion [65℄, as worked
out in the Landau and Lifshitz [66℄ or Gottfried [67℄ texts. The end result for the
time-ordered produ t in equation (3.60) be omes
UI(β) = exp
[
−g(tmin)nf
√2mω3
(1− e−βω
)a
]
exp
[
−g(tmin)nf
√2mω3
(eβω − 1
)a†]
× exp
[
−g2(tmin)n
f2
√2mω3
(eβω − 1− βω + µnf
)]
, (3.61)
whi h used the Campbell-Baker-Hausdor� theorem
eA+B = eBeAe12[A,B]−
(3.62)
for the ase when the ommutator [A,B]− is a number, not an operator, to get the
�nal expression. We substitute this result for the evolution operator into the tra e
50
over the bosoni states, expand the exponentials of the a and a† operators in a power
series, and evaluate the bosoni expe tation value to �nd
Zat = Trf
∞∑
m=0
∞∑
n=0
(n+m)!
n!m!m!
(g2(tmin)n
f2
2mω3
(eβω − 1 + e−βω − 1
))m
× exp
[
−βωn+ βµnf − g2(tmin)nf2
2mω3
(eβω − 1− βω
)]
. (3.63)
Next, we use Newton's generalized binomial theorem
∞∑
n=0
(n +m)!
n!m!zn =
1
(1− z)m+1, (3.64)
to simplify the expression for the partition fun tion to
Zat = Trf exp
[βg2(tmin)n
f2
2mω2+ βµnf
]1
1− e−βω. (3.65)
Performing the tra e over the fermioni states then yields
Zat =1
1− e−βω
{
1 + 2eβµ exp
[βg2(tmin)
2mω2
]
+ e2βµ exp
[2βg2(tmin)
mω2
]}
. (3.66)
We al ulate expe tation values following the same pro edure, but inserting the
relevant operators in the Heisenberg representation at the appropriate pla e, and
arrying out the remainder of the derivation as done for the partition fun tion. For
example, sin e the Fermi number operators ommute with the atomi Hamiltonian,
they are the same operator in the Heisenberg and S hroedinger representations, and
we immediately �nd that the ele tron density is a onstant in time and is given by
〈n↑ + n↓〉 =2eβµ exp
[βg2(tmin)
2mω2
]
+ 2e2βµ exp[2βg2(tmin)
mω2
]
1 + 2eβµ exp[βg2(tmin)
2mω2
]
+ e2βµ exp[2βg2(tmin)
mω2
] . (3.67)
We next want to al ulate 〈x(t)〉 and 〈x2(t)〉 − 〈x(t)〉2, where the operator is in
the Heisenberg representation, and given in equation (3.94). It is straightforward but
tedious to al ulate the averages. After mu h algebra, we �nd
〈x(t)〉 = 〈n↑ + n↓〉(
−g(tmin)
mω2cosωt− Re
{
ie−iωt
∫ t
0
dt′eiωt′ g(t′)
mω
})
. (3.68)
51
(Note that if we are in equilibrium, g(t) = g is a onstant, then one �nds 〈x(t)〉 =
−〈n↑ + n↓〉g/mω2, whi h has no time dependen e, as expe ted.) The �u tuation sat-
is�es
〈x2(t)〉 − 〈x(t)〉2 = [〈n↑〉(1− 〈n↑〉) + 〈n↓〉(1− 〈n↓〉) + 2〈n↑n↓〉 − 2〈n↑〉〈n↓〉]
×[g(tmin)
mω2cosωt+ Re
{
ie−iωt
∫ t
0
dt′eiωt′ g(t′)
mω
}]2
+1
2mωcoth
(βω
2
)
, (3.69)
whi h onsists of two terms: a time-dependent pie e (whi h be omes a onstant when
g is a onstant) that represents the quantum �u tuations due to the ele tron-phonon
intera tion and a phonon pie e that varies with temperature (and is independent of
g). The latter pie e be omes large when T → ∞ (being proportional to T at high T ),
whi h tells us that �u tuations generi ally grow with in reasing the temperature of
the system, so that one expe ts the zeroth moment of the self energy to in rease as
the temperature in reases, or if the system is heated up by being driven by a large
ele tri �eld. Note that if one expands the self-energy perturbatively, as in Migdal-
Eliashberg theory, then only the term independent of g survives, as the other term is
higher order in g and lies outside of the Migdal-Eliashberg result [68℄.
3.2.4 Dis ussion and appli ations of the sum rules
One of the most important re ent experiments in ele tron-phonon intera ting sys-
tems involves time-resolved angle-resolved photoemission (tr-ARPES), whi h an be
analyzed in su h a way that one an extra t information about the ele troni self-
energy [61, 68℄. If one assumes that the phonons form an in�nite heat apa ity bath,
then they are not hanged by the ex itation of the ele trons, and the �u tuations of
the phonon �eld remain a onstant as a fun tion of time. This leads to a self-energy
that an transiently hange shape as a fun tion of time, but does not hange its
52
spe tral weight. Re ent al ulations show pre isely this behavior [61, 68℄. One an
also understand it from the perturbation theory expansion, where a dire t evalua-
tion of the diagrams for the self-energy, and the sum rules for the ele troni Green's
fun tions, establish that the zeroth moment of the retarded ele troni self-energy is
a onstant [68℄. What is perhaps more interesting, is when one treats a fully self-
onsistent system where the ele trons and phonons both an ex hange energy with
one another, and the phonon bath properties hange transiently. In this ase, one
has to examine the self- onsisten y for both the ele trons and the phonons within
the perturbation theory, and the general form of the sum-rules hold. The al ulations
shown above in the atomi limit indi ate that it is likely that adding energy into
the phonon system in reases the phonon �u tuations and thereby reates a stronger
ele troni self-energy. One would expe t there to be os illations of the spe tral weight
as well. It is also likely that these ideas an be in orporated into the quantitative
analysis of experiments that we expe t to see o ur over the next few years.
3.2.5 Con lusions and future work
In this work, we have shown the simplest sum rules for ele trons intera ting with
phonons. These sum rules have been established in equilibrium for some time now, but
our work shows that they dire tly extend to nonequilibrium. We also established new
sum rules for the phonon propagator. In general, these sum rules are ompli ated to
use, be ause they require one to determine both the average phonon expe tation value
and its �u tuations, so they might �nd their most important appli ation to numeri s
as ben hmarking, assuming one an al ulate the relevant expe tation values with
the numeri al te hniques employed to solve the problem. But they also allow us to
examine the physi al behavior we expe t to see if we look at how the moments might
hange in time due to the e�e t of a transient light pump applied to the system.
53
For example, we expe t that as energy is ex hanged from ele trons to phonons, the
ele tron self-energy should in rease its spe tral weight, with the opposite o uring as
the phonons transfer energy ba k to the ele trons. This result is not one that ould
have been easily predi ted without the sum rules.
In the future, there are a number of ways these sum rules an be extended. One
an examine more realisti models, like the Hubbard-Holstein model and �nd those
sum rules. One an look into the e�e ts of anharmoni ity on the sum rules, and �nally,
one an arry out the al ulations to higher order, to examine more moments. We
plan to work on a number of these problems in the future.
3.3 Nonequilibrium sum rules for the Hubbard-Holstein model
Re ent developments in pump-probe spe tros opy have provided a powerful tool to
study nonequilibrium properties of a large variety of strongly orrelated systems with
oupled ele trons, phonons, and spin degrees of freedom within femtose ond time
s ales [72, 73, 74, 75, 76, 77, 78℄. This te hnique has been applied to study high Tc
uprates, whi h exhibit strong ele tron-ele tron and ele tron-phonon ouplings [59,
79, 80℄. Hen e, understanding the nonequilibrium dynami s of the ele tron-phonon
intera tion and it's interplay with the ele tron-ele tron intera tion still remains one
of the most intriguing topi s in ondensed matter physi s.
The pump is an ultra-strong and ultrashort ele tri �eld pulse, whi h an be used
to dire tly ex ite either the ele trons or the phonons. The resulting nonequilibrium
state an subsequently be explored by a probe, whi h is a weaker pulse that mea-
sures the temporal response of the system. Amongst di�erent materials that have
been studied by pump-probe spe tros opy, high temperature super ondu tors have
attra ted signi� ant attention, in part, be ause the role played by the ele tron-phonon
54
intera tion in these material's properties is still not well-understood. For example,
Zhang et al. [8℄ re ently investigated the ultrafast response of the self-energy of a
high-temperature super ondu tor in both the normal and super ondu ting states.
The most dire t eviden e of an ele tron-phonon oupling in the uprates (or, more
generally, an ele tron-boson oupling) is the universal ele tron self-energy renormal-
ization, whi h manifests itself as a kink in the photoemission spe tra that o urs below
the Fermi energy pre isely at the oupled phonon energy [9℄. The strength of the kink
is dire tly related to the strength of the ele tron-phonon oupling. Whether this
phenomenon is dire tly related to high temperature super ondu tivity still remains
un lear. One intriguing result from the pump/probe experiments [8℄ is that the kink
softens when in the super ondu ting state, even with a relatively weak pump. This
raises the question, is the pump dynami ally redu ing the ele tron-phonon oupling
in the super ondu ting state? It turns out sum rules an be employed to examine
this question. A numeri al study [68℄, shows that the kink softens when the system
is pumped, even if there is no dynami redu tion of the ele tron-phonon oupling, as
determined by the zeroth moment of the retarded self-energy as a fun tion of time.
Hen e, kink softening alone is insu� ient to tell whether there is a dynami redu tion
of the ele tron-phonon oupling.
We explain how one an use exa t sum rules to investigate the e�e t of a pump
on a system with both ele tron-ele tron and ele tron-phonon intera tions. We use the
Holstein-Hubbard model, whi h is one the simplest models to des ribe the interplay
between ele tron-ele tron and ele tron-phonon intera tions [81, 82, 83, 84, 85℄. The
sum rules also provide a powerful tool in the ben hmarking of omputational work to
he k the pre ision of numeri al solutions. The approa h was developed to al ulate
the �rst two moments of the spe tral fun tion in order to estimate the a ura y of
Monte-Carlo solutions of the Hubbard model in two-dimensions [54℄. Sin e then, the
55
appli ation of sum rules has extended to a variety of strongly orrelated systems
in equilibrium and nonequilibrium both for homogeneous and inhomogeneous ases
[10, 11, 12, 13℄. Sum rules for the retarded Green's fun tion through se ond order for
the Holstein model [86℄ and the zeroth-order self-energy sum rule for the Holstein-
Hubbard model [87℄ have also appeared in equilibrium. Preliminary work already
found the lowest-order sum rules in the nonequilibrium Holstein model [14℄. Here,
we fo us on the full Holstein-Hubbard model and derive the nonequilibrium spe tral
moments to one higher order.
The remainder of the paper is organized as follows. In Se . 3.3.1, we introdu e the
Holstein-Hubbard model and we derive the exa t sum rules for the spe tral fun tion
of retarded Green's fun tion up to the third moment. Periodi ity of the system is
not needed for these al ulations. In Se . 3.3.2, we derive the orresponding spe tral
moments for the retarded self-energy. For translationally invariant systems, we obtain
the moments in momentum spa e in Se . 3.3.3. To verify the expe tation values and
the �u tuations, we al ulate the atomi limit of the Holstein-Hubbard model in Se .
3.3.4 and ompare those exa t results to the moments. A summary and on lusions
are provided in Se . 3.3.5.
3.3.1 Formalism for the sum rules of the spe tral fun tion for the
Holstein-Hubbard model
The Holstein-Hubbard model has been widely used to des ribe the systems with both
ele tron-phonon and ele tron-ele tron intera tions [81, 82, 83, 84, 85℄. The Hamilto-
56
nian for the (inhomogeneous) Holstein-Hubbard model is given by
HH−H(t) = −∑
ijσ
tij(t)c†iσcjσ +
∑
i
Ui(t)ni↓ni↑
+∑
i
[gi(t)xi − µi(t)](ni↑ + ni↓)
+∑
i
1
2Mip2i +
∑
i
1
2κix
2i
, (3.70)
where niσ = c†iσciσ is the ele tron number at site i. The phonon oordinate and
momentum are denoted by xi and pi, respe tively. The ele tron hopping matrix tij(t)
is a (possibly time-dependent) Hermitian matrix and Ui(t) is the (possibly time-
dependent) on-site Hubbard repulsion. The ele trons are oupled to phonons by ou-
pling strength gi(t) whi h is parametrized by an energy per unit length (and may be
time-dependent). A lo al site energy µi is also in luded (it is the hemi al potential
if it is independent of i). This model aptures the features of variety of interesting
phenomena su h as the Mott transition, polaron and bipolaron formation. It also
has ordered phases to super ondu tivity, harge-density-wave order, and spin-density-
wave order. The dynami al mean-�eld theory (DMFT) has been applied to investigate
the model exa tly [81, 88, 89, 90, 91℄. We next use the formulas in Eqs. (3.24- 3.27), to
determine the sum rules for the nonequilibrium and inhomogeneous Holstein-Hubbard
model. To simplify the formulas, we introdu e the notation [O = O(Tave)℄ to indi ate
the operator is evaluated at the average time Tave, after taking the limit trel → 0. In
addition, we de�ne νiσ = µi(Tave) − Ui(Tave)〈niσ(Tave)〉 − gi(Tave)〈xi(Tave)〉 to make
the expressions more readable (we also use the notation σ = −σ). The zeroth moment
is trivial,
µR0ijσ(Tave) = δij , (3.71)
57
and higher moments are shown below, where we employ the fermioni operator iden-
tity n2iσ = niσ
µR1ijσ(T ) = − tij − νiσδij (3.72)
µR2ijσ(Tave) =
∑
k
tik tkj + tij νiσ + tij νjσ + ν2iσδij + U2i [〈niσ〉 − 〈niσ〉2]δij
+ g2i [〈x2i 〉 − 〈xi〉2]δij + 2Uigi[〈niσxi〉 − 〈niσ〉〈xi〉]δij, (3.73)
µR3ijσ(Tave) = −
∑
kl
tik tkltlj − νiσ∑
k
tik tkj −∑
k
tikνkσ tkj −∑
k
tik tkj νjσ
−ν2iσ tij − U2i [〈niσ〉 − 〈niσ〉2]tij − g2i [〈x2i 〉 − 〈xi〉2]tij − 2Uigi[〈niσxi〉 − 〈niσ〉〈xi〉]tij
−ν2jσ tij − U2j [〈njσ〉 − 〈njσ〉2]tij − g2j [〈x2j〉 − 〈xj〉2]tij − 2Uj gj[〈njσxj〉 − 〈njσ〉〈xj〉]tij
−νiσ tij νjσ − UiUj[〈c†iσ ciσ c†jσcjσ〉 − 〈niσ〉〈njσ〉]tij − gigj[〈xixj〉 − 〈xi〉〈xj〉]tij
−giUj [〈njσxi〉 − 〈njσ〉〈xi〉]tij − gjUi[〈niσxj〉 − 〈niσ〉〈xj〉]tij − ν3iσδij +giκiMi
〈xi〉
+U3i [〈niσ〉 − 〈niσ〉3]δij + g3i [〈x3i 〉 − 〈xi〉3]δij − 3µiU
2i [〈niσ〉 − 〈niσ〉2]δij
−3µig2i [〈x2i 〉 − 〈xi〉2]δij + 3U2
i gi[〈niσxi〉 − 〈niσ〉〈xi〉]δij + 3Uig2i [〈niσx
2i 〉 − 〈niσ〉〈x2i 〉]δij
−6µigiUi[〈niσxi〉 − 〈niσ〉〈xi〉] + Uiδij∑
kl
[tik tkl〈c†iσ clσ〉+ tkltli〈c†lσciσ〉 − 2tik tli〈c†lσ ckσ〉]
+Uiδij∑
k
[µk − µi][tki〈c†kσciσ〉+ tik〈c†iσ ckσ〉] + Uiδij∑
k
[tkigi〈c†kσciσxi〉+ tikgk〈c†iσ ckσxk〉]
+δijUi
∑
k
Uk[tki〈c†kσciσ c†kσckσ〉+ tik〈c†iσ ckσc†kσckσ〉] + UiUj [tji〈c†jσciσ c†jσciσ〉+ tij〈c†iσ cjσc†jσ ciσ〉]
+1
2Re i
∑
k
[dtikdTave
tkj − tikdtkjdTave
]− 1
2Re i
dtijdTave
[µi − µj] +1
2Re i[
dµi
dTave− dµj
dTave]tij
+1
2Re i
dtijdTave
[gi〈xi〉 − gj〈xj〉]−1
2Re i[
dgidTave
〈xi〉 −dgjdTave
〈xj〉]tij
+1
2Re i
dtijdTave
[Ui〈niσ〉 − Uj〈njσ〉]−1
2Re i[
dUi
dTave〈niσ〉 −
dUj
dTave〈njσ〉]tij
+1
4Re
d2tijdT 2
ave
+1
4Re δij [
dµ2i
dT 2ave
− dU2i
dT 2ave
〈niσ〉 −dg2idT 2
ave
〈xi〉]. (3.74)
These are the main results of this work.
58
Note that in the third moment, we have a term
giκi
Mi〈xi〉 whi h arises from the
phononi part of the Hamiltonian. The reason of absen e of su h terms in lower order
is as follows. The �rst moment is obvious be ause the ommutator of the fermioni
operator with phonon part of Hamiltonian is zero. Although in the se ond moment,
there are multiple terms whi h in lude ommutator of x operator with momentum
operator p, these terms be ome imaginary and onsequently do not ontribute in
the se ond moment and they only ontribute in the third moment. Furthermore, we
noti e that the sum rules depend on a number of di�erent expe tation values. Hen e,
they are not just fun tions of the parameters of the model, but also they depend on
the expe tation values of a number of di�erent operators. One expe tation value an
be immediately determined, namely 〈xi〉. This an be done by shifting the phonon
oordinate by xi → xi+ gi(t)(ni↑+ni↓)/κi = xi. The Hamiltonian is an even fun tion
of xi, so we must have 〈xi〉 = 0. This implies that 〈xi〉 = −gi(t)〈ni↑ + ni↓〉/κi.
Unfortunately, similar arguments will not allow us to determine higher power law
expe tation values of the oordinate xi.
3.3.2 Formalism for the sum rules for retarded ele troni self-
energy
Next, we derive the retarded self-energy moments. The self-energy does not vanish at
high frequen y, but approa hes a onstant value, whi h we denote as ΣRijσ(Tave, ω =
∞) and is real. The moments are de�ned from integrals over the imaginary part of
the self-energy via
CRnijσ = −1
π
∫
dωωnImΣijσ(ω) (3.75)
The zeroth moment gives the overall strength of the self-energy. These moments an be
obtained from the Dyson equation whi h onne ts the self-energy with the Green's
fun tion. For the nonequilibrium ase, it's useful to work in the Larkin-Ov hinkov
59
representation where the Green's fun tion and the self-energy ea h be ome 2 × 2
matri es [45℄. The omplete derivation of this Dyson equation for the nonequilibrium
self-energy has already been derived in Ref. [12℄ so we only report the �nal results.
These equations are used, in turn, to determine the moments of the self-energies, as
shown below. Note that the tilde here means the moments for the nonintera ting ase,
when all intera tions vanish [Ui(t) = 0 and gi(t) = 0℄:
µR0ijσ(Tave) = µR0
ijσ(Tave), (3.76)
µR1ijσ(Tave) = µR1
ijσ(Tave)∑
kl
µR0ikσ(Tave)Σ
Rklσ(Tave, ω = ∞)µR0
ljσ(Tave), (3.77)
µR2ijσ(Tave) = µR2
ijσ(Tave) +∑
kl
µR0ikσ(Tave)Σ
Rklσ(Tave, ω = ∞)µR1
ljσ(Tave)
+∑
kl
µR0ikσ(Tave)C
R0klσ(Tave)µ
R0ljσ(Tave) +
∑
kl
µR1ikσ(Tave)Σ
Rklσ(Tave, ω = ∞)µR0
ljσ(Tave)
, (3.78)
µR3ijσ(Tave) = µR3
ijσ(Tave) +∑
kl
µR0ikσ(Tave)Σ
Rklσ(Tave, ω = ∞)µR2
ljσ(Tave)
+∑
kl
µR0ikσ(Tave)C
R0klσ(Tave)µ
R1ljσ(Tave) +
∑
kl
µR0ikσ(Tave)C
R1klσ(Tave)µ
R0ljσ(Tave)
+∑
kl
µR1ikσ(Tave)Σ
Rklσ(Tave, ω = ∞)µR1
ljσ(Tave) +∑
kl
µR1ikσ(Tave)C
R0klσ(Tave)µ
R0ljσ(Tave)
+∑
kl
µR2ikσ(Tave)Σ
Rkl(Tave, ω = ∞)µR0
ljσ(Tave), (3.79)
where ΣRij(ω = ∞) is the high-frequen y limit of the self-energy, i. e., the real onstant
term of the self-energy. Using the fa t that
µR0ijσ(Tave) = µR0
ijσ(Tave) = δij , (3.80)
the self-energy moment sum rules an be expli itly determined after some algebra.
We �nd
ΣRijσ(Tave, ω = ∞) = [Ui〈niσ〉+ gi〈xi〉]δij, (3.81)
60
CR0ijσ(Tave) = U2
i [〈niσ〉 − 〈niσ〉2]δij + g2i [〈x2i 〉 − 〈xi〉2]δij + 2giUi[〈xiniσ〉 − 〈xi〉〈niσ〉]δij,
(3.82)
CR1ijσ(Tave) = −µig
2i [〈x2i 〉 − 〈xi〉2]δij − µiU
2i [〈niσ〉 − 〈niσ〉2]δij
+ g3i [〈x3i 〉 − 2〈x2i 〉〈xi〉+ 〈xi〉3]δij − giU2i [4〈xiniσ〉〈niσ〉+ 3〈niσ〉2〈xi〉+ 2〈niσ〉〈xi〉 − 3〈xiniσ〉]δij
+ U3i [〈niσ〉 − 2〈niσ〉2 + 〈niσ〉3]δij − Uig
2i [4〈xiniσ〉〈xi〉+ 5〈xi〉2〈niσ〉 − 3〈x2i niσ〉]δij +
giκiMi
〈xi〉
+ Uiδij∑
kl
[tik tkl〈c†iσclσ〉+ tkltli〈c†lσ ciσ〉 − 2tik tli〈c†lσckσ〉] + Uiδij∑
k
[µk − µi][tki〈c†kσciσ〉+ tik〈c†iσ ckσ〉]
+ Uiδij∑
k
[tkigi〈c†kσciσxi〉+ tikgk〈c†iσckσxk〉] + Uiδij∑
k
Uk[tki〈c†kσciσ c†kσckσ〉+ tik〈c†iσckσ c†kσckσ〉]
+ UiUj [tji〈c†jσciσ c†jσ ciσ〉+ tij〈c†iσ cjσc†jσciσ〉]
+1
2Re i
dtijdTave
[gi〈xi〉 − gj〈xj〉]−1
2Re i[
dgidTave
〈xi〉 −dgjdTave
〈xj〉]tij
+1
2Re i
dtijdTave
[Ui〈niσ〉 − Uj〈njσ〉]−1
2Re i[
dUi
dTave〈niσ〉 −
dUj
dTave〈njσ〉]tij
+1
4Re
d2tijdT 2
ave
+1
4Re δij [
dµ2i
dT 2ave
− dU2i
dT 2ave
〈niσ〉 −dg2idT 2
ave
〈xi〉]. (3.83)
Note that the zeroth moment is lo al (diagonal) even if the self-energy has
momentum dependen e, while the �rst moment an be nonzero only for lo al terms
(i = j) and for terms where the hopping is nonvanishing (tij 6= 0). In parti ular, if
we use the zeroth moment to determine the strength of the e�e tive ele tron-phonon
intera tion, then for a pure Holstein model, the only way the ele tron-phonon inter-
a tion is dynami ally hanged is if the orrelation fun tion of the phonon oordinate
hanges as a fun tion of time. This an happen, for example, if energy �ows into the
phonon bath, but is likely to be delayed due to the bottlene k for energy �ow from
ele trons to phonons. S reening e�e ts, whi h an hange the net ele tron-phonon
oupling, are not in the Holstein-Hubbard model, and require a more omplex model
to be properly des ribed.
61
3.3.3 Spe tral sum rules in momentum spa e
When the system is translationally invariant, it is more onvenient to work in
momentum spa e instead of real spa e. So, we next examine the situation where
tij is a periodi hopping matrix and the lo al hemi al potential, ele tron-phonon
oupling, and Hubbard intera tion are all uniform throughout the latti e. This al-
ulation requires us to make an appropriate Fourier transformation. The al ulations
are tedious, but straightforward. To begin, we start with the de�nition of the retarded
Green's fun tion in the momentum spa e,
GRkσ(t, t
′) = −iθ(t, t′)〈{ckσ(t), ckσ(t′)}〉, (3.84)
where k denotes the momentum. The orresponding reation and annihilation oper-
ators in momentum spa e an be obtained by performing a Fourier transform, ckσ =∑
i eik·Riciσ/N , and c†
kσ =∑
i e−ik·Ric†iσ/N . Here, N is the number of latti e sites.
Substituting the inverse Fourier transformation into the formula for the real-spa e
moments, then yields
µRnkσ (Tave) =
1
N
∑
ij
e−ik·(Ri−Rj)µRnijσ(Tave). (3.85)
The momentum-based sum rules then be ome
µR0kσ(Tave) = 1, (3.86)
µR1kσ(Tave) = ǫ
k
− νσ, (3.87)
where ǫk
= −∑{δ} ti i+δeik·δ
, {δ} is the set of all of the translation ve tors for whi h
the hopping matrix is nonzero (the index i+ δ s hemati ally denotes the latti e site
orresponding to site Ri+δ), and νσ = µ(Tave)−U(Tave)〈nσ(Tave)〉−g(Tave)〈x(Tave)〉.
62
Note, that in a paramagneti solution, the �lling will be independent of the spin σ.
The higher moments be ome the following:
µR2kσ(Tave) = ǫ2
k
− 2ǫk
νσ + ν2σ + U2[〈nσ〉 − 〈nσ〉2] + g2[〈x2〉 − 〈x〉2] + 2U g[〈nσx〉 − 〈nσ〉〈x〉],
(3.88)
µR3kσ(Tave) = ǫ3
k
− 3ǫ2k
νσ + ǫk
ν2σ + 3ǫk
U2[〈nσ〉+ 3〈nσ〉2] + ǫk
g2[〈x2〉 − 〈x〉2]
+ 6ǫk
U g[〈nσx〉 − 〈nσ〉〈x〉]− ν3σ + U3[〈nσ〉 − 〈nσ〉3] + g3[〈x3〉 − 〈x〉3]− 3µU2[〈nσ〉 − 〈nσ〉2]
− 3µg2[〈x2〉 − 〈x〉2] + gκ
M〈x〉+ 3U2g[〈nσx〉 − 〈nσ〉〈x〉] + 3U g2[〈nσx
2〉 − 〈nσ〉〈x2〉]
− 6µgU [〈nσx〉 − 〈nσ〉〈x〉] + 2U gǫ2k
〈nσx〉+ U2∑
q,p′,q′
(ǫq+q′−p + ǫ
p
′−q−q′)〈c†q
′+q′−p′cq
c†p
′cq
′〉
+ U2∑
q,p′,q′
(ǫp−q′ + ǫ
q−q′)〈c†k+q′cqc
†q−kcq′〉+
1
4Re
d2ǫ2k
dT 2ave
+1
4Re [
dµ2
dT 2ave
− dU2
dT 2ave
〈nσ〉 −dg2
dT 2ave
〈x〉].
(3.89)
Similarly, we an obtain the sum rules for the retarded self-energy,
ΣRkσ(Tave, ω = ∞) = U〈nσ〉+ g〈x〉, (3.90)
CR0kσ (Tave) = U2[〈n〉 − 〈n〉2] + g2[〈x2〉 − 〈x〉2] + 2gU [〈xn〉 − 〈x〉〈n〉], (3.91)
CR1kσ (Tave) = −µg2[〈x2〉 − 〈x〉2]− µU2[〈nσ〉 − 〈nσ〉2] + g3[〈x3〉 − 2〈x2〉〈x〉+ 〈x〉3]
− gU2[4〈xnσ〉〈nσ〉+ 3〈nσ〉2〈x〉+ 2〈nσ〉〈x〉 − 3〈xnσ〉] + U3[〈nσ〉 − 2〈nσ〉2 + 〈nσ〉3]
− U g2[4〈xn〉〈x〉+ 5〈x〉2〈nσ〉 − 3〈x2nσ〉] + 2U gǫ2k
〈nσx〉+giκ
M〈x〉
+ U2∑
q,p′,q′
(ǫq+q′−p + ǫ
p
′−q−q′)〈c†q
′+q′−p′cq
c†p
′cq
′〉+ U2∑
q,p′,q′
(ǫp−q′ + ǫ
q−q′)〈c†k+q′cqc
†q−kcq′〉
+1
4Re
d2ǫ2k
dT 2ave
+1
4Re [
dµ2
dT 2ave
− dU2
dT 2ave
〈nσ〉 −dg2
dT 2ave
〈x〉]. (3.92)
These forms of the di�erent sum rules may be more useful for most al ula-
tions, whi h work with translationally invariant systems. Note that, as expe ted, the
63
moments either have no momentum dependen e, or inherit a momentum dependen e
from the bandstru ture, be ause the o�-diagonal moments always had a dependen e
on the hopping matrix element. As noted before, the higher moments require many
di�erent expe tation values to be known in order to properly employ them. If one is
using methods like quantum Monte Carlo simulation, where one an measure su h
expe tation values in addition to determining the Green's fun tion and self-energy,
then one an employ these results as a he k on the a ura y of the al ulations.
Similarly, if one has an approximation method that is employed for the Holstein-
Hubbard model, then by al ulating these di�erent expe tation values within the
approximation, one an test the overall self- onsisten y of the approximation to see if
it satis�es these exa t relations. Of ourse, if everything is evaluated with an approx-
imate solution, there is no guarantee that the approximation is a urate even if it
self- onsistently satis�es these sum rules. But if it does not satisfy them, then you
immediately know they are in error.
We also want to emphasize that these results hold in a wide range of di�erent
nonequilibrium situations and are quite general. This makes them quite valuable
be ause there often are few exa t results known about nonequilibrium solutions. We
hope the ommunity will regularly use these sum rules to he k the a ura y of
di�erent al ulations, espe ially those in nonequilibrium.
Unfortunately, we have not been able to �nd su� iently a urate numeri al al u-
lations on the full Holstein-Hubbard model, along with the al ulation of the required
expe tation values, to readily he k the results of these sum rules against state-of-the-
art al ulations. So, as a substitute, we examine the atomi limit next, whi h allows
us to he k the pie es of the sum rule that do not depend on the hopping.
64
3.3.4 Atomi limit of the Holstein-Hubbard model
We an solve the atomi limit (tij → 0) exa tly, and thereby he k all terms in the
moments that survive when the hopping vanishes. The atomi Hamiltonian be omes
HatH−H(t) = U(t)n↑n↓ + [g(t)x− µ](n↑ + n↓) +
p2
2M+
1
2κx2, (3.93)
where, for simpli ity, we assume the hemi al potential is independent of time, be ause
time dependen e with respe t to the hemi al potential an be trivially handled due
to the fa t that the total ele tron number operator ommutes with the Hamiltonian.
Using the equation of motion for the raising and lowering operators, in the Heisenberg
representation, we �nd the phonon oordinate operator be omes
xH(t) =ae−iω(t−tmin) + a†eiω(t−tmin)
√2mω
− Re
{
ie−iωt
∫ t
tmin
dt′eiωt′
g(t′)
}n↑ + n↓mω
. (3.94)
Note that be ause the ele tron number operator now ommutes with the Hamiltonian,
it has no time dependen e. The Hilbert spa e is a dire t produ t of the harmoni
os illator number states given by the number operator representation
|n〉 = 1√n!
(a†)n |0〉, (3.95)
and the four fermioni states
|0〉, | ↑〉 = c†↑|0〉, | ↓〉 = c†↓|0〉, | ↑↓〉 = c†↑c†↓|0〉. (3.96)
The partition fun tion is
Zat =∑
0,↑,↓,↑↓
∞∑
nb=0
〈nb, nf | exp[−β{(gx− µ)(nf↑ + nf
↓) + Unf↑ n
f↓ + ω(nb +
1
2)}]|nb, nf〉.
(3.97)
where nb and nf are the bosoni and fermioni number operators, respe tively. Note
that the partition fun tion is al ulated in the initial equilibrium state, so g = g(tmin)
65
and U = U(tmin) have no time dependen e. To al ulate the partition fun tion, we
need to go to the intera tion representation with respe t to the bosoni Hamiltonian
in imaginary time. The steps of the al ulation are similar to what we have already
done for the Holstein model in Ref. [11℄ with the modi� ation to in lude the Hubbard
intera tion. Here, we will only report the �nal results and we refer the interested
reader to Ref. [11℄ for the details. The �nal results for the partition fun tion and the
other observables be omes,
Zat =1
1− e−βω
{
1 + 2eβµ exp
[βg2(tmin)
2mω2
]
+ eβ(2µ−U(tmin)) exp
[2βg2(tmin)
mω2
]}
,(3.98)
and
〈n↑ + n↓〉 =2eβµ exp
[βg2(tmin)
2mω2
]
+ 2eβ(2µ−U(tmin)) exp[2βg2(tmin)
mω2
]
1 + 2eβµ exp[βg2(tmin)
2mω2
]
+ eβ(2µ−U(tmin)) exp[2βg2(tmin)
mω2
] , (3.99)
〈x(t)〉 = 〈n↑ + n↓〉(
−g(tmin)
mω2cosω(t− tmin)− Re
{
ie−iωt
∫ t
tmin
dt′eiωt′ g(t′)
mω
})
.
(3.100)
We noti e that, in equilibrium, g(t) = g be omes a onstant and the average
phonon oordinate be omes time-independent, 〈x(t)〉 = −〈n↑ +n↓〉g/mω2, as we saw
previously. Additionally, the �u tuation satis�es,
〈x2(t)〉 − 〈x(t)〉2 = [〈n↑〉(1− 〈n↑〉) + 〈n↓〉(1− 〈n↓〉) + 2〈n↑n↓〉 − 2〈n↑〉〈n↓〉]
×[g(tmin)
mω2cosω(t− tmin) + Re
{
ie−iωt
∫ t
tmin
dt′eiωt′ g(t′)
mω
}]2
+1
2mωcoth
(βω
2
)
. (3.101)
One an see that the �u tuation is onsists of both quantum and thermal parts.
The �rst term denotes the quantum �u tuation due to the interation in the system.
The role of ele tron-phonon intera tion in the �u tuation has manifested itself by
66
a lear time dependent of oupling fa tor. However, we noti e that the ele tron-
ele tron intera tion is somewhat hidden in the oe� ient of the square term whi h
is a fun tion of ele tron density given by equation 3.99. One noti es that by setting
U = 0, we re over the results for the Holstein model in Ref. [14℄ as we expe ted. Then,
the se ond term denotes the thermal �u tuation whi h is temperature dependent and
it grows as temperature in reases. To omplete the al ulation of the atomi limit,
we also need to al ulate the following expe tation value,
〈x(t)n↓(t)〉 = 〈(n↓(t) + n↑(t))2〉(
−g(tmin)
mω2cosω(t− tmin)− Re
{
ie−iωt
∫ t
tmin
dt′eiωt′ g(t′)
mω
})
.
(3.102)
Re all that the Pauli ex lusion prin iple implies that 〈nσ(t)2〉 = 〈nσ(t)〉 and onse-
quently,
〈x(t)n↓(t)〉 = (〈n↓(t)〉+ 〈n↑(t)〉+ 2〈n↓(t)n↑(t)〉)
×(
− g(tmin)
mω2cosω(t− tmin)− Re
{
ie−iωt
∫ t
tmin
dt′eiωt′ g(t′)
mω
})
.
(3.103)
By substituting the expe tation value of position from equation 3.100, we get
〈x(t)n↓(t)〉 = 〈x(t)〉+ 2〈n↓(t)n↑(t)〉(
− g(tmin)
mω2cosω(t− tmin)− Re
{
ie−iωt
∫ t
tmin
dt′eiωt′ g(t′)
mω
})
,
(3.104)
whi h expli itly shows that di�erent orrelations are dependent to ea h other be ause
of the ele tron-phonon oupling. This identity be omes more valuable as the ele tron
density orrelation for up and down spin an be evaluated easily for ases su h as
half-�lling whi h implies a simple to evaluate ele tron harge and phonon oordinate
orrelation. Before ending this se tion, we omment about the zero moment of the
self-energy in the atomi limit. Putting together all three terms in equation (3.82) that
67
we have evaluated in atomi limit, we observe the following behaviors: First, the zero
moment of the self-energy in nonequilibrium is time dependent through dependen e
to original time-dependent parameters. Se ond, the zero moment of the self-energy
hanges with temperature whi h omes from the �u tuation term of phonons. This
temperature dependen e an be reated by an external driving �eld, for example , it
an be the ele tri �eld of the pump in the pump-probe tr-Arpes experiment. Our
result is onsistent with a re ent study performed by Kemper and his ollaborates
where they have performed numeri al simulation to study the e�e t of the pump
on the kink softening in the Holstein-Hubbard model[68℄. Their result indi ates that,
although the self-energy an hange with time, the spe tral weight of self-energy stays
onstant in all times.
We end this se tion with proposing a slightly di�erent method to verify the sum
rule result in the atomi limit. First, we al ulate the c(t1) and c†(t2) in the Heisenberg
representation in the atomi limit:
c(t1) = T e−i∫ t1t0
[−U(t)n(t)+µ(t)−g(t)x(t))]dtc,
c†(t2) = T ei∫ t2t0
[−U(t)n(t)+µ(t)−g(t)x(t))]dtc†. (3.105)
Although x(t) does not ommute at di�erent times, the time ordered produ t an be
evaluated as this ommutator be omes a simple fun tion of time and by transforming
in the intera tion representation we an obtain the following produ t after performing
long algebra, see Ref. [11℄ for more details,
c(t1) = e−i∫ t1t0
[−U(t)n(t)+µ(t)]dteig2(t1)
2mω3 (ωt1−sinωt1) ×−g(t1)√2mw3
[(eiωt1−1)a†+(1−e−iωt1)a]c,(3.106)
c†(t2) = ei∫ t2t0
[−U(t)n(t)+µ(t)]dte−ig2(t2)
2mω3 (ωt2−sinωt2) ×−g(t2)√2mw3
[(e−iωt2−1)a+(1−eiωt2 )a†]c†.(3.107)
68
Then, by using the equation 3.4, we an rewrite the spe tral moment in the atomi
limit as,
µRnat = (∂t1 − ∂t2)
nGRat(t1, t2)|(t2−t1)→0 (3.108)
where
GRat(t1, t2) = −iθ(t1, t2)〈{c(t1), c†(t2)}〉. (3.109)
By substituting the fermioni operators we get:
GRat(t1, t2) = −iθ(t1, t2)e−i
∫ t1t2
[−U(t)n(t)+µ(t)]dt e−i
2mw2 [g2(t1)t1−g2(t2)t2]e−
i2mw3 [g(t1) sin(ωt1)−g(t2) sin(ωt2)]
×(
〈e[−g(t1)√2mw3
(eiωt1−1)a†+(1−e−iωt1)a]e[− g(t2)√
2mw3(e−iωt2−1)a+(1−eiωt2 )a†]
c c†〉
+〈e[−g(t2)√2mw3
(e−iωt2−1)a+(1−eiωt2 )a†]e[− g(t1)√
2mw3(eiωt1−1)a†+(1−e−iωt1 )a]
c† c〉)
. (3.110)
In the next step, we an evaluate the expe tation value, similar to what we did for
the other ones:
GRat(t1, t2) = −iθ(t1, t2)
1
Z∑
0,↑,↓,↑↓
∞∑
nb=0
e−βωn+βµnf− g2(tmin)nf2
2mω3 (eβω−1−βω)
e−i∫ t1t2
[−U(t)n(t)+µ(t)]dt e−i
2mw2 [g2(t1)t1−g2(t2)t2]e−
i2mw3 [g(t1) sin(ωt1)−g(t2) sin(ωt2)]
×(
〈nb, nf |eA1a eA2a†eA3a+A4a†eA5a+A6a†c c† + eA1a eA2a†eA4a+A6a†eA3a+A5a†c† c|nb, nf 〉,(3.111)
where, the oe� ients are as follows
A1 = −g(τmin)nf
√2mw3
(1− e−ωβ),
A2 = −g(τmin)nf
√2mw3
(eωβ − 1),
A3 = − g(t1)√2mw3
(eiωt1 − 1),
A4 = − g(t1)√2mw3
(1− e−iωt1),
A5 = − g(t2)√2mw3
(eiωt2 − 1),
A6 = − g(t2)√2mw3
(1− e−iωt2). (3.112)
69
After rearranging di�erent terms by using the identity eαA+βB = eαAeβBe−12αβ[A,B]
,
and evaluating taking the tra e with respe t to both fermioni and bosoni operator,
we have
GRat(t1, t2) = −iθ(t1, t2)
1
(1− e−βω)ZF1F2[e−A4A5 + 2eβµ+
g2β
2mω2 eA6A3
+e2(βµ−U(τmin))+g2β
mω2 (2e−A6A3 − e−A4A5)]. (3.113)
where
F1 = e−i∫ t1t2
[−U(t)n(t)+µ(t)]dt e−i
2mw2 [g2(t1)t1−g2(t2)t2]e−
i2mw3 [g(t1) sin(ωt1)−g(t2) sin(ωt2)],
F2 = e−A2A3−A4A5− 12A3A4− 1
2A5A6 e
1
1−e−βω [A1A4+A1A6+A3A2+A3A6+A5A2+A5A4+A5A6].(3.114)
At this stage, we need to evaluate the time derivative of the above expression. For
example, the zero moment is straightforward and it an be obtained by taking the
limit (t2 − t1) → 0 whi h be omes
µR0at = GR
at(t1, t2)|(t2−t1)→0 = 1. (3.115)
3.3.5 Dis ussion and on lusion
In this paper, we have derived a general formalism whi h enables us to evaluate the
nth derivative of a time dependent operator in the Heisenberg representation. We have
noti ed that, this identity an be useful in some of other studies su h as full ounting
statisti s problem [92℄ or al ulating the dynami al algebra of the bosons [93℄ where
one needs to evaluate the derivative of di�erent operators. These are beyond the s ope
of the urrent study and we will postpone the deep studies for future works.
Then, we have used this identity to evaluate a sequen e of spe tral moment sum
rules for retarded Green's fun tion and orresponding self-energy in the normal state.
70
These spe tral fun tions have been obtained for the nonequilibrium and generi non-
homogenous ase. The spe tral fun tions provide an exa t formalism in nonequilib-
rium whi h an be used to he k the self- onsisten y he k of both experimental
and omputational results. We have al ulated the spe tral moment for the retarded
Green's fun tion whi h indi ate that both ele tron-phonon and ele tron-ele tron inter-
a tion has ontribution in the spe tral fun tion and only will hange with time if the
original parameter su h as tij , U , and g are fun tion of time.
Although we an not on lude whether the ele tron-ele tron or ele tron-phonon
intera tion has main ontribution in spe tral fun tion of retarded Green's fun tion,
the results for the spe tral moment of the self-energy are more interesting. In fa t,
our results in the atomi regime has provided an intuitive pi ture for studying the
behavior of the spe tral moment of the zero moment of the self-energy in the presen e
of external deriving. This result learly shows a temperature dependen e through
phonon �u tuation whi h arises from both quantum and thermal �u tuation and
provides an strong eviden e that the zero moment of the spe tral fun tion in reases
with in reasing temperature. Furthermore, this result agrees with a re ent numeri al
study in Ref. [68℄, and suggest that the softening of the kink in tr-Arpes of pump-probe
experiment an be indu ed from the applied pump without any hange in ele tron-
phonon oupling. This an be understand as following: although the self-energy itself
hanges with time whi h indu es a spe tral redistribution, however this redistribution
will ompensate ea h other at di�erent interval of frequen y and the integral over the
frequen y will remain onstant as we expe t it from the sum rule.
All the results presented in this paper have been al ulated for the normal state.
So, one important question is how the sum rules are hanged in the super ondu ting
state. To answer this question one needs to arefully extend the de�nition of the
spe tral fun tion into the super ondu ting state. One possible pro edure would be
71
using the Nambu-Gorkov formalism whi h ontains both the normal and anoma-
lous Green's fun tion. Therefore, for the self-energy one needs also determine the
proper Dyson equation in the super ondu ting state. In addition to expanding the
sum rules into super ondu ting state, a re ent study has indi ated an enhan ed tran-
sient ele tron-phonon oupling in a driven bilayered graphene whi h attra ted a lot of
attention[94, 95℄. However, the me hanism for su h enhan ement of ele tron-phonon
oupling remain un lear. One of the possible me hanism is the possibility of non-
linear ele tron-phonon oupling. So it would be interesting to investigate the role
of the non-linear ele tron-phonon oupling in the presen e of the external �eld by
using the spe tral moment sum rules. We hope that we will be able to address these
issues and expand the sum rules results to both super ondu ting state and the ase
of Hubbard-Holstein model with nonlinear oupling in the presen e of the external
�eld.
72
4
Current-voltage profile of a strongly orrelated materials
heterostru ture using non-equilibrium dynami al mean field theory
�The approximate omputations are a fundamental part of physi al s ien e.�
� Steven Weinberg
The multilayer devi e of our interest has onsisted of a number of planes built on
top of ea h other. Depending on the physi al system, the layers an be insulators or
metals. These planes have a small thi kness in one dire tion (here we onsider the z
dire tion) while they are in�nite in the other two dimensions. A �nite number of layers
an be onsidered as the barrier region where the interesting physi al phenomena
o ur and it's onne ted to the bulk via a �nite number of metalli planes, see Figure
4.1, where the bulk will determine the boundary ondition of the system. We will use
the Fali ov-Kimball [2℄ model to des ribe our multilayer devi e, the Hamiltonian of
the system in the general ase in the presen e of an ele tri �eld an be written as
HFK(t) = −∑
αl
t⊥αα+1[eieAα(t)c†αlcα+1l + e−ieAα(t)c†α+1lcαl]
−∑
αlδ
t‖αc†αlcαl+δ +
∑
αl
(−µα + Uαwαl)c†αlcαl (4.1)
73
where t⊥αα+1 is the hopping from plane α to α+1 and it equals to t⊥α+1α for a Hermitian
Hamiltonian. t‖α is the hopping within ea h plane between nearest neighbors, j = i+δ
with δ being a latti e onstant. µα = µ+Vα and Uα denotes the hemi al potential and
the intera tion at ea h plane, where they an hange for di�erent planes depending
on the intera tion and the voltage a ross the plane. Moreover wαl = f †αlfαl whi h an
be 0 or 1 for the heavy parti le. The Aα(t) and Vα are the time dependent ve tor
potential and s alar potential to des ribe the ele tri �eld,
Eα(t) = −∇Vα − ∂Aα(t)
∂t(4.2)
In the next se tion, we will explain the gauge that we will hoose for the ve tor
potential and s alar potential. As we know regardless of the hoi e of the gauge one
should obtain the same results for physi al quantities.
74
L
Bulk
Bulk
R
αα− 1 α+ 1
tαα+1
E
Metal
Insulator
Figure 4.1: S hemati pi ture of a multilayer devi e. The devi e onsists of a di�erent
number of metalli and insulator layers. In this paper, we have a single insulator layer
in the enter of the devi e, whi h is atta hed to the bulk via metalli leads at both
sides. The di�erent planes are indi ated by a Greek index α and tαα+1 denotes the
hopping from plane α to α+ 1 in z dire tion. An extra ele tri �eld is applied in the
barrier region to ompensate for the s attering.
4.1 Non equilibrium DMFT formalism for a multilayer devi e
In this se tion, we explain the main formalism for studying the steady-state transport
through a multilayer devi e. The most ommon method to obtain the urrent-voltage
pro�le is based on a voltage biased method where the left and right leads are separated
from the barrier and their hemi al potential is shifted by an opposite voltage on
opposite sides µl,r = ±V/2 [69, 70℄. Then, after turning on the hopping between
the leads and the barrier, the urrent drives into the system and after some time
eventually rea hes the steady state. But here, we will use a di�erent formalism whi h
has been proposed by Freeri ks et al, the so- alled urrent biased approa h [17℄. In
this formalism, the left and right leads are always onne ted to the barrier. We assume
75
that the system is in equilibrium at temperature T in the in�nite past. Then we apply
an ele tri �eld to generate urrent in the system and after that we turn it o� and
we let the system evolve so that the ve tor potential be omes Aα(t) = −A0. Sin e
the metalli planes are ballisti , the urrent will ontinue to �ow in the system even
after the ele tri �eld has been turned o�. Sin e the system rea hes a steady state,
and we an do the Fourier transform to the frequen y domain, whi h makes a huge
simpli� ation in our al ulation. Both urrent and �lling on ea h plane are onstant,
but due to the s attering in the insulator, the urrent will drop unless we add an
external ele tri �eld at the barrier to ompensate the s attering in those planes. In
pra ti e, this an be done in two ways, one with the time-dependent ve tor potential,
whi h leads to an ele tri �eld Eα = −∂Aα(t)∂t
. The se ond method is to apply a voltage
bias whi h gives rise to a s alar potential, and we get an ele tri �eld on the planes
whi h experien e the hange in the s alar potential E = −∇Vα. Here, we will work
with ve tor potential gauge, but one an show that the result is identi al to the se ond
gauge.
4.2 Equation of motion
In this se tion, we will derive the equation of motion whi h leads to a set of iterative
equations in the multilayer devi e whi h has been previously obtained in Ref. [71℄.
To solve the iterative equations one needs to know the boundary ondition that we
will explain in the next se tion. We start with the ontour ordered Green's whi h is
generalized to the multilayer system,
gcαβ ij(t, t′) = − i
ZTr[Tce−βH(−∞)cαi(t)c
†βj(t
′)], (4.3)
where Z = Tr[e−βH(−∞)] is a partition fun tion with H(−∞) being the Hamiltonian
in the in�nite past at equilibrium. As usual, cα i and c†β j are reation and annihilation
76
operators with indi es α and β denoting di�erent planes, while i and j denotes the
di�erent sites at ea h plane. Tc is the time ordering and it depends on where two
times t, t′ lie on the Keldysh ontour, see �gure 2.1. Sin e the ele tri �eld a ts in
the z dire tion, we an transfer to the mixed basis, where we an perform a Fourier
transform for the xy degrees of freedom into momentum spa e, while keeping the z
dire tion in the real spa e basis, so the ontour ordered Greens fun tion be omes,
gcαβ(k‖; t, t′) = − i
ZTr[Tce−βH(−∞)cαk‖(t)c
†βk‖
(t′)]. (4.4)
To obtain the equation of motion, we take the time derivative of the Green's fun tion
with respe t to t,
∫
c
dt{[(−i∂t + µα − ǫ‖α(k‖))δc(t, t)− Σc
α(t, t)]gcαβ(k
‖; t, t′)
+ [t⊥αα+1eiAα(t)gcα+1β(k
‖; t, t′) + t⊥α−1αe−iAα(t)gcα−1β(k
‖; t, t)]δc(t, t′)}
= δαβδc(t, t′), (4.5)
where we have introdu ed the planar band stru ture de�ned as ǫ‖α(k‖) = −t‖α
∑
δ eik‖.δ
with δ being the nearest neighbor translation ve tor and the ontour ordered delta
fun tion is de�ned as
∫
cdt′δc(t, t′)f(t′) = f(t). Moreover, Σc
α(t, t′) denotes the ontour
ordered self-energy at the α's plane. As we an see, due to the shift of indi es of the
Green's fun tion appearing in equation 4.5, we will get two sets of re ursive equations.
Before deriving the re ursive equations, we assume that we have �xed the α and β
indi es and the inverse will be done with respe t to time. To derive these re ursive
equations, we �rts multiply equation 4.5 by gc −1αβ (k‖; t, t′) from the right side and we
substitute β → α and α→ α− n
77
0 = [−i∂t + µα−n − ǫ‖(k‖)]δc(t, t′)− Σc
α−n(t, t′)
+ t⊥α−nα−n+1eiAα−n(t)
∫
c
dtgcα−n+1α(k‖; t, t)gc −1
α−nα(k‖; t, t′)
+ t⊥α−n−1α−neiAα−n−1(t)
∫
c
dtgcα−n−1α(k‖; t, t)gc −1
α−nα(k‖; t, t′). (4.6)
Now we introdu e the ontour-ordered left fun tion Lcα−n de�ned as
Lcα−n(k
‖;µα−n; t, t′) = −t⊥α−nα−n+1e
iAα−n(t)
∫
c
dtgcα−n−1α(k‖; t, t)gc −1
α−nα(k‖; t, t′). (4.7)
Substituting equation 4.7 in 4.6, we an rewrite it as
Lcα−n(k
‖;µα−n; t, t′) = [−i∂t + µα−n − ǫ
‖α−n(k
‖)]δc(t, t′)− Σα−n(t, t
′)− t⊥2α−n−1α−n
×e−iAα−n−1(t)Lc −1α−n−1(k
‖;µα−n−1; t, t′)eiAα−n−1(t′). (4.8)
Next, we an perform similar pro edure to obtain another re ursive relation, we let
α→ α+ n and β → α then we get
Rcα+n(k
‖;µα+n; t, t′) = −t⊥α+n−1α+ne
−iAα+n−1(t)
∫
c
dtgcα+n−1α(k‖; t, t′)gc −1
α+nα(k‖; t, t′),(4.9)
and the re ursion relation be omes
Rcα+n(k
‖;µα+n; t, t′) = [−i∂t + µα+n − ǫ
‖α+n(k
‖)]δc(t, t′)− Σc
α+n(t, t′)− t⊥2
α+n+1α+n
×eiAα+n(t)Rc −1α+n+1(k
‖;µα+n+1; t, t′)e−iAα+n(t′). (4.10)
Using the de�nition of the left and right fun tions and setting α = β, the equation of
motion an be written as
gc −1αα (k‖; t, t′) = [−i∂t + µα − ǫ‖α(k
‖)]δc(t, t′)− Σc
α(t, t′)
− t⊥2α−1αe
−iAα−1(t)Lc −1α−1 (k
‖;µα−1; t, t′)eiAα−1(t′)
− t⊥2αα+1e
iAα(t)Rc −1α+1 (k
‖;µα+1; t, t′)e−iAα(t′). (4.11)
78
Noti e that the sum of the inverse of left and right part an be onsidered as a
ontribution to the self-energy of the leads. Using the left and right iterative equations,
we an rewrite the above equation in the following form,
gc −1αα (k‖; t, t′) = −[−i∂t + µα − ǫ‖α(k
‖)]δc(t, t′) + Σc
α(t, t′)
+ Lcα(k
‖;µα; t, t′) +Rc
α(k‖;µα; t, t
′). (4.12)
After rea hing the steady state, the system re overs the time translational invarian e
and the Green's fun tion, self-energy, left and right fun tion, all will depend only
on the relative time in the presen e of onstant urrent �ow, so that we an Fourier
transform into the frequen y domain. But prior to that, we take an extra step to
simplify the above equation as all are ontour ordered obje ts. So we transfer into
the Larkin-Ov hinkov representation [45℄. By solving the matrix equation, we get 3
equations for the retarded, advan ed and Keldysh fun tions. Now lets look at their
respe tive de�nitions
gRαβ(k‖; t, t′) = −iθ(t− t′)〈{cαk‖(t), c
†β k‖
(t′)}+〉, (4.13)
gAαβ(k‖; t, t′) = iθ(t′ − t)〈{cαk‖(t), c
†β k‖
(t′)}+〉, (4.14)
and
gKαβ(k‖; t, t′) = −i〈[cα k‖(t), c
†β k‖
(t′)]−〉, (4.15)
then we an obtain the following identities,
gR∗αβ(k
‖; t, t′) = gAβ α(k‖; t′, t), (4.16)
and
gK∗αβ(k
‖; t, t′) = −gKβ α(k‖; t′, t). (4.17)
79
By performing a Fourier transform into the frequen y domain, we get,
gR∗αβ(k
‖;ω) = gAβ α(k‖;ω), (4.18)
and
gK∗αβ(k
‖; t, t′) = −gKβ α(k‖;ω), (4.19)
whi h shows that the Keldysh Green's fun tion is purely imaginary in the frequen y
domain. Using the above identities, we only need to al ulate retarded and Keldysh
Green's fun tion. To do so, we �rst start with the left and right fun tion in the
frequen y domain,
LRα−n(k
‖;µα−n;ω) = ω + µα−n − ǫ‖(k‖)− ΣRα−n(ω)
− t⊥2α−nα−n−1
LRα−n−1(k
‖;µα−n−1;ω + Eα−n−1), (4.20)
where we have introud ed the ele tri �eld by hoosing non-zero ve tor potential
whi h leads to Aα(t) = −A0 − Eαt. Similarly, we obtain the Keldysh left fun tion as
well,
LKα−n(k
‖;ω) = −ΣKα−n(ω) +
t⊥2α−nα−n−1L
Kα−n−1(k
‖;ω + Eα−n−1)
| LRα−n−1(k
‖;ω + Eα−n−1 + Eα−n−1) |2. (4.21)
We an write similar equations for the right fun tions,
RRα+n(k
‖;ω) = ω + µα+n − ǫ‖α+n(k
‖)− ΣRα+n(ω)
− t⊥2α+n+1α+n
RRα+n+1(k
‖;ω −Eα+n), (4.22)
and
RKα+n(k
‖;ω) = −ΣKα+n(ω) +
t⊥2α+n+1α+nR
Kα+n+1(k
‖;ω − Eα+n)
| LRα+n+1(k
‖;ω − Eα+n) |2. (4.23)
In the next se tion, we will show how one an �x the boundary ondition in the bulk
to use the re ursive equation for obtaining the retarded and Keldysh Green's fun tion.
80
4.3 Fixing the boundary ondition
As we an see, the retarded and advan ed fun tions an be al ulated independently
from the Keldysh part. First, we obtain L−∞ and R∞. Using the fa t that the self-
energy is zero for ballisti metals, ΣR±∞(ω) = 0 and also, t⊥αα−1 = t⊥ for the limit
α→ ±∞, we get
LR2−∞(k‖;ω)− [ω + µα − ǫ‖(k‖)]LR
−∞(k‖;ω) + t2⊥ = 0 (4.24)
whi h leads to
LR2−∞(k‖;ω) =
ω + µα − ǫ‖(k‖)
2± 1
2
√
[ω + µα − ǫ‖(k‖)]2 − 4t2⊥. (4.25)
The sign needs to be hosen in su h a way that the imaginary part of LRis greater than
zero. In a similar way, we �nd the solution for R∞ and we observe that, L−∞ = R∞.
These results for the left and right fun tions are the same as for equilibrium. As we
expe t, in the absen e of an ele tri �eld, even in the urrent arrying state, the states
are un hanged and only the momentum hanges whi h leads to hanges in o upan y
of states. Within the formalism, we �x a number of planes at both side of the barrier.
And here we onsider 30 planes. As we expe t, the planes far from the barrier will
not be a�e ted and they equal the bulk values, so we onsider L1 = L−∞ on the
left side and R61 = R∞. In the se ond step, we an use equations 4.20 and 4.22
to evaluate the left and right fun tion at all planes, then we an al ulate the lo al
Greens fun tion from equation 4.12 whi h we need to integrate over two-dimensional
density of states,
gR,Aαα (ω) =
∫
dǫ‖αρ2Dα (ǫ‖α)g
R,Aαα (ǫ‖α, ω), (4.26)
where
81
gRαα(k‖, ω) =
1
−[ω + µα − ǫα(k‖)− ΣRα (ω)] + LR
α (k‖;ω) +RR
α (k‖;ω)
. (4.27)
The advan ed fun tion an be found by taking the omplex onjugate of the retarded,
as we derived in equation 4.17. The retarded Greens fun tion provides information
about the density of states. To obtain information about the �lling and ultimately to
al ulate the urrent, we need to evaluate the Keldysh Green's fun tion. But we an't
al ulate the Keldysh part in the same way that we did for the retarded. The reason
being the di�eren e between the re ursive equations got the retarded and Keldysh
Green's fun tions. As we an see, for zero self-energy, the Keldysh re ursive equation
4.21 be omes homogeneous and it an not be de�ned uniquely, as we an multiply
both sides by any value, and we rea h the trivial solution of 0 = 0. However, we an
take a di�erent path. Starting from the de�nition of the left and right fun tions 4.7
- 4.10 , we an derive the following identities for retarded
LRα−1(k
‖;ω) = −t⊥α−1αeiAα−1(t)
gRαα(k‖;ω)
gRα−1α(k‖;ω)
, (4.28)
RRα+1(k
‖;ω) = −t⊥αα+1e−iAα(t)
gRαα(k‖;ω)
gRα+1α(k‖;ω)
, (4.29)
and Keldysh fun tions,
LKα−1(k
‖;ω) = −LRα−1(k
‖;ω)gKα−1α(k
‖;ω)
gAα−1α(k‖;ω)
− t⊥α−1αeieAα−1(t)
gKαα(k‖;ω)
gAα−1α(k‖;ω)
, (4.30)
RKα+1(k
‖;ω) = −RRα+1(k
‖;ω)gKα+1α(k
‖;ω)
gAα+1α(k‖;ω)
− t⊥α+1αe−ieAα+1(t)
gKαα(k‖;ω)
gAα+1α(k‖;ω)
. (4.31)
This means that to evaluate the left and right Keldysh fun tion we need to al ulate
the lo al and nearest neighbour Green's fun tion in bulk. In the following, we will
82
derive the retarded and Keldysh Green's fun tion in the presen e of a onstant ve tor
potential Aα(t) = −A0 + Eα(t) . This has been al ulated in [11℄ and here we only
write down the results,
gR(k; t, t′) = −iθ(t− t′)e−i[ǫ‖(k‖)−2t cos(kz+A0)−µbulk ](t−t′), (4.32)
where kz is the momentum in the z dire tion and both t and t′ are mu h larger
ompare to −∞. By performing the Fourier transform we get,
gR(k;ω) =1
ω + µbulk − ǫ‖(k‖) + 2t cos(kz + A0) + i0+. (4.33)
To make sure this gives the same results as we found for the self-energy of the leads,
we need to transform into the mixed basis by performing just a Fourier transform on
kz into real spa e, where we get
gRαβ(k‖;ω) =
1
2π
∫ π
−π
dkze−i(zα−zβ)kzgR(k;ω). (4.34)
By hanging the variables ω+µbulk− ǫ‖(k‖)− i0+ = 2tγ and k′z = k+A0, the integral
an be al ulated by the residue theorem. As the details have been shown in [12℄, we
only report the results here for the lo al Green's fun tion,
gRαβ(k‖;ω) =
1
±√
[ω + µbulk − ǫ‖(k‖)− i0+]2 − 4t2⊥, (4.35)
the sign of the retarded Green's fun tion an be hosen from the ausality posterior,
whi h implies that the Green's fun tion has negative imaginary part when it is om-
plex. This result is onsistent with what we have obtained in equation 4.27. Similarly,
we an al ulate the nearest neighbor Green's fun tion,
gRαα+1(k‖;ω) = e−iA0
−ω+µbulk−ǫ‖(k‖)2t
±√
[ω+µbulk−ǫ‖(k‖)]2
4t2− 1
±√
[ω + µbulk − ǫ‖(k‖)− i0+]2 − 4t2, (4.36)
83
where the sign needs to be hosen a ording to the same argument as above. By
substituting k′z → −k′z we an also evaluate the other integral whi h be omes,
gRαα−1(k‖;ω) = eiA0
−ω+µbulk−ǫ‖(k‖)2t
±√
[ω+µbulk−ǫ‖(k‖)]2
4t2− 1
±√
[ω + µbulk − ǫ‖(k‖)− i0+]2 − 4t2. (4.37)
We an he k that by substituting equation 4.35 into 4.33, we re over equation 4.25.
Next we need to al ulate the Keldysh Green's fun tion. In the bulk, it be omes [12℄,
gK(k; t, t′) = ifK [ǫ‖(k‖)− 2t cos(kz)− µbulk]e−i[ǫ‖(k‖)−2t cos(kz+A0)−µbulk ](t−t′). (4.38)
In the steady state, we an perform the Fourier transform into frequen y spa e,
gK(k;ω) = 2πifK [ǫ‖(k‖)− 2t cos(kz)− µbulk]
× δ[ω + µbulk + ǫ‖(k‖) + 2t cos(kz + A0)]. (4.39)
To go ba k into mixed basis, we perform another Fourier transform from kz into the
mixed-basis Green's fun tion,
gKαβ(k‖;ω) =
1
2π
∫ π
−π
dkze−i(zα−zβ)kzgK(k;ω). (4.40)
This integral an be done, as gK(k;ω) is proportional to a delta fun tion, but one needs
to divide the integrand by the absolute value of the derivative of the argument inside
the delta fun tion, as delta fun tion be omes zero for two di�erent roots. Moreover,
the Keldysh fun tion is non-zero only inside the band,
| ω + µbulk − ǫ‖(k‖)
2t|≤ 1. (4.41)
84
Similar to the retarded ase, we an do this integral with th residue theorem and the
�nal result be omes,
gKαβ(k‖;ω) = i
{
fK [ω+(k‖;A0)][
− ω + µbulk − ǫ‖(k‖)
2t− i
√
1−(ω + µbulk − ǫ‖(k‖)
2t
)2]|zα−zβ |
+fK [ω−(k‖;A0)][
− ω + µbulk − ǫ‖(k‖)
2t+ i
√
1−(ω + µbulk − ǫ‖(k‖)
2t
)2]|zα−zβ |}
× 1√
4t2 − (ω + µbulk − ǫ‖(k‖))2ei(zα−zβ)A0 , (4.42)
where ω±is de�ned as follows:
ω+(k‖;A0) = ω cos(A0)− (µbulk − ǫ‖(k‖))(1− cos(A0)
±√
4t2 − [ω + µbulk − ǫ‖(k‖)] sin(A0). (4.43)
Then the lo al Keldysh Green's fun tion be omes
gKαα(k‖;ω) = i
fK [ω+(k‖;A0)] + fK [ω−(k‖;A0)]√
4t2 − (ω + µbulk − ǫ‖(k‖))2. (4.44)
where fK(ω) = 11+eβω is the Fermi-Dira distribution. Similarly, we an �nd the
nearest neighbor Keldysh Green's fun tion,
85
gKαα+1(k‖;ω) = i
{
fK [ω+(k‖;A0)][
− ω + µbulk − ǫ‖(k‖)
2t− i
√
1−(ω + µbulk − ǫ‖(k‖)
2t
)2]
+fK [ω−(k‖;A0)][
− ω + µbulk − ǫ‖(k‖)
2t+ i
√
1−(ω + µbulk − ǫ‖(k‖)
2t
)2]}
× e−iA0
√
4t2 − (ω + µbulk − ǫ‖(k‖))2, (4.45)
and
gKαα+1(k‖;ω) = i
{
fK [ω+(k‖;A0)][
− ω + µbulk − ǫ‖(k‖)
2t+ i
√
1−(ω + µbulk − ǫ‖(k‖)
2t
)2]
+fK [ω−(k‖;A0)][
− ω + µbulk − ǫ‖(k‖)
2t− i
√
1−(ω + µbulk − ǫ‖(k‖)
2t
)2]}
× eiA0
√
4t2 − (ω + µbulk − ǫ‖(k‖))2. (4.46)
Now, we verify that when A0 = 0, we re over the equilibrium results,
gKαα(k‖;ω) =
2ifK(ω)√
4t2 − (ω + µbulk − ǫ‖(k‖))2. (4.47)
Now we an go ba k to equations 4.30 and 4.31 to al ulate the left and right Keldysh
fun tions. After doing some long algebra we obtain,
LK−∞(k‖;ω) = −ifK [ω+(k‖;A0)]
√
4t2 − (ω + µbulk − ǫ‖(k‖))2 (4.48)
86
and
RK∞(k‖;ω) = ifK [ω−(k‖;A0)]
√
4t2 − (ω + µbulk − ǫ‖(k‖))2. (4.49)
After al ulating the left fun tion at the �rst plane and the right fun tion at last
plane, we an use equations 4.21 and 4.23 to al ulate the left and right Keldysh
fun tions at other planes. Then we integrate over the two dimensional density of
states to get the lo al Keldysh fun tion. At the �nal step, we need to al ulate the
impurity Green's fun tion, whi h we will explain in next se tion.
4.4 Impurity solver
One of the most hallenging parts of nonequlibrium dynami al mean-�eld theory is
to �nd the proper impurity solver. There have been few di�erent methods to �nd the
impurity solver for various models. Most of the impurity solvers, su h as the ontin-
uous time quantum Monte Carlo, weak oupling and strong oupling perturbation
theory are based on diagrammati expansions. However, the Fali ov-Kimball model
has spe ial advantages, as the impurity solver is exa tly solvable,
gcimp = (1− w1)gc0 + w1(g
c−1
0 − Uδc)−1. (4.50)
Similar to what we did before, using the Larkin-Ov hinkov transform, we an get the
Keldysh and retarded (or advan ed) parts separately,
gRimp = (1− w1)gR0 + w1(g
R−1
0 − U)−1, (4.51)
and
gKimp = (1− w1)gK0 + w1(g
R−1
0 − U)−1gR−1
0 gK0 gA−1
0 (gA−1
0 − U)−1. (4.52)
87
4.5 Cal ulating the urrent in multilayer devi e
We start with explaining how one an determine the urrent operator for a parti -
ular model. Let's onsider a system with an external ele tri �eld E(r, t). Then, the
equation of ontinuity, whi h onne ts number urrent to harge density is
∂ρ(r, t)
∂t+∇. j(r, t) = 0. (4.53)
Introdu ing the ele tri polarization P(r, t) = rρ(r, t), we take the partial derivative
with respe t to time to get, ∂P(r, t)/∂t = r∂ρ(r, t)/∂t. Substituting ∂ρ(r, t)/∂t with
−∇. j(r, t) from equation 4.53 and integrating over spa e, we get
j(r, t) =∂P(r, t)
∂t. (4.54)
Using the Heisenberg equation of motion that we showed in hapter 2, the time
derivative of operators an be repla ed by ommutators with Hamiltonian, so we an
write
j(r) = i[H,P(r)], (4.55)
where on the latti e, the polarization operator an be written as Pj = Rjc†jcj . Now
we try to evaluate the urrent operator for the multilayers. Sin e the potential term
depends on the number operator, we just need to al ulate the ommutator of the
polarization operator with the hopping terms. Noti e that the polarization operator
in this ase will depend on the site index j and the plane index α. We have listed the
ommutators that need to be al ulated below:
[c†γlcγ+1l,Rαjc†αjcαj] = Rαj(c
†γlcαjδγ+1αlj − c†αjcγlδγαlj)
(4.56)
88
[c†γ+1lcγl,Rαjc†αjcαj ] = Rαj(c
†γ+1lcαjδγαlj − c†αjcγlcγlδγ+1αlj)
(4.57)
[c†γlcγl+δl,Rαjc†αjcαj ] = Rαj(c
†γlcαjδγαjl+δ − c†αjcγlcγlδγαlj)
. (4.58)
So the �nal answer be omes
j = i{−t⊥∑
γl
(eieAγ(t)Rγ+1lc
†γlcγ+1l − eieAγ(t)
Rγlc†γlcγ+1l
+ e−ieAγ(t)Rγlc
†γ+1lcγl − e−ieAγ(t)
Rγ+1lc†γ+1lcγl)
− t‖∑
γlδ
(Rγl+δc†γlcγl+δ −Rγlc
†γlcγl+δ). (4.59)
Using the nearest neighbors ve tors δ = Rγl+δ −Rγl and δ′ = Rγl −Rγ+1l, we get
j = i{−t⊥∑
γl
δ′(eieAγ(t)c†γlcγ+1l + e−ieAγ(t)c†γ+1lcγl)− t‖∑
γlδ
δc†γlcγl+δ} (4.60)
where we will al ulate the perpendi ular part whi h is the harge transport through
the multilayer. Using the de�nition of the lesser Green's fun tion, we get
j = −it⊥∑
γl
δ′[eieA0g<γ+1 γ + e−ieA0g<γ γ+1]. (4.61)
So in order to al ulate the urrent, one has to al ulate the nearest neighbor lesser
Green's fun tion. We start from the formula derived in 4.28- 4.31 for the left and
right fun tions.
LRα−1 = −te−iA0
gRααgRα−1α
, (4.62)
89
RRα+1 = −teiA0
gRαα
gRα+1α
, (4.63)
where we get,
gRαα+1 = −te−iA0gRα+1α+1
LRα
, (4.64)
gRα+1α = −teiA0gR,Aα,α
RRα+1
. (4.65)
Similarly starting from the left and right fun tions for the Keldysh fun tion, we �nd
LKα−1 = −LK
α−1
gKα−1α
gAα−1α
− te−ieA0gKααgAα−1α
, (4.66)
RKα−1 = −RK
α+1
gKα+1α
gAα+1α
− teieA0gKααgAα+1α
. (4.67)
After performing some simple algebra and substituting gAαα+1 and gAα+1α, we get,
gKαα+1 = t[eiA0LK
α gAα+1α+1 − e−iA0LA
αgKα+1α+1
LRαL
Aα
], (4.68)
and
gKα+1α = t[e−iA0RK
α+1gAαα − eiA0RA
α+1gKαα
LRα+1L
Aα+1
]. (4.69)
Then the �nal equation for the urrent be omes,
j = −it2⊥∑
k‖
∫ ∞
−∞dω[ RK
α+1gAαα
LRα+1L
Aα+1
+LAαg
Kα+1α+1
LRαL
Aα
− e−2iA0gKα+1α+1
LRα
− e2iA0gKαα
LRα+1
]
(4.70)
Now that we have derived the formula for the urrent a ross the multilayer devi e,
in the rest of this se tion, we will present the results whi h we have obtained from
the zipper algorithm for the lo al density of states and urrent both in bulk and at
di�erent layers.
90
4.5.1 Results of the lo al density of states
In this se tion, we present some results that have been already reported in Ref. [71℄.
We have onsidered total number of planes Nplane = 61 with a single insulator in the
middle N31 as the barrier plane. First, we start with the lo al density of states, whi h
is − Im[GR]π
. In Fig. 4.2, We present the results for di�erent values of the intera tion
as U = 1 for the dirty metal, U = 4 for the near Mott insulator and, U = 16 deep
in the insulating phase. These results have been already reported in Ref. [17℄ and we
only report it for the pedagogi al purposes.
-15 -10 -5 0 5 10 15ω
0
0.05
0.1
0.15
0.2
-Im
[GR]/
π U=1
U=4
U=16
(a)
-15 -10 -5 0 5 10 15ω
0
0.05
0.1
0.15
0.2
-Im
[GR]/
π
U=1
U=4
U=16
(b)
Figure 4.2: Retarded Green's fun tion for di�erent values of the intera tion in the
(a) bulk, ompared to (b) barrier plane for di�erent intera tions at A0 = π/20 and
T = 0.1. In both �gures, the red, blue, and green urves indi ate U = 1, U = 4 and
U = 16, respe tively.
As we an see, for a small value of intera tion U = 1, the density of the states is
smooth and there is no gap, while with in reasing the intera tion, for example, lose to
the Mott insulator U = 4, we see that the density of states develops a dip and �nally
for a large value of intera tion at U = 16 a wide gap appears. Now, we look at the
density of states of the barrier plane. We observe that the general hara teristi s are
similar to the bulk ase, while the features be ome sharper. Moreover, an interesting
91
phenomenon is happening in the insulating regime, where we an dete t nonzero
density of states entered around zero frequen y. This an be explained by a proximity
e�e ts of metalli planes, where some ele trons an tunnel through the barrier plane
be ause it is so thin. The lo al density of state in the �rst plane is shown in Fig. 4.3
-15 -10 -5 0 5 10 15ω
0
0.05
0.1
0.15
0.2
-Im
[GR]/
π
U=1U=4U=16
(a)
-2 -1 0 1 2ω
0.135
0.14
0.145
0.15
-Im
[GR]/
π
U=1U=4U=16
(b)
Figure 4.3: Retarded Green's fun tion for di�erent values of the intera tion at the
�rst plane for A0 = π/20 and T = 0.1. In (b) we have shown the blow out, in order
to show the Friedel os illations whi h are the e�e t of the barrier plane.
whi h may relax ba k to the bulk values as one in reases the number of planes.
Although, the �rst plane is far from the barrier, however one an see the e�e t of
the barrier by small os illations whi h is known as Friedel os illation. In Fig. 4.4, we
have shown the result of the lo al density of states for planes lose to barrier. The
presen e of an insulating plane auses a broadening of the lo al density of states in
the adja ent plane to it, however, it gets weaker on the se ond adja ent plane and
one would expe t to ompletely disappear in the next planes.
92
-15 -10 -5 0 5 10 15ω
0
0.05
0.1
0.15
0.2-I
m[G
R]π
U=1U=4U=16
(a)
-15 -10 -5 0 5 10 15ω
0
0.05
0.1
0.15
0.2
-Im
[GR]/
π
U=1U=4U=4
(b)
Figure 4.4: Retarded Green's fun tion for di�erent values of the intera tion at one
and two adja ent planes to the barrier for A0 = π/20 and T = 0.1. In (a), one an
learly observe the broadening in the density of states due to the e�e t of barrier,
while the e�e ts gets weaker for the next adja ent planes as it is shown in (b).
4.5.2 Results of the urrent in bulk
Again, we start with reprodu ing the results of urrent reported in Ref. [17, 71℄. After
we al ulate the retarded Green's fun tion, we al ulate the lesser Green's fun tion,
whi h enables us to al ulate the urrent in the multilayer devi e. Here, we show the
results for the urrent in bulk. We an al ulate the urrent in bulk in the presen e
of a ve tor potential whi h shifts the momentum by −A0z,
Jz(t, A0) = −i∑
k
vk+A0g<(k; t, t′) (4.71)
where vk = dǫ(k)/dz = 2t⊥ sin(k) is the band velo ity and using g<(k; t, t′) =
if<(ǫ(k)− µ) leads us,
Jz(t, A0) =∑
k
2t⊥ sin(kz + A0)f<(ǫ(k)− µ). (4.72)
93
-6 -4 -2 0 2 4 6ω/t
0
0.1
0.2
0.3
0.4
0.5Im
glo
c<(ω
)/2π
Bulk DOS
A0=4π/20
A0=π/20
A0=0
(a)
(a)
-6 -4 -2 0 2 4 6ω/t
0
0.1
0.2
0.3
0.4
0.5
Im g
loc<
(ω)/
2π
0 π/2 π0
0.2
0.4
A0=πA
0=17π/20
A0=13π/20
A0=π/2
Current
A0
Bulk DOS
(b)
T=0.01
(b)
Figure 4.5: Lo al lesser Green's fun tion for various urrent arrying states in the
bulk at low temperature T = 0.01. Inside inset at (b), we plotted the urrent as a
fun tion of the ve tor potential, whi h shows sinusoidal behavior. For omparison, we
also plotted the minus of imaginary part of lo al retarded Green's fun tion divided
by π/2.
Jz(t, A0) = −1
3
∑
k
ǫ(k)f<(ǫ(k)− µ) sin(A0). (4.73)
This shows that the urrent has sinusoidal dependen e to the ve tor potential and it
is also proportional to the average of the kineti energy in equilibrium. In Fig. 4.5,
we plotted the urrent for di�erent values of A0 at a temperature T = 0.01. As we
an see for A = 0, the lesser Green's fun tion simpli�es into the Fermi distribution,
as we expe ted. We also plotted the retarded Green's fun tion divided by π/2 for
omparing to the data. Then, as we in rease the ve tor potential, the higher states
start to get o upied so that to keep the �lling onserved, the �lling of lower energy
state should hange as well.
4.5.3 Results of the urrent a ross the multilayer devi e
As our goal is to evaluate the urrent-voltage pro�le of a multilayer devi e, here, we
present the results of urrent a ross the devi e as it is been al ulated at ea h layer.
We have used Eq. 4.70, whi h relates the urrent to nearest neighbor lesser Green's
94
fun tion. In Fig. 4.6, we present the results for dirty metal regime U = 1, for di�erent
initial ve tor potential. As it is shown in Fig. 4.6, the urrent drops as it passes
5 10 15 20 25 30 35 40 45 50 55 60Plane
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Cur
rent
A=0, π/2
A=π/20
A=3π/20
A=2π/20
A=4π/20
A=5π/20
A=6π/20
A=7π/20A=8π/20A=9π/20
(a)
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60Plane
0.6
0.615
0.63
2.24
2.32
2.4
Cur
rent
3.6
3.8
4
4.2
A=π/20
A=4π/20
A=9π/20
(b)
Figure 4.6: Current through the multilayer devi e for di�erent values of A0 for U = 1and T = 0.1. The urrent drops as it passes through the barrier region, and the loss
of urrent be omes bigger as one in reases the initial ve tor potential. (b) To larify
the amount of urrent drop a ross the barrier, we have shown the urrents for three
di�erent values of the ve tor potentials A = π/20,A = 4π/20, and A = 9π/20 from
bottom to top, respe tively. One an learly see that, the amount of drop in urrent
for the di�erent ve tor potentials are approximately 0.02,0.12 and, 0.35 respe tively
from the bottom plot to top plot.
through the barrier. This is the result of s attering in the insulator plane. For the
bigger values of ve tor potential, whi h drives a bigger amount of urrent through the
devi e, the loss of urrents be omes bigger. We noti e that for A = 0 and, A = π/2
the urrent a ross the devi e is zero whi h is onsistent with our formula. To keep
the onservation of urrent and �lling, we will apply the lo al ele tri �eld a ross the
barrier.
The most simple pro edure is applying the ele tri �eld only on the barrier region.
We have examined this idea for a few di�erent values of ve tor potential and the
results are shown below. As it is shown Fig. 4.7, the urrent starts to re over as one
95
applies the ele tri �eld. However, the e�e t of the ele tri bias on di�erent planes are
di�erent and they ompete with ea h other. For example, for a single ele tri �eld,
although the urrent at the barrier improves, however, the plane before the barrier
shows the opposite e�e t and the urrent starts to drop dramati ally as one in reases
the ele tri �eld. Similar behavior also happens for some other ases, where one applies
ele tri �eld on the other planes before and after barrier. Of ourse, sin e the other
two layers or not insulating, we need a mu h smaller ele tri �eld ompared to the
barrier region. Furthermore, one noti es that there is a threshold in whi h in reasing
the ele tri �eld will make the results worse. In Fig. 4.11 and 4.9, we present the
results for both urrent and �lling for (U = 1, A = 6π/20) and U = 4, A = 2π/20. As
one an see, for the larger intera tion value, the loss of urrent be omes magni� ently
bigger, whi h indi ate higher s attering probability in the barrier plane.
Some of other approa hes that one may try is to hange the right ve tor potential
or in rease the numbers of planes with nonzero ele tri �eld. In fa t, we have examined
all of this pro edures and, as one might have guess from the above results, this
pro edure be omes a daunting task and even after performing al ulation, there is
no guarantee that one might have found the best value of parameters for the urrent
and �lling onservation. For this reason, we have taken di�erent approa h by using
the Newton method as a optimization pro edure whi h provides a systemati way of
obtaining the best values for the ele tri �eld and other possible parameters. This
will be the subje t of the next se tion.
96
0 5 10 15 20 25 30 35 40 45 50 55 60Plane
1.17
1.18
1.19
1.2
1.21
1.22
1.23
1.24
1.25
1.26
Cur
rent
E=0E
31=-0.04
E31
=-0.05
E31
=-0.06
E31
=-0.07
E31
=-0.08
E31
=-0.09
(a)
0 5 10 15 20 25 30 35 40 45 50 55 60Plane
1.18
1.2
1.22
1.24
1.26
Cur
rent
E30
=E31
=0.0
E30
=-0.02, E31
=-0.04
E30
=-0.02, E31
=-0.05
E30
=-0.02, E31
=-0.06
E30
=-0.02,E31
=-0.07
E30
=-0.2,E31
=-0.08
E30
=-0.02,E31
=-0.09
(b)
0 10 20 30 40 50 60Plane
1.18
1.2
1.22
1.24
1.26
Cur
rent
E31
=E32
=0.0
E31
=-0.04,E3=-0.02
E31
=-0.05,E32
=-0.04
E31
=-0.07,E32
=-0.04
E31
=-0.08,E30
=-0.04
E31
=-0.09, E32
=-0.04
( )
0 10 20 30 40 50 60Plane
1.18
1.19
1.2
1.21
1.22
1.23
1.24
1.25
Cur
rent
E30=E31=E32=0.0
E30=-0.02, E31=E32=0.0
E32=-0.02, E31=E32=0.0
E30=-0.02, E31=-0.04, E32=0.0E31=-0.04, E30=E32=0.0
E30=E32=-0.2, E31=0.0
E30=0.0, E31=-0.04, E32=-0.02
E30=E32=-0.02, E31=-0.04
(d)
Figure 4.7: Current of multilayer devi e in the presen e of ele tri �eld lose to
barrier region for U = 1, A = π/10, and T = 0.1. For all di�erent ases, the urrentstarts to re over as one applies the ele tri �eld. (a) Shows the results of urrent
for di�erent values of ele tri �eld for single plane(E31 6= 0).(b) Shows the results of urrent for di�erent value of ele tri �eld at the barrier plane and one plane before the
barrier (E30 6= 0 and E31 6= 0 ). ( )Shows the results of urrent for di�erent value of
ele tri �eld at the barrier plane and a plane after barrier(E31 6= 0 and E32 6= 0 ). (d)Shows the results of urrent for di�erent values of ele tri �eld for triple ase.(E30 6= 0,E31 6= 0, E32 6= 0).
97
5 10 15 20 25 30 35 40 45 50 55 60Plane
3
3.05
3.1
3.15
3.2
3.25
3.3
3.35
3.4
Cur
rent
E30
=E31
=E32
=0.0
E30
=-0.04,E31
=-0.06,E32
=0.0
E30
=-0.10, E31
=-0.16,E32
=-0.08
E30
=-0.11,E31
=-0.17,E32
=0.0
E30
=-0.11,E31
=-0.18,E32
=0.0
E30
=-0.12,E31
=-0.19,E32
=0.0
(a)
0 10 20 30 40 50 60Plane
0.485
0.49
0.495
0.5
0.505
0.51
0.515
Filli
ng
E30
=E31
=E32
=0.0
E30
=-0.04,E31
=-0.06,E32
=0.0
E30
=-0.10,E31
=-0.16,E32
=-0.08
E30
=-0.11,E31
=-0.17,E32
=0.0
E30
=-0.11,E31
=-0.18,E32
=0.0
E30
=-0.12,E31
=-0.19,E32
=0.0
(b)
Figure 4.8: Current and �lling through the multilayer devi e in the presen e of
ele tri �eld lose to barrier region for U = 1, A = 6π/20, and T = 0.1.
0 5 10 15 20 25 30 35 40 45 50 55 60Plane
0.6
0.7
0.8
0.9
1
1.1
Cur
rent
E30
=E31
=E32
=0.0
E30
=-0.04,E31
=-0.06,E32
=0.0
E30
=-0.04,E31
=-0.10,E32
=0.0
E30
=-0.04,E31
=-1.10,E32
=0.0
E30
=-0.10,E31
=-1.30,E32
=0.0
E30
=-0.10,E31
=-1.50,E32
=0.0
E30
=-0.10,E31
=-1.60,E32
=0.0
E30
=-0.10,E31
=-1.70,E32
=0.0
(a)
0 5 10 15 20 25 30 35 40 45 50 55 60Plane
0.46
0.47
0.48
0.49
0.5
0.51
0.52
0.53
0.54
Filli
ng
E30
=E31
=E32
=0.0
E30
=-0.04,E31
=-0.06,E32
=0.0
E30
=-0.10,E31
=-1.5,E32
=0.0
E30
=-0.10,E31
=-1.6,E32
=0.0
E30
=-0.10,E31
=-1.7,E32
=0.0
(b)
Figure 4.9: Current and �lling through the multilayer devi e in the presen e of
ele tri �eld lose to barrier region for U = 4, A = 2π/20, and T = 0.1. (b) For the�lling, we have only plotted some of the ases as they get very lose to ea h other.
98
4.6 Optimizing of the DMFT-Zip algorithm
As we mentioned, in order to ompensate the urrent and �lling loss a ross the barrier,
one needs to apply an external lo al ele tri �eld lose to barrier region. The idea is
to �nd the best values for an external ele tri �eld on a number of planes lose to the
barrier plane whi h leads to values that improve simultaneously the urrent and �lling
onservations. This method is based on the Newton root �nding method in whi h one
�nds the root for the matrix equation. Below, we explain the optimization pro edure
with arbitrary number of parameters and we explain the details of the al ulation for
the ase with three ele tri parameters.
1. Set the initial value for parameters for ea h parameters.
2. Cal ulate urrent and �lling at ea h plane as Ji, and ρi respe tively.
3. Obtain the di�eren e of the urrent and �lling from the target value at ea h
plane:
∆Ji = Jtarget − Ji
∆ρi = ρtarget − ρi (4.74)
4. Add small shift to ea h values of the parameters and evaluate the hanges in urrent
and �lling at ea h plane,
δJi(δJ)Parameterj
=J
′
i − Ji(δJ)Parameterj
δρi(δρ)Parameterj
=ρ
′
i − ρi(δρ)Parameterj
. (4.75)
5. Solve the following matrix equation,
∑
j
[δJi
(δJParameter)j
,δρi
(δρ)Parameterj
]δEj = [∆Ji,∆ρi], (4.76)
The above matrix equation an be solved by singular value de omposition(SVD)
whi h is required to �nd the pseudo-inverse of a re tangular matrix. Then, one repeats
99
the pro edure to rea h onvergen e. For the ase with three ele tri �elds, the vari-
ational matrix is 122 × 3, as there are 61 values for urrent and 61 values for �lling
and there are 3 olumns, regarding the applied ele tri �eld at 3 di�erent planes. The
target ve tor in right hand side is 122 and onsequently, by using the SVD, we �nd
the best values for the ele tri �eld at ea h plane. Sin e this method, is based on the
Newton root �nding method, we expe t that the iterations to onverge fast. Before
presenting our results for di�erent ases, we omment on adding more parameters.
For example, one an add more ele tri �elds or even varying the ve tor potential
or hemi al potential. In any of these ases, for ea h parameter, one needs to al u-
late both urrent and �lling for ea h parameter whi h adds a olumn to variational
matrix in left hand side, and the solution of the SVD will have orresponding number
of values.
In Fig. 4.10, we present the results of the optimization for di�erent ve tor poten-
tials for single ele tri �eld on the barrier. We have set the total riteria of the
onvergen e by introdu ing a new variable, whi h is the sum over all ∆J for the new
set of variables found by SVD. We have also repeated the same pro edure for double
and triple ele tri �eld, in whi h, we have obtained the optimized value for ea h ase.
Below in Fig. 4.11, we ompare the results with the original results in Fig. 4.6 whi h
makes lear how the loss in the �lling gets ompensated with the external ele tri �eld
a ross barrier. Of ourse, the results are mu h better for a smaller ve tor potential,
as one may guess that one needs to add further parameters in order to fully re over
the onservation. We have also, reported the best values obtained for ele tri �eld on
the barrier plane and the plane before it.
100
10 20 30 40 50 60Plane
0.6
0.605
0.61
0.615
0.62
0.625
0.63
Cur
rent
E30
=0.0,E31
=0.0,E32
=0.0
E30
=0.0,E31
=-0.01,E32
=0.0
E30
=0.0,E31
=-0.02,E32
=0.0
E30
=0.0,E31
=-0.03,E32
=0.0
E30
=0.0,E31
=-0.04,E32
=0.0
E30
=0.0,E31
=-0.05,E32
=0.0
E30
=0.0,E31
=-0.06,E32
=0.0
A=π/20
10 20 30 40 50 60Plane
1.65
1.7
1.75
1.8
1.85
Cur
rent E
30=0.0,E
31=0.0,E
32=0.0
E30
=0.0,E31
=-0.025,E32
=0.0
E30
=0.0,E31
=-0.050,E32
=0.0
E30
=0.0,E31
=-0.075,E32
=0.0
E30
=0.0,E31
=-0.100,E32
=0.0
E30
=0.0,E31
=-0.125,E32
=0.0
E30
=0.0,E31
=-0.150,E32
=0.0
A=3π/20
10 20 30 40 50 60Plane
2.2
2.25
2.3
2.35
Cur
rent
E30
=0.0,E31
=0.0,E32
=0.0
E30
=0.0,E31
=-0.030,E32
=0.0
E30
=0.0,E31
=-0.060,E32
=0.0
E30
=0.0,E31
=-0.090,E32
=0.0
E30
=0.0,E31
=-0.120,E32
=0.0
E30
=0.0,E31
=-0.150,E32
=0.0
E30
=0.0,E31
=-0.180,E32
=0.0
A=4π/20
10 20 30 40 50 60Plane
3
3.05
3.1
3.15
3.2
3.25
3.3
Cur
rent
E30
=0.0,E31
=0.0,E32
=0.0
E30
=0.0,E31
=-0.04,E32
=0.0
E30
=0.0,E31
=-0.08,E32
=0.0
E30
=0.0,E31
=-0.12,E32
=0.0
E30
=0.0,E31
=-0.16,E32
=0.0
E30
=0.0,E31
=-0.20,E32
=0.0
E30
=0.0,E31
=-0.24,E32
=0.0
A=6π/20
10 20 30 40 50 60Plane
3.2
3.3
3.4
3.5
3.6
3.7
Cur
rent
E30
=0.0,E31
=0.0,E32
=0.0
E30
=0.0,E31
=-0.045,E32
=0.0
E30
=0.0,E31
=-0.090,E32
=0.0
E30
=0.0,E31
=-0.135,E32
=0.0
E30
=0.0,E31
=-0.170,E32
=0.0
E30
=0.0,E31
=-0.215,E32
=0.0
E30
=0.0,E31
=-0.260,E32
=0.0
A=7π/20
10 20 30 40 50 60Plane
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
Cur
rent
E30
=0.0,E31
=0.0,E32
=0.0
E30
=0.0,E31
=-0.065,E32
=0.0
E30
=0.0,E31
=-0.130,E32
=0.0
E30
=0.0,E31
=-0.185,E32
=0.0
E30
=0.0,E31
=-0.245,E32
=0.0
E30
=0.0,E31
=-0.305,E32
=0.0
E30
=0.0,E31
=-0.370,E32
=0.0
A=9π/20
Figure 4.10: Optimization of urrent for di�erent values of ve tor potential with
single layer nonzero ele tri �eld for U = 1 and T = 0.1. As the ve tor potential
in reases, the solution of the optimization pro edure, leads to bigger value for ele tri
�eld as one would have expe ted.
101
5 10 15 20 25 30 35 40 45 50 55 60Plane
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Cur
rent
A=0,π/2
A=π/20
A=2π/20
A=3π/20
A=4π/20
A=5π/20
A=6π/20
A=7π/20A=8π/20A=9π/20
[E30=-0.01052 , E31=-0.03419]
[E30=-0.02036 , E31=-0.04348]
[E30=-0.01858 , E31=-0.08108]
[E30=-0.03662 , E31=-0.13436]
[E30=-0.05914 , E31=-0.15200]
[E30=-0.01149 , E31=-0.19980]
[E30=-0.07885 , E31=-0.22037][E30=-0.06787 , E31=-0.20016]
[E30=-0.068571 , E31=-0.2237]
(a)
Figure 4.11: Comparing the urrent obtained from the optimization algorithm with
the one at zero ele tri �eld. The results for ele tri �eld at ea h plane has been been
reported for ea h ve tor potential for U = 1 and T = 0.1.
4.7 Con lusion: Current-Voltage (I-V) profile
As we mentioned earlier, one of the most intriguing problem is to des ribe di�erent
properties of a multilayer devi e atta hed to metalli leads. One of the most popular
methods is to onsider left and right reservoir at both side of the metalli leads whi h
are set a voltage ±V/2, respe tively. Then, the tunneling matrix between the metalli
leads and the reservoir are hanged slowly in whi h the system eventually rea hes the
steady state. Consequently, the urrent through the leads an be measured whi h
allows one to obtain the urrent-voltage pro�le. Instead, in this hapter, we have
presented a di�erent method based on urrent biasing the multilayer devi e, as we
hange the ve tor potential a ross the devi e to drive the urrent in devi e. In the
previous se tion, we have obtained the optimized value for the ele tri �eld a ross the
barrier region. In Fig. 4.12 , we present the IV pro�le for the single and double layers
with the external ele tri �eld for U = 1. As it is shown in above, the I-V pro�le
102
0 0.05 0.1 0.15 0.2 0.25 0.3Voltage
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Cur
rent
Single
(a)
0 0.05 0.1 0.15 0.2 0.25 0.3Voltage
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Cur
rent
Double
(b)
Figure 4.12: Current-Voltage pro�le for a multilayer devi e. (a) it shows the pro�le
whi h is obtained where we have only applied ele tri �eld on the barrier plane while
in (b) we have obtained the pro�le for the double ele tri �eld applied on barrier
and a plane before the barrier. We noti e that for larger values of ve tor potential
the result starts deviating from the linear behavior with slope passing the zero as is
expe ted for the dirty metal ase with U = 1 and at T = 0.1.
starts deviating from the linear behavior for the larger values of ve tor potential, this
is expe ted for the dirty metal regime. The next step would be al ulating the I-V
pro�le for insulating regime. This step is in progress and we hoping to get results
soon.
103
5
Designing mixtures of ultra old atoms to boost Tc by using
dynami al mean field theory solution
�One of the joys of physi s is the startlingly wide lasses of situations we
an treat with essentially identi al analyses.�
� Ri hard Feynman
Re ent development in ultra old atoms in opti al latti es has reated a great
potential to probe di�erent exoti phenomenas with various degrees of ontrol whi h
are not dire tly a essible in onventional strongly orrelated systems. As we men-
tioned in hapter 1, one of the simplest models to study strongly orrelated systems
is the Hubbard model whi h des ribes the hopping of ele trons in a latti e and with
Coulomb repulsion as two parti les sit at the same site. For reminder, we rewrite the
Hubbard Hamiltonian from Eq. whi h is 5.1: is de�ned as follows:
HHub = −∑
ijσ
tij(c†iσcjσ + c†jσciσ) + U
∑
i
ni↑ni↓ (5.1)
Even, this simple model has a ri h phase diagram. Here, we will only fo us on the
Antiferromagneti (AFM) part of the phase diagram, see �gure 5.1. Although, the
ooling and trapping of fermioni gases are more hallenging, as the Pauli ex lusion
prin iple prevents the s-wave s attering of fermioni spe ies, the Mott insulator of
fermioni atoms has been observed in
40K atoms in ultra old opti al latti es [19℄,
see �gure 5.1. However, the temperature required to a ess the long-range AFM is
104
lower than urrent available te hniques. However, the antiferromagneti orrelation
of quantum degenerate of
6Li fermi gas has been observed in 3 dimensional Hubbard
model, were the temperature lowered to 1.4 Tc, where Tc is the riti al temperature of
Neel order [20℄. In addition, re ent development in site-resolved imaging for fermion
quantum gases has provided a dire t tool to dete t the orrelations in fermioni
gasses [96℄. However, the experiment performed on 2 dimensional Hubbard model
of quantum degenerate of
6Li fermioni gases demonstrates an in rease of orrelation
fun tion in lower temperature and the ability to measure AFM orrelation at any
distan e [96℄, while a ording to Mermin-Wagner theorem there is no phase transition
in 2D. In this hapter, we dis uss the possibility of boosting the riti al temperature
Figure 5.1: Enhan ement of quantum ordering riti al temperature for di�erent
values of degenera y of trapped fermioni spe ies.
by in reasing the degenera y of the light spe ies (fermion) in the mixture of heavy
and light mixture of atoms in ultra old opti al latti es. The heavy-light mixture an
be des ribed by the Fali ov-Kimball model where one needs to onsider the heavy
parti le as a lo alized parti le [2℄. As we have shown in hapter 2, the DMFT solution
of the Fali ov-Kimball model is exa t in in�nite dimension. The idea of enhan ement
of the riti al temperature omes from the relation between the riti al temperature
105
and degenera y of spe ies [18℄. The riti al temperature of the phase transition an
be found by al ulating the di�erent sus eptibilities as they diverge at a riti al
temperature. It turns out that the harge and spin sus eptibilities for heavy-light
parti les an be des ribed as [18℄
χcf =1
1− ξ.dT
=1
1− Tc
T
, (5.2)
whi h indi ates that the riti al temperature should be in reasing as the degenera y
of fermioni spe ies is in reases. Consequently, the main riteria to a hieve a higher
phase transition temperature would be �nding light(fermioni ) atoms with higher
degenera y. In �gure 5.1, we have depi ted the enhan ement of the riti al tempera-
ture for N = 3, and N = 4. A ording to our al ulation, for a fermioni spe ies with
N = 4 degenera y, the enhan ement of riti al temperature happens and one an
dete t the AFM in higher temperature, whi h is a essible with urrent ooling te h-
niques. The idea of riti al temperature enhan ement and required riteria for �nding
an appropriate mixture will be the topi of this hapter. The rest of this hapter is
organized as follows. In se tion 5.1, we will review the basi s of the intera tion of
light with atoms, whi h is ru ial to understand the trapping and ooling of atoms
in opti al latti es. In se tion 5.2, we will brie�y explain the DMFT solution for the
mixture of heavy and light parti les in opti al latti e that has been al ulated in
[18℄. Then, we list the riteria for �nding the right mixture of ultra old atoms whi h
present this e�e t and �nally, in the last se tion, we present our published results on
numeri al al ulation and we dis uss the possibility of di�erent mixture whi h satisfy
the riteria required for riti al temperature enhan ement [21℄.
106
5.1 Intera tion of light and matter
Understanding the intera tion between light and atoms is the most important tool for
building the opti al latti es and trapping the atoms. The intera tion between laser
and atom onsists of two parts, the so alled absorptive and dispersive for e. The
absorptive part of intera tion arises from the momentum transfer during a photon
absorption and emission into another mode, whi h is widely used as ooling method if
the frequen y of light is lose to an atomi resonan e. First, we explain the dispersive
part of intera tion. We all the detuning parameter δ = ω − ω0, where ω is the
frequen y of light and ω0 is the transition frequen y of atom. The dispersive part
arises from the dipole for e. When an atom is pla ed into laser light, the ele tri �eld
of light E indu es an atomi dipole moment in the atom, whi h leads to intera tion
of the atom with light whi h is proportional to the intensity of light. By detuning
far away from the transition frequen y we an enter the regime where we an ignore
the s attering and the dipole for e be omes dominant. In this way we an build an
arti� ial latti e where we an ontrol the shape and the depth of the latti es.
First, we explain the me hanism of intera tion of laser light and atom with a
lassi al model. As we know when we pla e a lassi al dipole inside ele tri �eld, it
tends to align with the ele tri �eld, where it stores the potential energy as
Vdip = −p.E, (5.3)
where p is the dipole moment. In the following we use lassi al os illator model to
al ulate the dipole for e. Let's onsider that the atom is onsists of two ele troni
levels with the frequen y transition des ribed by ω0 and the ele tri �eld and the
107
dipole moment an be des ribed as,
E = E e−iωtk + c.c,
p = p e−iωtk+ c.c.
The dipole moment for atom inside ele tri �eld is [97℄
p = α(ω)E, (5.4)
where α(ω) is the wavelength dependent omplex polarizability of the atom. The
e�e tive potential is given by
Vdip = −1
2〈p.E〉. (5.5)
Sin e the light is rapidly os illating we take the average over one os illation period 〈.〉
and the 1/2 omes from the fa t that dipole moment is indu ed and is not permanent.
Vdip = − 1
2ǫ0cRe(α)I(r), (5.6)
where intensity I(r) = 2ǫ0c|E|2 and ǫ0 is the ele tri permeability. We also are inter-
ested to al ulate the s attering rate whi h is proportional to the imaginary part,
ΓSc =1
~ǫ0cIm(α)I(r), (5.7)
using the lassi al os illator model, we an al ulate the polarizability [97℄:
α(ω) = 6πǫ0c3 Γ/ω3
0
ω20 − ω2 − i ω3
ω20Γ
, (5.8)
where Γ denotes the linewidth of the atomi transition. If we assume ∆ = ω−ω0 > Γ,
then we get,
Vdip = −3πc2
2ω30
( Γ
ω0 − ω+
Γ
ω0 + ω
)
I(r), (5.9)
108
Γsc =3πc2
2ω30
( ω
ω0
)3( Γ
ω0 − ω+
Γ
ω0 + ω
)2
I(r). (5.10)
Within the rotating wave approximation ∆ << ω, we an simplify the equations
Vdip(r) = −3πc2
2ω30
Γ
∆I(r), (5.11)
and
Γsc(r) =3πc2
2~2ω30
Γ2
∆2I(r). (5.12)
This is the basi equation whi h des ribes the hara teristi of trapping of atoms
in opti al latti es. If the frequen y of light is smaller than the transition frequen y
∆ < 0, the dipole for e F = −∇Vdip will attra t atoms toward the high intensity, the
so alled the red detuned, while for ∆ > 0 the atoms feel for e toward the intensity
minima. It is easy to generalize the the above equation for the ase of multi-level
atoms. In this ase we need to sum the dipole potential and s attering rate for ea h
individual transitions multiplied by their line strength fa tors f , for example for a
J → J ′�ne stru ture transition, the line width γ be omes [97, 98, 99℄,
Γ =e2ω2
0
2πǫ0mec32J + 1
2J ′ + 1f. (5.13)
Opti al latti es is onsists of laser beams whi h a t like periodi dipole potential
to trap ultra old atoms in a similar way to the Coulomb potential a ting on atoms
in real latti e. A ommon way of reating opti al latti es is using linear polarized
mono hromati laser with wavelength λ beams whi h is retro-re�e ted to reate the
standing waves,
Vlat(x) = −V0sin2(kx), (5.14)
where the waveve tor is k = 2πλand the latti e depth V0 is typi ally given in the units
of re oil energy Er = ~2k2
2m. Moreover in the simple ase of retro-re�e ted, the sites
109
are separated by a latti e spa ing d = λ/2. The above potential resembles the latti e
stru ture in a rystal.
Now we explain the absorptive intera tion that an be used in ooling pro edure.
When an atom absorbs a photon from a single resonant laser, it gains a moment ~k,
where k is the wavelength of laser. Then the atom will will emit a photon so that the
net for e after many absorption-emission y les be omes,
Fsc = σΓ.~k, (5.15)
where Γ is the natural line width and σ(∆, I) is the o upation probability of the
ex ited state. Noti e that the for e depends on detuning ∆ = ω−ω0 and the intensity
of laser light.
5.2 DMFT solution for mixture of heavy and light parti les in
opti al latti e
The se ond order phase transition in physi al systems manifest itself by diver-
gen e of physi al observable su h as spe i� heat and magneti sus eptibility at
riti al temperature Tc. The Fali ov-Kimball model also undergoes di�erent phase
transitions. Here, we will fo us on the so alled he kerbord phase transition. The
he kerboard-like ele troni modulation has been dete ted in di�erent physi al sys-
tems whi h involves the instability of low dimensional harge leading to the harge
density wave(CDW) [100℄. As we mentioned before, the DMFT solution for the
Fali ov-Kimball model predi ted that the riti al temperature is proportional with
the degenera y of spe ies. Di�erent harge and spin sus eptibility have been al-
ulated from the DMFT solution for the Fali ov-Kimball model by Freeri ks and
Zlati¢(1998) [18℄ whi h expli itly shows the degenera y dependen e of the riti al
110
temperature. Here, we brie�y review the derivation of a stati sus eptibility for the
mixture of heavy and light atoms whi h an be des ribed by the Fali ov-Kimball
Hamiltonian. First, we start with the harge sus eptibility for itinerant ele trons.
The CDW phase transition arises from the distortion of harge, so the harge sus-
eptibility an be de�ned as the derivative of the harge density with respe t to h as
one adds a �eld −∑i hini, whi h ause the symmetry breaking,
χccij =
1
2N + 1
∫ β
0
dτ [Tr〈e−βHnc
i(τ)ncj(0)〉
Z − Tr〈e−βHnc
i〉Z Tr
〈e−βHncj〉
Z ], (5.16)
where Z is the partition fun tion and nci =
∑2N+1σ=1 c†iσciσ is the harge density whi h
an be presented in the Heisenberg representation as
ni(τ) = exp[τ(H− µN )]nci exp[−τ(H− µN )]. (5.17)
Sin e nci = T
∑
nGi(iωn), we an rede�ne the sus eptibility as
χccij =
T
2N + 1
∑
nσ
dGjjσ(iωn)
dhi= − T
2N + 1
∑
nσ
∑
lm
Gjlσ(iωn)dG−1
lmσ(iωn)
dhiGmjσ(iωn),
(5.18)
where G−1lmσ(iωn) = [iωn + µ + hl − Σllσ(iωn)]δlm − tlm. Using the hain rule for the
derivative, we get
χccij =
T
2N + 1
∑
nσ
[−Gijσ(iωn)Gjiσ(iωn) +∑
lmσ′
Gjlσ(iωn)GljσdΣllσ(iωn)
dGllσ′
dGllσ′(iωn′ )
dhi].(5.19)
Now, we an perform a Fourier transform into momentum spa e and we de�ne Γcclm
Γcclm =
1
T
1
2N + 1
∑
σσ′
dΣσ(iωl)
dGσ′(iωm), (5.20)
and we have de�ned,
χccl (k) =
1
L
1
2N + 1
∑
j−q,σ
dGjjσ(iωn)
dhqeik.(j−q), (5.21)
111
and
χcc0l = − 1
L
1
2N + 1
∑
j−q,σ
Gjqσ(iωn)Gqjσ(iωn)eik.(j−q), (5.22)
where L is the number of latti e sites. Then, the harge sus eptibility be omes,
χccl (k) = χ
cc0l (k)− T
∑
m
χccl0(q)Γ
cclmχ
ccm(k), (5.23)
whi h is the Dyson equation for sus eptibility and known as the Bethe-Salpeter equa-
tion. Now, we need to ompute the vertex Γccl0 term. For the Fali ov-Kimball model,
we have obtained the self-energy in hapter 2 as
Σ(iωn) = − 1
2G(iωn)+U
2± 1
2G(iωn)
√
1− 2(1− 2w1)UG(iωn) + U2G2(iωn)
(5.24)
whi h expli itly has G(iωn) dependen e and impli it w1 dependen e, so we have
Γccl0 =
1
(2N + 1)T
∑
σσ′
{
(∂Σlσ
∂Glσ
)w1δσσ′δlm + (∂Σlσ
∂w1
)Glσ(∂w1
∂Gmσ′
)}
. (5.25)
With introdu ing new variable γ(k) de�ned as
γ(k) =∑
l
χccl (k)
∑
σ
(∂w1
∂Glσ), (5.26)
we get,
χccl (k) = χ
ccl0(k)
1− (∂Σlσ1/∂w1)Glσ1γ(k)
1 + χccl0(k)(∂Σlσ1/∂Glσ1)w1
(5.27)
where σ1 is a parti ular spin state that derivative has been al ulated. Now, if we mul-
tiply both sides by Σσ(∂w1/∂Glσ) and perform the summation over l, we should get
the equation for γ(k). Using the hain rule ∂w1/∂Glσ =∑
m[∂w1/∂Zm][∂Zm/∂Glσ],
where Zlσ = iωn + µ− λlσ = G−1lσ + Σ, we have
∂w1/∂Glσ = −∂w1
∂Zl[1−G2
l (∂Σl
∂Gl)w1]
G2l [1−
∑
m∂w1
∂Zm)(∂Σm
∂w1)Gl
]. (5.28)
112
Substituting in γ(k) equation, we get
γ(k) =
∑
lσ ∂w1/∂Zlσ[1−G2l (∂Σl/∂Gl)w1]/[1 +Glηl(k)−G2
l (∂Σl/∂Gl)w1]
1−∑lσ ∂w1/∂ZlσGlηl(k)(∂Σl/∂w1)Gl/[1 +Glηl(k−G2
l (∂Σl/∂Gl)w1],(5.29)
where we have introdu ed new de�nition,
ηl(k) = Gl[−1
G2l
− 1
χccl0(k)
]. (5.30)
Now, the harge sus eptibility be omes
χcc(k) = −T∑
l
[1− γ(k)(∂Σl/∂w1)Gl]G2
l
1 +Glηl(k)−G2l (∂Σ1/∂Gl)w1
. (5.31)
Now, using the identity that we have obtained for w1 In the Fali ov-Kimbalmodel, one
an ompute the derivative terms. Similarly, one an al ulate the harge sus eptibility
for lo alized parti les and other orrelations, for more details see Ref. [18℄.
5.3 Enhan ing quantum order with fermions by in reasing spe ies
degenera y
While the o�-diagonal long-range order in old bosoni atomi gases has been
observed many years ago, quantum magnetism in fermioni gases is still a hallenge
for experimentalists. Despite the possibility to ontrol the intera tion between spin
states of atoms in an opti al latti e [101℄, the temperatures required to obtain mag-
neti ordering remain lower than those a hievable with urrent te hniques. Therefore,
it is easier to demonstrate the presen e of magneti orrelations, before the true
long-range magneti order is established. Using the spin-sensitive Bragg s attering
of light, antiferromagneti orrelations in a two-spin- omponent Fermi gas, magneti
This se tion is reprinted from Khadijeh Naja�, M. M. Ma±ka, Kahlil Dixon, P. S. Juli-
enne, J. K. Freeri ks, Copyright Physi al Review A 96 (5), 053621.
113
orrelations have been observed at a temperature 40% higher than the putative tem-
perature for the transition to the antiferromagneti state in three dimensions [20℄. In
this experiment, the two lowest hyper�ne ground states of fermioni
6Li atoms in a
simple ubi opti al latti e were labeled as spin-up and spin-down states. The repul-
sive intera tion between atoms in these states was ontrolled by a magneti Feshba h
resonan e. Sin e the magneti superex hange intera tion is given by J = 4t2/U , the
experiment ontrolled the value of J and in a parti ular regime, it measured anti-
ferromagneti orrelations as extra ted from the spin stru ture fa tor. Very re ently
it was demonstrated that spin (and harge) orrelations an be dete ted also with
the help of site-resolved imaging. In Refs. [96, 102, 103℄ quantum gas mi ros opy
was used to determine spatial orrelations for fermioni atoms in a two-dimensional
opti al latti e. While there is no phase transition in 2D, the measurements have
shown an in rease of the orrelation length as the temperature was lowered. Similar
antiferromagneti orrelations extending up to three latti e sites have also been
observed in a 1D system [104℄.
The simplest many-body model that has a nonzero phase transition in two dimensions
is the Ising model [105℄. Its fermioni analog, the Fali ov-Kimball model [2℄, also
displays a nonzero transition temperature in two dimensions, whi h behaves Ising-like
when the intera tion strength be omes large. This system an be easily simulated
with mixtures of old atoms on opti al latti es, be ause it involves mobile fermions
intera ting with lo alized fermions [106℄. One simply needs to have the hopping of
the two atomi spe ies to be drasti ally di�erent. The simplest ase of one trapped
atomi state for ea h of the fermioni spe ies maps onto the spinless version of the
Fali ov-Kimball model. This model has been solved exa tly in in�nite dimensions via
114
dynami al mean-�eld theory (DMFT) [107, 108℄ and numeri ally in two dimensions
with Monte Carlo (MC) [109℄.
Our motivation for this work stems from the DMFT solution to the problem.
There, one an derive a ondition for the transition to an ordered phase with a
he kerboard pattern [107, 108℄, whi h takes the form 1 =∑
n γ(n), with the sum
running over all integers [whi h label fermioni Matsubara frequen ies iωn = πi(2n+
1)T , with T the temperature℄. The fun tion γ(n) is a ompli ated fun tion that
is onstru ted from the mobile fermion Green's fun tion, its self-energy, the on-site
interation between the lo alized and mobile fermions U and the density of the lo alized
fermions w1. The important point to note, is that if we in rease the degenera y of the
mobile fermions (while enfor ing that they do not intera t with themselves), then the
Tc equation is modi�ed by γ(n) → Nγ(n), where N is the number of degenerate states
for the mobile fermions [108℄. Sin e one an immediately show that
∑
n γ(n) → C/T
for T → 0 and
∑
n γ(n) → C ′/T 4for T → ∞ [107℄, we expe t that the transition
temperature for the degenerate system will initially grow linearly in N and then turn
over to a slower in rease, proportional to N1/4for larger N . It is the rapid growth with
degenera y for small N , whi h makes these e�e ts so spe ta ular. (These ideas are
further supported by the observation that in reasing spe ies degenera y lowers the
�nal temperature after the opti al latti e is ramped up in alkaline-earth systems [110℄)
The argument that Tc grows linearly with the degenera y at low temperature an
be made more general. We start with the Hamiltonian for the Fali ov-Kimball model
on a latti e Λ that has |Λ| latti e sites. The Hamiltonian for a given on�guration of
115
the heavy atoms {w} is
H({w}) = −t∑
〈ij〉
N∑
σ=1
c†iσciσ + U∑
i
N∑
σ=1
niσwi
=
N∑
σ=1
Hσ({w}), (5.32)
where σ denotes the N di�erent ��avors� of the mobile fermions and wi = 1 or 0,
denotes whether site i has a lo alized fermion on it, or not, respe tively (the lo alized
fermions ontinue to be spinless). The hopping matrix is hosen to be nonzero only
for nearest neighbors, and we set t = 1 as our energy unit (we also set kB = 1). We
de�ne Ei ≡ εi − µ, with µ the hemi al potential and {εi} the set of (degenerate)
eigenvalues of Hσ({w}), whi h is independent of the spe i� value of σ be ause the
mobile fermions are nonintera ting amongst themselves, and they share the same
intera tion with the lo alized fermions. Here, the index i runs over i = 1, . . . , |Λ| (we
will be working on a square latti e of edge L whi h then has |Λ| = L× L).
The orresponding grand partition fun tion is given by
Z =∑
{w}
|Λ|∏
i=1
[1 + e−βEi({w})]N , (5.33)
with β = 1/T the inverse temperature. Introdu ing the free energy F , Eq. (5.33) an
be rewritten as
Z =∑
{w}e−βF({w}), (5.34)
where
F({w}) = −Nβ
∑
i
ln[1 + e−βEi({w})]
= N∑
i
Eiθ [−Ei({w})]
− N
β
∑
i
ln[1 + e−β|Ei({w})|] , (5.35)
116
and θ(. . .) is the Heaviside unit step fun tion. In the low-temperature limit the se ond
term on the RHS vanishes. Inserting the limiting form of F into Eq. (5.34) yields
Z =∑
{w}e−βN
∑
i Eiθ[−Ei({w})]. (5.36)
Note, that this result an be re ognized to be the ondition for the �lling of mobile
fermions into the Fermi sea determined by the bandstru ture orresponding to the
parti ular on�guration of the lo alized fermions, as given by the on�guration {w}.
Sin e, in the low-temperature limit F does not depend on temperature, the partition
fun tion depends on temperature only through the term βN . This means that the
thermodynami s of the system depends only on the ratio T/N , with initial orre tions
expe ted to be small as T rises (be ause they will be proportional to T/TF with some
suitably large Fermi temperature TF ). As a result the riti al temperature Tc in the
low-temperature limit will ne essarily in rease linearly with in reasing degenera y N .
This is an exa t result, independent of the details of the latti e or the dimensionality�
it only requires there to be a phase transition.
There are two assumptions that went into this anaylsis, whi h turn out not to
hold when we a tually al ulate the maximal Tc as a fun tion of N . First, the lowest
Tc values are not so low, so the linear regime fairly rapidly rosses over to a slower
in reasing behavior and se ond, the intera tion value Umax(N), where the maximal
Tc,max(N) o urs, a tually hanges with N (see the inset in Fig. 5.3), so the arguments
about the pre ise fun tional dependen e of the Tc,max(N) on N turns out not to hold
in the a tual data; our arguments assumed we ompared systems with the same U .
The �rst e�e t is to redu e how Tc in reases with N , while the se ond enhan es how
Tc in reases with N .
Corre tions to the linear dependen e of Tc on N ome mostly from states lose
to the Fermi level [Ei ≈ 0, see the se ond term in the RHS of Eq. (5.35)℄. Therefore,
117
Figure 5.2: (Color on-line) Comparison of the 2D DMFT (solid red line) and MC
(blue dots, onne ted with a solid line as a guide to the eye) riti al temperatures to
the he kerboard density wave at half �lling for both spe ies on a square latti e with
N = 1. The lines marked as "Ising mean-�eld" (green) and "Ising exa t" (bla k) show
the riti al temperatures for the orresponding Ising model, whi h be ome exa t for
the respe tive theories when U → ∞. Left panel shows the riti al temperature for
N = 1 whi h is the below of the urrent a hievable temperature at experiment(1.4TN).Middle panel shows the enhan ement of the riti al temperature for N = 3 whi h is
lose to urrent a hivable temperature and the right panle for N = 4, manifest the
enhan ement whi h is well above the urrent a hivable temperature
we an expe t that the linear se tion of the Tc(N) urve an be longer for bipartite
latti es for whi h the density of states is redu ed lose to the Fermi energy, e.g., for a
hexagonal latti e. Also in 3D, where there is no Van Hove singularity (the singularity
for the square latti e is redu ed by the intera tion with the heavy atoms) the linear
part an persist to even higher temperatures.
In Fig. 5.2, we plot the transition temperature to the he kerboard density wave
on a square latti e with N = 1. The top urve is for the DMFT approximation, while
the bottom urve is for the exa t MC results. Note that the intera tion strength for
the peak of the urve lies in the range of U ≈ 4−5 with the maximal U value slightly
118
Figure 5.3: (Color on-line.) The maximal riti al temperature Tc plotted as a fun -
tion of mobile fermion degenera y N (as al ulated with MC). The dashed line shows
the orresponding DMFT Tc al ulated at U = Umax(DMFT). The solid lines show
MC Tc's for di�erent values of U . The bla k dotted line shows Tc(DMFT)×0.75, whi hagrees well with nearly all the MC results. In the inset, Umax(DMFT) is plotted as a
fun tion of degenera y N , indi ating it hanges signi� antly with N .
higher for DMFT versus MC. The DMFT results are semiquantitative, and learly
overestimate the Tc, but the overall error is not that large.
As N in reases, we �nd that the maximum Tc in reases as does the value of
the intera tion strength where the Tc(U) urve is maximized. The full urve out to
N = 100 is plotted in Fig. 5.3. The DMFT results are al ulated for ea h N by
�rst �nding the intera tion strength at the maximum of the Tc urve. For the MC
results, we work with �xed U , varying N and then onstru ting the �maximal hull�
of the data. It turns out that these MC results are nearly perfe tly �t to the DMFT
results when the latter are renormalized by a fa tor of 0.75. The DMFT urve initially
grows linearly with N , but then settles into an in rease that grows proportional to
√N − 1.7, whi h is in between our linear and 0.25 power results, as we expe ted, due
to the fa t that Umax in reases with N .
119
We �nd the enhan ement of the maximal Tc for higher N versus N = 1, given
by Tc,max(N)/Tc,max(1) satis�es: 1.98 (MC, N = 2), 1.899 (DMFT, N=2); 2.84 (MC,
N = 3), 2.651 (DMFT, N = 3); and 3.60 (MC, N = 4), 3.287 (DMFT, N = 4).
Sin e the maximal Tc(DMFT) for the Fali ov-Kimball model is about one half the
maximal Tc(DMFT) for the orresponding Hubbard model, we need to be able to have
a degenera y of N ≥ 3 before this e�e t will have a high enough Tc that it an rea h
urrent experimentally a essible values for the 3D ase. We fo us the remainder of
this letter on dis ussing possible experimental realizations for su h higher degenera y
mixtures.
The Fali ov-Kimball model has zero intera tion between the mobile fermions. One
an argue, on rather general grounds, that the modi� ation of Tc due to a nonzero
intraspe ies intera tion u will have orre tions to Tc of order u2. Hen e, if u is small,
the e�e t we dis uss here should ontinue to hold, with only slight redu tions. This
allows us to formulate our sear h riterion for physi al systems that will show this
degenerate spe ies e�e t.
In sear hing for appropriate mixtures, we want to �nd systems that (i) an have
a degenera y of three or more for the light fermioni spe ies, (ii) have a similar inter-
spe ies intera tion U between the mobile and lo alized fermions, whi h will be tuned
either via an interspe ies Feshba h resonan e, or via the depth of the trapping poten-
tial for the light spe ies; and (iii) have a small intraspe ies intera tion u between the
mobile fermions. We also note, that as long as the lo alized parti le is nondegenerate,
then it an a tually be either Bose or Fermi, sin e its statisti s does not enter the
analysis be ause it does not move. (However, if the heavy parti le is a boson, we do
need its intraspe ies intera tion to be large and positive, so it generi ally forms a
Mott insulator with at most one parti le per site and it does not Bose ondense on
the latti e.)
120
We start with examining some prototypi al systems whi h have already been
demonstrated to be trapped on opti al latti es. The �rst hoi e to examine is mixtures
of
40K (mobile fermion) and
87Rb (lo alized boson) [111℄. If we ould trap the mF =
−5/2,−7/2, and −9/2 states of K, we would have an N = 3 mixture. This system
is ni e, in the sense that it has a tunable interspe ies intera tion via a Feshba h
resonan e, and the intraspe ies intera tions for K have a s attering length on the
order of 100 a0 (in some ases one of the pairs an be tuned to zero s attering length).
The hallenge is that the Rb-Rb intera tion is too small (on the order of 100 a0), and
is not tunable, whi h would make it di� ult to satisfy the required onditions for
this e�e t. If we instead try
133Cs (lo alized boson) [112℄, we �nd that the Cs-Cs
intera tion is large, with a s attering length near 2000 a0 at B ≈ 260 G, but the
interspe ies intera tion is small (≈ −40 a0) and not simultaneously tunable for all
three K spe ies.
Moving on to other possibilities, if we use mixtures of
171Yb or
173Yb (mobile
fermion) [113, 114℄ and
133Cs (lo alized boson) [115, 116℄, we only have a degenera y
of N = 2 for 171Yb, even though its intraspe ies s attering is small, while for
173Yb the
intraspe ies s attering length is ≈ 200 a0, whi h is still viable, given the potentially
large Cs-Cs s attering length, but it would require a tunable Cs-Yb s attering length
that is large, and although this has not yet been measured, we do not anti ipate
that there is any reason why it should be parti ularly large. If we tried Rb as the
lo alized boson [117℄, it su�ers from the same issues as with K-Rb�namely, the Rb-
Rb s attering is too small.
Using
6Li as the mobile fermion appears attra tive [118, 119℄. However, the inter-
spe ies s attering length is only small for low �elds, and when a mixture is formed
from the N = 3 trappable state, at least one intraspe ies intera tion will be large
(although the other two an be lose to zero). So, this ase is suboptimal.
121
Next, we onsider mixtures of
87Sr (light fermion) whi h has up to N = 10 and a
Sr-Sr s attering length on the order of 100 a0 [120, 121℄. If we use Cs as the (lo alized
boson), then if the Cs-Cs s attering length an be set to the order of a few 1000 a0,
and the Sr-Cs s attering length is on the order of 500 a0, then this system might work
to illustrate this degenerate spe ies e�e t, and it has the potential to be spe ta ularly
large.
The remaining hoi es that might be workable seem to be longshots, but annot
yet be ruled out be ause we do not have enough information about their interspe ies
intera tions. We dis uss some of these possibilities next.
43Ca is a fermion with a nu lear spin of 7/2 [122, 123℄,
25Mg is a fermion with
a nu lear spin of 5/2 [124℄, Ba has two spin 3/2 fermioni spe ies [125℄, and
201Hg
is also spin 3/2 [126℄. It is unknown what the intraspe ies intera tions are amongst
these di�erent spin states, how many an be trapped, and what their interspe ies
intera tions are with potential heavy parti les. So they all are possible, but at this
stage quite di� ult systems to work with. Finally, there are all of the magneti -dipole
systems, like Er [127, 128℄, Dy [129, 130℄, and Cr [131, 132, 133℄. These systems
often have haoti intraspe ies intera tions due to a huge number of resonan es, but
they might show some small intera tions at low �elds, and hen e may also be viable
andidates for the light fermions.
In summary, we have illustrated the idea that by enhan ing spe ies degenera y,
one an enhan e Tc for fermioni neutral atoms trapped on opti al latti es su h that
their Tc to an ordered state an be raised high enough that they would be a essible
to explore with urrent experimental te hnology in ooling. This idea omes at this
problem from a di�erent angle than the many di�erent ooling strategies that have
been proposed, and ould provide the ability to truly study spatially ordered quantum
phases. The hallenge is to �nd the right mixture of atoms where this e�e t an be fully
122
exploited. We have suggested some possible systems, with Yb-Cs and Sr-Cs mixtures
as the most promising, but it is lear the experiments will be hallenging to arry out.
We want to end by ommenting that similar work has examined SU(N) symmetri
Hubbard models. The repulsive ase a tually sees a de rease in the antiferromagneti
Tc with in reasing N [134℄, while the attra tive ase sees an enhan ement similar to
what we see for the density-wave instability [135℄, we do not know of any large N > 3
systems with attra tive intera tions. Furthermore, there are hallenges with �nding
atomi systems with a small enough U value (for large N), sin e a maximal hopping
is required to have an a urate single-band des ription.
123
6
Nonequilbrium dynami s of XY hain
�A Physi al law must posses mathemati al beauty.�
� Paul Dira
Understanding the nonequilibrium dynami s of a many body quantum systems
has be ome one of the most intriguing and fas inating problems in ondensed matter
physi s and has attra ted the attention of physi ists from di�erent perspe tives . On
the one hand, understanding the time evolution of an intera ting quantum system and
the me hanism in whi h the systems thermalize, is one of the fundamental questions
in quantum me hani s. Sin e 1929, when von Neumann proposed the equilibration
of a quantum system, the problem remained unsolved [136℄ until re ently, where the
advent of ultra old atoms has opened a new horizon on realizing isolated quantum
systems with highly ontrollable parameters and paved the path for investigating the
dynami s of a quantum systems [137℄. On the other hand, �nding a systemati way to
hara terize the nonequilibrium dynami s of quantum system plays a ru ial role in
new te hnologies su h as a quantum simulator whi h is a type of quantum omputer
[26℄. In the following, we will elaborate on the importan e of this problem from both
fundamental point of view and it's appli ation in te hnology.
From the theoreti al side, one of the main goals of quantum me hani s is to
understand the time evolution of a generi quantum system starting from its initial
nonequilibrium state. A simple paradigm of su h problem is known as the quen h
124
problem; where one studies the dynami s of the system often by hanging one of
the system's parameters. In this hapter, we will onsider the XY spin hain in one
dimension whi h an be onsidered as a simple laboratory to study di�erent orre-
lation fun tions after a quantum quen h. The XY spin hain is one of the basi and
most studied systems that an shed light on fundamental on epts su h as dynami s
and propagation of information in spin systems [27, 195℄. The XY spin hain be omes
even more important as the spin hain is one the primary andidates for onstru ting
and building quantum simulators [26, 28, 29℄.
Furthermore, the XY hain manifests a quantum phase transition [22℄. The
quantum phase transition happens at zero temperature where the thermal �u tua-
tions are absent and the ground state energy of a system varies abruptly as one of the
ouplings su h as the external magneti �eld hanges [23℄. Similar to thermal phase
transition, at riti al points, some of the properties hange so drasti ally that they
annot be des ribed analyti ally [138℄. Furthermore, in the vi inity of the riti al
point, the orrelation length diverges and the system be omes s ale invariant in that
it an be des ribed by a simple �eld theory in two-dimensional Eu lidean spa e
(x, τ), where τ = it is the imaginary time. Due to this s ale invarian e, the system
manifests universal behavior and onsequently, various quantities su h as di�erent
sus eptibilities and orrelation fun tions an be des ribed by a power law behavior.
For example, lose to riti al point, the orrelation length diverges as ξ = |λ− λc|−ν
with ν orresponding to a universal riti al exponent. Another important quantity is
known as the hara teristi energy s ale ∆ that vanishes at a riti al point and s ales
as ∆ ≈ |λ − λc|zν, where z is known as the dynami al exponent. Su h a universal
behavior is independent of the mi ros opi properties of the system and only depends
on underlying symmetries, dimensionality and the range of intera tion in systems.
125
In addition, it turns out that systems that are invariant under s ale transforma-
tion, translation, and rotation are also invariant under a onformal transformation.
The onformal invarian e be omes more ru ial in two dimensions as the onformal
�eld theory be omes invariant under an in�nite-dimensional symmetry group whi h
makes the onformal �eld theory (CFT), a powerful tool to des ribe orrelation fun -
tion lose to riti al points. The details of how CFT an be used to des ribe systems
at riti al points is beyond the s ope of this thesis and we refer the interested reader
to Ref. [139℄.
The rest of this hapter is organized as follows. First, in se tion 6.1 , we will
introdu e the XY model as the simplest model to des ribe a spin intera tion whi h
manifests a quantum phase transition. We will further explain the mapping onto
the free fermioni system and des ribe the diagonalization pro edure. In se tion 6.2,
we will explain some of the orrelation fun tions that we are interested to study
and mention some of the relevant experiments. Finally, we will present some of our
published results on the in�uen e of the initial state on the revival probability and
the light- one velo ity, and the formation probabilities and Shannon information in
last three se tions, respe tively [21, 31, 32℄.
6.1 XY spin hain
The problem of an intera ting spin hain is one of the oldest problems in quantum
me hani s whi h has been proposed by Heisenberg and Dira in order to explain the
magneti properties of materials [141, 142℄. The Heisenberg Hamiltonian is de�ned
as
H = J∑
i
Si.Si+1, (6.1)
126
whi h des ribes the e�e tive intera tion between nearest neighbor spins with oupling
J . In 1931, Hans Bethe solved the Heisenberg spin hain by onstru ting its many
body fun tion [143℄. Although his solution opened a new path into a �eld of exa tly
solvable models, the solution is still remains quite involved. Here, instead, we will use
the XY model as a simple paradigm of an exa tly solvable model where it an be
simply realized by turning o� the intera tion in the z dire tion and onsequently the
Hamiltonian be omes
HXY = −L∑
j=1
[
(1 + a
2)σx
j σxj+1 + (
1− a
2)σy
jσyj+1 + hσz
j
]
, (6.2)
where σx,y,zj are the Pauli matri es. We have hosen the oupling onstant equal to
+1 for a ferromagneti intera tion. Moreover, a indi ates the anisotropy intera tion
between spins, and h denotes the transverse magneti �eld. We will use the Jordan-
Wigner transformation and introdu e fermioni operators as cj =∏
m<jσzm
σxj −iσy
j
2and
N =∏L
m=1σzm = ±1 whi h maps the Hilbert spa e of a quantum hain of a spin 1/2
into the Fo k spa e of spinless fermions [22, 144℄. Then, the Hamiltonian be omes
H =
L−1∑
j=1
[
(c†jcj+1 + ac†jc†j+1 + h.c.)− h(2c†jcj − 1)
]
+N (c†Lc1 + ac†Lc†1 + h.c.). (6.3)
where c†L+1 = 0 and c†L+1 = ±c†1 for open and periodi boundary onditions respe -
tively. The phase diagram of the XY- hain is shown in �gure (6.4) where di�erent
phases arise from di�erent values of a and h.
In the following we will show a generi way to diagonalize a free fermion system
whi h, we will losely follow the method in Ref. [22, 144℄. First, we onsider a generi
(real) free fermion Hamiltonian:
H =∑
ij
[c†iAijcj +1
2c†iBijc
†j +
1
2ciBjicj ]−
1
2TrA (6.4)
where c†i and ci are fermioni reation and annihilation operators and in our ase A
and B are real matri es. By hoosing the proper matri es A and B, one an re over
127
a
h
1
−1
1
criticalX
X
critical XY
critical XY
Ising
critical Ising
Figure 6.1: (Color online)Di�erent riti al regions in the quantum XY hain. In the
riti al XX h = 0 while for riti al XY, we have a = ±1.
the XY Hamiltonian from the free fermioni Hamiltonian. We will present the A
and B below. To diagonalize the Hamiltonian, we use the following unitary anoni al
transformation
†
= U
†
η
η†
. (6.5)
with
U =
g h
h g
, (6.6)
Then, we an write the diagonalized from of the Hamiltonian as follows
HD =∑
k
|λk|η†kηk + const, (6.7)
128
This requires that [ηi, H ] = |λi|ηi, or in other words
(
λg λh
)
=(
g h
)
A B
−B −A
, (6.8)
we emphasize that λ is the diagonal matrix made of positive eigenvalues |λk|. Then,
by onsidering the real A and B matri es, we have g and h an be derived from the
following equations:
g =1
2(φ+ψ), (6.9)
h =1
2(φ−ψ). (6.10)
where we have
(A+B)φk = |λk|ψk, (6.11)
(A−B)ψk = |λk|φk. (6.12)
or
(A−B)(A+B)φk = |λk|2φk, (6.13)
(A+B)(A−B)ψk = |λk|2ψk. (6.14)
When λk 6= 0, φk and λk an be al ulated by solving the eigenvalue equation (6.13),
then ψk an be determined using equation (6.11). When λk = 0, φk and ψk an be
dedu ed dire tly from equations (6.11) and (6.12). To evaluate the onstant term, we
need to take the tra e of the Hamiltonian in the c and η representations
H =∑
k
|λk|η†kηk −1
2Trλ, (6.15)
It's worth to omment about a few important issues whi h one needs to take into
a ount properly. In this method, all λk's are onsidered positive. This is due to the
fa t that in this way one an �nd the ground state |G〉 by a ting with an η ve tor,
ηk|G〉 = 0 (6.16)
129
Subsequently, one an build the higher ex ited states by a ting with η†k.
6.2 Dynami s of observables in XY spin hain
Most of interesting quantities in physi s an be expressed in the form of orrelation
fun tions. While a one point orrelation fun tion su h as magnetization, provides
information about the lo al properties of system, higher orrelation fun tions an
provide information about the non-lo al properties of many body systems. In this
hapter, we will study the dynami s of di�erent quantities and orrelators. We start
with two point orrelation fun tion whi h is de�ned as
Cln ≡ 〈c†l cn〉,
C†ln ≡ 〈c†ncl〉,
Fln ≡ 〈clcn〉,
F †ln ≡ 〈c†l c†n〉, (6.17)
Noti e that for onserved number of parti les, we have:Fln = F †ln = 0. By introdu ing
new operators ai = c†i + ci and bi = c†i − ci, one may re�ne the following fun tions,
Gbaij ≡ 〈biaj〉,
Gabij ≡ 〈aibj〉,
Gaaij ≡ 〈aiaj〉,
Gbbij ≡ 〈bibj〉, (6.18)
whi h are known as G matri es. We will introdu e the G matrix orresponding dif-
ferent boundary onditions and geometry in the next se tion. In addition, we will
show how one an use the G matrix to obtain higher orrelation fun tions and other
interesting quantities su h as formation probabilities, Shannon information, omplete
130
revival probabilities, and the light one velo ity. We have performed all of our al-
ulation dire tly in the on�guration basis, whi h are the eigenstate of σz and it has
a few advantages. First, by dire tly working in the on�guration basis, we are able
to map the omputational basis of a free fermioni system into a fermioni oherent
state whi h provides a powerful tool to arry out the al ulations. Se ond, most of
the experiments are performed in the omputational basis. Consequently, all of the
di�erent quantities that we introdu e in this hapter an be dire tly be ompared
with orresponding experimental results.
One of the earliest non-lo al orrelation fun tions that has been studied vastly is
an emptiness formation probability whi h is the probability of �nding a onse utive
sequen e of up spins with size l in a spin hain. The analyti al result for this quantity
has been derived for up and down spin on�gurations in Refs. [145, 146, 147℄. We have
generalized the study of the formation probability to various number of on�gurations
whi h we all them � rystal on�gurations�. The results of this study will be presented
in the next se tion [21℄. In fa t, our method, allows us to dire tly measure the prob-
ability of all on�gurations and onsequently, we were able to study the Shannon
information and the it's time evolution after a quantum quen h. Furthermore, as we
mentioned above our al ulations an be dire tly ompared with experimental data.
For example, in a re ent study Zhang and his ollaborators in Monroe's group at JQI,
were able to dete t the many body dynami al phase transition (DPT) in a quantum
simulator with 53 qubits [25℄.
In this experiment, they dete ted the DPT after a quantum quen h in the trans-
verse Ising model with long range intera tions whi h is intra table with onventional
omputational methods, see �gure 6.2. To dete t the DPT, they have studied the
average magnetization and espe ially the average two spin orrelators. Furthermore,
they were able to study higher order orrelation fun tions as the probability of a
131
Figure 6.2: S hemati set up of quen h of trapped ion experiment. a) A one dimen-
sional spin hain is prepared in a produ t state, then the quen h pro edure is per-
formed by swit hing the spin-spin intera tion and transverse magneti �eld and �nally,
the magnetization an be measured. Di�erent olors of spin indi ates di�erent spin
states. For example, the blue spins indi ate the initial spin dire tion, and red olor
indi ate the down spin for whi h the proje tive measurement is performed. Other
olors indi ate the superposition of spins. b) The time evolution of the average of
the spin magnetization is shown in the Blo h sphere. The ompetition between Ising
intera tion along the x axis and external magneti �eld along the z axis, ause the
os illation and the relaxation. The average magnetization for small and large trans-
verse �eld is shown in blue and green olor respe tively. Figure taken from Ref. [25℄.
132
domain of up spins whi h we have proposed in our study [21℄, see �gure 6.3. We have
Figure 6.3: Distribution of domain size of a one dimensional spin hain with 53 spins.
The top �gure indi ates the bright state or all up spin state (| ↑〉x). The other threespin on�gurations in the top and bottom, presents di�erent on�gurations omposed
of up and down spins. In the enter, the statisti of domains sizes is shown for di�erent
values of transverse magneti �eld. The boxes in the image indi ate two large domain
sizes orresponding to di�erent values of the transverse �eld. Noti e that, how for the
bigger value of transverse �eld, the probability of �nding a large size domain is very
low. The dashed lines indi ates a exponential �t. Figure taken from Ref. [25℄
also studied some other quantities su h as post measurement entanglement entropy
in a quantum spin hain and full ounting statisti s and the distribution of subsystem
energy in a free fermioni system whi h are beyond the s ope of this thesis and the
interested reader is referred to Refs. [33, 34℄.
6.3 On the possibility of omplete revivals after quantum quen hes
to a riti al point
This hapter is reprinted from This hapter is reprinted from K Naja�, MA Rajabpour,
On the possibility of omplete revivals after quantum quen hes to a riti al point, Phys.
Rev. B 96:014305, Copyright(2017)
133
The quantum me hani al version of Poin aré re urren e theorem guarantees that
any system with dis rete energy eigenstates, after a su� iently long but �nite time,
will return to a state whi h is very lose to it's initial state. Although this seems a
natural expe tation for many body systems, it usually takes astronomi al times to see
an (almost) omplete revival. However, in some systems, partial revivals are possible
whi h makes the problem a very interesting subje t, see for example[149, 150, 151,
152, 153, 154, 155, 156, 157, 158, 159, 160, 161℄. The problem of revivals or related
phenomena also appears in many other on epts su h as dynami al transition and
quantum speed limit [162, 163, 164, 165℄. Quite naturally, one usually is interested
in the problem of revivals when the number of parti les is limited or in other words
when there is a �nite size e�e t. The presen e of the �nite size e�e t usually makes
the exa t al ulations di� ult, however, it has the bene�t of being a essible by the
numeri al means.
In a re ent letter [148℄, the author has made an interesting analyti al al ulation
by using onformal �eld theory for a �nite system size and observed that the omplete
revivals are possible for Los hmidt amplitude in riti al systems in a period whi h
is a fra tion of the system size. This is quite unexpe ted be ause any full revival
requires a nearly perfe t �ne-tuning of phase onditions. To the best of our knowledge,
this is the �rst example of the predi tion of full revivals in many-body quantum
systems in an a essible time. In this brief arti le, we make a loser look to this
phenomena in mi ros opi systems. In parti ular, we show that the full revivals of
[148℄ are impossible in mi ros opi realizations of those onformal �eld theories that
were studied in [148℄. In the next se tion, �rst, we brie�y review the arguments in favor
of and against the presen e of omplete revivals in riti al systems. Then in se tion
three, we study the Los hmidt e ho (�delity) in a quantum XY hain and �nd an
exa t determinant formula for a parti ular initial state. In se tion four, we al ulate
134
the �delity at the riti al point of the periodi (open) transverse �eld Ising hain
analyti ally (numeri ally), and show that the omplete revivals dis ussed in [148℄
are absent. In se tion �ve, we explore the other parts of the phase diagram of the
XY hain. In parti ular, we study the quasi-parti le pi ture for di�erent post-quen h
Hamiltonians and show that the pi ture an determine the periods of the revivals
just in some part of the phase diagram. Finally, in the last se tion, we on lude our
paper.
6.3.1 Revivals in onformal field theories
Consider a one-dimensional periodi quantum hain of length L and the Hamiltonian
H . Assume an initial state whi h is very lose to a onformally invariant state |B〉.
Sin e the onformal states are non-normalizable, one needs to introdu e a parameter
β whi h is alled extrapolation length and then one an write the initial state as
|ψ0〉 ≈ e−β4H |B〉. The extrapolation length is usually of the order of a few latti e sites,
in other words, we have L ≫ β. The parameter β an be estimated by al ulating
the expe tation value of the Hamiltonian as 〈ψ0|H|ψ0〉 = πcL6β2 , see [148℄. To show the
omplete revival, the referen e [148℄ al ulates the �delity de�ned as
F (t) = |〈ψ0|e−iHt|ψ0〉| (6.19)
using the well-established re ipe, see for example [161℄. Based on his argument, for
minimal models at large
Lβ, there must always be omplete revivals at multiples of
t = M L2, where M ∼ 24
1−c. Not surprisingly in the regime L ≫ β, there is no e�e t
of the extrapolation length in the revival times. Although there is nothing wrong in
the CFT al ulations in [148℄, this e�e t an not be seen in a mi ros opi quantum
hain. There are two good reasons: The �rst reason, whi h is already noti ed in [161℄,
is related to the presen e of the ex ited states in any global quantum quen h whi h in
135
prin iple an not be des ribed by CFT. The se ond reason is that [148℄ assumes that
there is a one to one orresponden e between an initial state in a mi ros opi system
and onformal boundary states whi h in general is not true. There are many dis rete
initial states, probably exponentially growing with the system size, that �ow to the
same onformal boundary states either with the same β or di�erent extrapolation
lengths. This means that although in a CFT setup the system omes ba k to itself
with probability one, in the dis rete model it an be in a state whi h is ompletely
di�erent but still with the same ontinuum des ription. Although in prin iple, this
problem an be resolved by onsidering all the possible irrelevant perturbations of the
CFT, see for example [148, 166℄ it will eventually a�e t the omplete revivals anyway.
The referen e [148℄ dis usses, in parti ular, a quen h from the disordered phase in the
transverse �eld Ising hain and shows that there should be omplete revivals at times
t = nL2for even n, while for odd n the omplete revivals are suppressed. In the next
se tion, we will show that the omplete revivals are absent in the riti al transverse
�eld Ising hain.
6.3.2 Los hmidt e ho in quantum XY hain
In this se tion, we study the revivals in the quantum spin hain when the initial state
is the ase with all spins σzare up or down. The Hamiltonian of XY- hain is as
follows:
HXY = (6.20)
−JL∑
j=1
[
(1 + a
2)σx
j σxj+1 + (
1− a
2)σy
j σyj+1
]
− h
L∑
j=1
σzj .
Di�erent phases of the model for J = 1 are shown in the Figure 6.4. The line
a = 1 is the transverse �eld Ising hain. The h = 1 line is riti al for all the values of
136
a and we all it XY riti al line. On the ir le a2 + h2 = 1, the wave fun tion of the
ground state is fa torized into a produ t of single spin states [167℄.
a
h
1
1
CriticalX
XCritical XY
Ising
Critical Ising
Figure 6.4: Di�erent regions in the phase diagram of the quantum XY hain. The
riti al XX hain has entral harge c = 1 and riti al XY line has c = 12. The region
a2 + h2 < 1 is depi ted with the yellow olor.
After using the Jordan-Wigner transformation c†j =∏
l<j σzl σ
+j , whi h maps the
Hilbert spa e of a quantum hain of a spin 1/2 into the Fo k spa e of spinless fermions,
the new Hamiltonian be omes
H = J
L−1∑
j=1
(c†jcj+1 + ac†jc†j+1 + h.c.)−
L∑
j=1
h(2c†jcj − 1) (6.21)
+NJ(c†Lc1 + ac†Lc†1 + h.c.),
where c†L+1 = 0 and c†L+1 = N c†1 for open and periodi boundary onditions respe -
tively with N =∏L
j=1 σzj = ±1. The above Hamiltonian an be written as:
H =
†.A. +1
2
†.B. † +1
2 .BT . − 1
2TrA, (6.22)
137
with appropriate A and B matri es as:
A =
−2h J 0 . . . NJ
J −2h J 0 0
0 J −2h J 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
NJ 0 . . . J −2h
,
B =
0 aJ 0 . . . −aJN
−aJ 0 aJ 0
0 −aJ 0 aJ 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
aJN 0 . . . −aJ 0
. (6.23)
To al ulate the Los hmidt e ho, �rst we de ompose e−iHtusing the Balian-Brezin
formula [168℄ as:
e−iHt = e12
†X
†
e †Y e−
12TrYe
12 Z , (6.24)
where X, Y, Z an be al ulated from the blo ks of matrix T de�ned as
T = e
−it
A B
-B -A
=
T11 T12
T21 T22
, (6.25)
Then we have
X = T12(T−122 ), Z = (T−1
22 )T21, e-Y = T
T22. (6.26)
Note that we always have Z = −X. Finally, the �delity for the desired initial state
(all the spins up) will be:
F (t) = |〈0|e−iHt|0〉| = | det(T22)|12 . (6.27)
138
The same formula is also valid for the ase when the initial state is all the spins down,
in other words when all the sites are �lled with fermions |1〉. In the next subse tions,
we will use the above formula to study the revivals in di�erent phases of the quantum
spin hain. Note that although we will keep the J oupling expli itly in some of the
formulas for numeri al al ulations we always take J = 1.
6.3.3 Revivals in riti al transverse field Ising hain
In this se tion, �rst we al ulate the �delity for the periodi riti al Ising point exa tly,
then we will study the open hain numeri ally.
Periodi riti al transverse field Ising hain
Consider a periodi riti al Ising model Hamiltonian in the Ramond se tor after
Jordan-Wigner transformation, equation (6.36) with N = J = a = h = 1. Sin e in
this ase the matri es A and B ommute, one an al ulate the �delity exa tly. Note
that in this ase the matrix
A B
-B -A
is a ir ulant matrix whi h guarantees the
exa t al ulation of the eigenvalues of the matrix T22 with lassi al methods. After
expanding (6.25) we have
T22 = T
∗11 = cosh(2t
√A) +
i√A
2sinh(2t
√A), (6.28)
T12 = −T21 = −itB[sinh(2t
√A)
2t√A
]. (6.29)
Although it is not needed for our future dis ussion, we also report the exa t form of
the matri es X:
X =−iB
2√A coth[2t
√A] + iA
, (6.30)
139
Sin e the eigenvalues of the matrix A are λj = −2 + 2 cos 2πjL, where j = 1, 2, ..., L ;
logarithmi �delity for the riti al periodi Ising hain an be written expli itly as
ln[F (t)] =1
4
L−1∑
j=0
ln[1− cos2[πj
L] sin2[4t sin[
πj
L]])]. (6.31)
In Figure 6.5, one an see that although there are partial revivals at multiples of t = L4,
whi h an be understood with the quasi-parti le pi ture, the omplete revivals do not
happen. Of ourse if one waits enough time, there will be always almost omplete
revivals but they are usually expe ted to happen in mu h larger times that are usually
ina essible. Note that for the onsidered initial state, we have 〈ψ0|H|ψ0〉 = L whi h
means that β =√
π12
or in other words we are in a regime that
Lβis very large whi h
is well inside the regime onsidered in [148℄.
0 25 50 75 100 125 150 175 200 225 250t
0
0.002
0.004
0.006
0.008
0.01
F(t)
L = 100
Figure 6.5: (Logarithmi �delity for the periodi riti al Ising hain starting from
the all aligned spins σzinitial state.
Open riti al transverse field Ising hain
In this subse tion, we repeat the analyses of the previous se tion for the open hains
to see the e�e t of the boundary ondition on the revivals. Unfortunately, we were
not able to provide an exa t result in this ase so the al ulations are based on a
numeri al evaluation of the determinant in the equation (6.27). The main reason for
140
our failure at the riti al Ising point an be tra ed ba k to this fa t that in this ase
the two matri es A and B do not ommute and so the expansion method gets too
ompli ated after few steps. Also note that in this ase the matrix
A B
-B -A
is not a
ir ulant matrix and so the ommon methods of diagonalization an not be applied.
The numeri al results depi ted in Figure 6.6 on�rms the absen e of the omplete
revivals introdu ed in [148℄ and the usefulness of the quasi-parti le pi ture. We will
ome ba k to a more detailed study of the quasi-parti le pi ture in the next se tion.
0 25 50 75 100 125 150 175 200 225 250t
0.0
5.0×10-6
1.0×10-5
1.5×10-5
2.0×10-5
F(t)
L=100
Figure 6.6: Logarithmi �delity for the open riti al Ising hain starting from the
all aligned spins σzinitial state.
6.3.4 Revivals and quasi-parti le pi ture
In this se tion, we extend the analyses of the previous se tion to the other parts of
the phase diagram of the XY hain. In addition, we also examine the appli ability of
the quasi-parti le pi ture in determining the periods of the revivals in the Los hmidt
e ho. First, we make a brief omment on the quasi-parti le pi ture, see [211℄. Based
on this semi- lassi al pi ture the pre-quen h state has more energy than the post-
quen h Hamiltonian ground state and so onsequently, the initial state plays the role
of a sour e of quasi-parti les. The quasi-parti les with the maximum group velo ity
usually are the ones that an be onne ted to the saturation of the entanglement
141
entropy [212℄ or the revivals in the Los hmidt e ho [161℄. The dispersion relation and
the group velo ity of the Hamiltonian (6.36) are
ǫk = J√
(cosφk − h)2 + a2 sin2 φk, (6.32)
vg = J sin φk2a2 cos φk − cosφk + h
√
(cosφk − h)2 + a2 sin2 φk
. (6.33)
where φk = 2πL(k + N−1
4) with k = 0, ..., L − 1. In Figure 6.7, we depi ted vg for
di�erent values of a and h. Note that for su� iently large L, there is no signi� ant
di�eren e between open and periodi ases. Using the above formula one an derive
0 π/2 π 3π/2 2πϕ
-1.5
0
1.5
ν g
a=0.2a=0.4a=0.6a=0.8a=1.0
-1.5
0
1.5
ν g
a=0.2a=0.4a=0.6a=0.8a=1.0
-2
0
2
ν g h=0.2h=0.4h=0.6h=0.8h=1.0
h=1
h=0.8
a=1
Figure 6.7: Group velo ity vφ with respe t to φ for di�erent values of a and h.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1a
1.61.71.81.9
22.1
ν gmax
h=1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1a
0
0.5
1
1.5
2
ν gmax
h=0.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2h
0
0.5
1
1.5
2
ν gmax a=1
Figure 6.8: Maximum group velo ity for di�erent values of a and h.
the maximum group velo ity vmaxg for di�erent values of a and h, see Figure 6.7.
142
Having these group velo ities for quasi-parti les, one an guess the following periods
for the appearan e of revivals in the Periodi and Open quantum hains:
Tp =L
2vg, To =
L
vg, (6.34)
where vg is usually the velo ity of the fastest quasi-parti les vmaxg . However, this is
not a rule and sometimes other quasi-parti les an arry more information than the
fastest quasi-parti les as it was dis ussed in the ontext of entanglement entropy in
[171℄ and in the ontext of Los hmidt e ho in [161℄. In those ases vg in the equation
(6.34) will be di�erent from vmaxg . We are not aware of a riterion whi h one an
use a priori to de ide what is the most important group velo ity. In the next three
subse tions, we study the revivals and the quasi-parti le pi ture in di�erent regimes.
Ising line: a = 1
In Figure 6.9, we depi ted the Los hmidt e ho for di�erent values of h. Two omments
are in order: �rst of all, at non- riti al points similar to the riti al point we have
partial revivals. Apart from the period of the revivals, there is no signi� ant di�eren e
in the form of the Los hmidt e ho at and outside of the riti al point. Se ondly, the
period of the revivals an be understood by taking the maximum group velo ity
vmaxg = 2h as the relevant velo ity.
6.3.5 Criti al XY line: h = 1
This is a riti al line whi h it is in the same universality lass as the riti al Ising
hain. The results for the Los hmidt e ho on di�erent points are shown in Figure 6.10.
The interesting fa t is that in this ase, the relevant group velo ity is learly |vfg | = 2a
whi h is the Fermi velo ity. As it was already dis ussed in the ontext of the Los hmidt
e ho after lo al quen hes in [161℄, it is not the maximum group velo ity for a <√32.
143
0 20 40 60 80 100 120 140 160 180 200t
0
0.0006
0.0012
0.0018h=1
0.0004
0.0008 h=0.75
0.002
0.004h=0.5
0.003
0.006 h=0.25
a=1F(
t)
0 20 40 60 80 100 120 140 160 180 200t
0
0.00012
0.00024h=1
6e-05
0.00012 h=0.75
0.00015
0.0003 h=0.5
0.0002
0.0004 h=0.25
a=1
F(t)
Figure 6.9: Los hmidt e ho on the Ising line for periodi and open hains in left and
right panel, respe tively(L = 80). The revivals are ompatible with quasi-parti le
pi ture with maximum group velo ity vmaxg = 2h.
This is an interesting example of a ase whi h the relevant group velo ity is di�erent
from the maximum group velo ity.
0 20 40 60 80 100 120 140 160 180 200t
0
0.0006
0.0012
0.0018a=1
0.005
0.01 a=0.75
0
0.004
0.008 a=0.5
0.05
0.1a=0.25
h=1
F(t)
0 20 40 60 80 100 120 140 160 180 200t
0
0.0001
0.0002 a=1
8e-06
1.6e-05 a=0.75
0.0016
0.0032 a=0.5
0.03
0.06a=0.25
h=1
F(t)
Figure 6.10: Los hmidt e ho on the XY riti al line for periodi and open hains
in left and right panel, respe tively(L = 80). The revivals are ompatible with quasi-
parti le pi ture with the Fermi group velo ity vfg = 2a.
144
Non- riti al regime: 0 < h, a < 1
Based on the results of the above two subse tions one might be tempted to guess that
sin e this region is non- riti al, the quasi-parti le pi ture with the maximum group
velo ity should be appropriate to guess the form of the partial revivals. However,
strikingly as one an see in Figure 6.11, there are two ompletely di�erent regimes
with very di�erent behaviors. In the region a2 + h2 > 1 the quasi-parti le pi ture
with the maximum group velo ity works perfe tly, however, in the region a2+h2 ≤ 1
there seems to be no lean way to attribute the revivals to quasi-parti les with �xed
velo ities. One might understand it as a regime that there are more than one type of
important quasi-parti les whi h their velo ity di�eren e kill lean periodi revivals. It
is absolutely un lear why the line a2 + h2 = 1 should separate these two regimes. To
keep the �gure simple, we have only depi ted four di�erent points, however we have
he ked many other di�erent points and on�rmed numeri ally that indeed this line
is at the border between the two di�erent regimes.
0 20 40 60 80 100 120 140 160 180 200t
0
0.0012
0.0024 a=1
0
0.0004
0.0008 a=0.8
0.005
0.01a=0.5
0.04
0.08 a=0.25
h=0.8
F(t)
0 20 40 60 80 100 120 140 160 180 200t
0
3e-05
6e-05 a=1
0
4e-05
8e-05 a=0.8
0
0.0003
0.0006a=0.5
0
0.006
0.012a=0.25
h=0.8
F(t)
Figure 6.11: Los hmidt e ho on the line h = 0.8 for periodi and open boundary
ondition. The revivals are ompatible with quasi-parti le pi ture with maximum
group velo ity (see Figure 6.8 far as a2 + h2 > 1. For the region a2 + h2 < 1
145
6.3.6 Con lusions
In this paper, we studied revivals in the XY hain starting from an initial state with all
the spins σzin the dire tion of the transverse �eld. Our on lusions are the following:
�rst of all, we proved that omplete revivals in the times introdu ed by [148℄, annot
happen in the mi ros opi quantum riti al hains. Se ondly, for the onsidered initial
state we showed that there are three interesting regimes. For a2+h2 > 1, one an use
the quasi-parti le pi ture to predi t the period of the partial revivals. On the riti al
XY line h = 1, one must use the Fermi velo ity vfg = 2a to al ulate the revivals.
However, for the other points, the maximum group velo ity vmaxg is the important
group velo ity. For the region a2 + h2 ≤ 1, the revivals do not follow a lean periodi
stru ture whi h indi ates the presen e of more than one type of important quasi-
parti les. It will be interesting to study the stru ture of revivals in also other models,
espe ially intera ting models su h as the XXZ hain. Los hmidt e ho in the non-
riti al phase of the XXZ hain has been already studied in [172, 173℄, however, it
seems that the problem of revivals in the riti al regimes of intera ting models has
not been studied in full detail so far. In parti ular, it is very important to study the
e�e t of the initial state in these models.
6.4 Light- one velo ities after a global quen h in a non-intera ting
model
In a seminal ontribution Lieb and Robinson [174℄ proved that in a non-relativisti
quantum system, the operator norm of the ommutator between two lo al observ-
ables A(x, t) and B(y, 0) is exponentially small as long as |x − y| ≥ vLRt, for
This hapter is reprinted from K.Naja�, M.A. Rajabpor, J.Vitt, Arxiv 1803.03856,Copy-
right(2018)
146
some vLR > 0. We refer to [175, 176, 177℄ for a more pre ise mathemati al state-
ment or re ent viewpoints. The Lieb-Robinson theorem holds in any dimension and
for a translation invariant Hamiltonian with arbitrary but �nite ranged intera -
tions. The parameter vLR is alled Lieb-Robinson velo ity and depends only on the
Hamiltonian [174, 175, 176, 177℄ driving the time evolution; in parti ular it is state-
independent. The existen e of a �nite vLR implies that information annot propagate
arbitrarily fast [174℄.
The Lieb-Robinson result represents nowadays a powerful tool to prove rigorous
bounds for orrelation and entanglement growth [178℄. Moreover, formidable exper-
imental [179, 180, 181, 182, 183℄ and theoreti al (for an overview [184, 185, 186℄)
progress in many-body quantum dynami s, have further underlined its striking phys-
i al impli ations. Among them, the emergen e of light- one e�e ts in orrelation fun -
tions of time-evolved lo al observables for systems that are not manifestly Lorentz
invariant, like quantum hains [187, 188, 189, 190, 191, 192, 193, 194℄. For instan e,
the Lieb-Robinson theorem has been invoked to explain the ballisti spreading of or-
relation fun tions in paradigmati ondensed matter models su h as the XXZ spin
hain [195℄ or the Hubbard model [196℄ after a global quen h; see also [197℄ for a ped-
agogi al survey. Similar onsiderations apply to the linear growth of the entanglement
entropies [198, 199, 200, 201, 202, 203, 204, 205, 206℄. Quantitative predi tions on the
spreading of orrelations and the entanglement entropy are also of lear experimental
interest; see [207, 208, 209, 210℄ for experimental veri� ations of light- one e�e ts
in many-body systems. A omplementary physi al interpretation of these emergent
phenomena is based on the idea [211, 212℄ that in a quen h problem the initial state
a ts as a sour e of pairs of entangled quasi-parti les. The quasi-parti les have oppo-
site momenta and move ballisti ally; see [213, 214, 215℄ for appli ations to integrable
models. Re ently it has also been pointed out that light- one e�e ts an be re ov-
147
ered from �eld theoreti al arguments without relying on parti ular properties of the
post-quen h quasi-parti le dynami s [216℄.
It is important to stress that the Lieb-Robinson theorem proves the existen e
of a maximal velo ity for orrelations to develop. However the observed propagation
velo ity is a tually state-dependent and non-trivially predi table. In other words, it is
not dire tly related to vLR that rather furnishes an upper bound. For one-dimensional
spin hains, a state-dependent light- one velo ity was noti ed �rst in [195℄. In a on-
text of integrability, it was pointed out how the dispersions of the quasi-parti les
in the stationary state [217℄ (the so- alled Generalized Gibbs Ensemble [218, 219℄)
was initial-state dependent. A predi tion for their velo ities was then proposed and
numeri ally tested. This idea has lead to many subsequent ru ial developments in
the �eld [213, 214, 220, 221, 222℄.
In the simpler setting of a non-intera ting model, the dispersion of the quasi-
parti les after the quen h annot be state-dependent, however as we will dis uss the
symmetries of the initial state and in parti ular translation invarian e an introdu e
additional sele tion rules on their momenta. As a matter of fa t, the light- one velo ity
an be state-dependent also in absen e of intera tions. A ir umstan e that, to our
best knowledge, has been not noti ed so-far.
The paper is organized as follows. In Se . 6.4.1 we introdu e the XY spin hain and
dis uss a spe ial lass of initials states for whi h time-evolution of lo al observables
an be al ulated easily. In Se . 6.4.2 and 6.4.4 we analyze the propagation velo ity
both for physi al two-point orrelation fun tions and the entanglement entropy. As
a by-produ t of our analysis we dis uss in Se . 6.4.3 an approximation of fermioni
orrelation fun tions at the edge of the light- one valid for large spa e-time separa-
tion. Su h an approximation shows that typi ally orrelations s ale as integer powers
of t−1/3. A similar statement was made previously [192℄ in the ontext of prethermal-
148
ization and weak breaking of integrability. After the on lusions in Se . 6.4.5, two
appendi es omplete the paper.
6.4.1 XY hain: notations and set up
In this se tion, we brie�y remind the XY hain, its fermioni representation and we
then introdu e the initial states dis ussed in the rest of the paper. The Hamiltonian
of the XY- hain [223℄ is
HXY =− JL∑
j=1
[
(1 + γ
2)σx
j σxj+1 + (
1− γ
2)σy
j σyj+1
]
− JhL∑
j=1
σzj , (6.35)
where the σαj (α = x, y, z) are Pauli matri es and J, γ, h are real parameters; in
parti ular J > 0 and onventionally h is alled magneti �eld. The XY model redu es
to the Ising spin hain for γ = 1 and it is the XX hain when γ = 0. We will also hoose
periodi boundary onditions for the spins, i.e. σαj = σα
j+L. Introdu ing anoni al
spinless fermions through the Jordan-Wigner transformation, c†j =∏
l<j σzl σ
+j , (6.35)
be omes
HXY = JL∑
j=1
(c†jcj+1 + γc†jc†j+1 + h.c.)− Jh
L∑
j=1
(2c†jcj − 1) (6.36)
where c†L+1 = −N c†1. Here N = ±1 is the eigenvalue of operator
∏Lj=1 σ
zj that is
onserved fermion parity. The above Hamiltonian an be written as
HXY =
†A +
1
2
†B
† +1
2 B
T − 1
2TrA, (6.37)
with appropriate real matri es A and B that are symmetri and antisymmetri ,
respe tively. In the se tor with an even number of fermions (N = 1), the so- alled
Neveau-S hwartz se tor, the Hamiltonian an be diagonalized in Fourier spa e by a
unitary Bogoliubov transformation. In parti ular, there are no subtleties related to
149
the appearan e of a zero-mode and one has HXY =∑L
k=1 εk(b†kbk − 1
2). The anoni al
Bogoliubov fermions b's have the following dispersion relation and group velo ities
εk = 2J√
(cos φk − h)2 + γ2 sin2 φk, (6.38)
v(φk) = 2J sinφkγ2 cosφk − cosφk + h
√
(cos φk − h)2 + γ2 sin2 φk
, (6.39)
where φk = 2πL(k − 1
2) and k = 1, ..., L . We will assume L even from now on. The
diagonalization pro edure will be also brie�y revisited in Se . 6.4.2.
In this paper, we are interested to study the time evolution (global quantum
quen h [198℄) of su h a system from initial states that are eigenstates of the lo al
σzj operators. For example, the initial state |ψ0〉 an be a state with all spins pointing
up (no fermions) or down (one fermion per latti e site); other possibilities an be
the Néel state | ↓↑↓↑ ... ↓↑〉 and alike. Moreover, all the states that we study have
a periodi pattern in real spa e with a �xed number of spin up. We label our rys-
talline initial states |ψ0〉 as (r, s), where r is the spin up (fermion) density and s is
the number of spin up in the unit ell of the rystal. For example, the Néel state is
labelled by (12, 1) and the state | ↓↓↑↑↓↓ ... ↓↓↑↑〉 will be (1
2, 2). It is also onvenient to
de�ne p ≡ srthat for simpli ity we restri t to be a positive integer (i.e. s is a multiple
of r). Although the lass of initial state onsidered is not omprehensive, it turns out
to be enough for the up oming dis ussion.
6.4.2 Evolution of the orrelation fun tions
Light- one e�e ts an be studied, monitoring the onne ted orrelation fun tion of
the z- omponent of the spin Sz = σz/2,
∆ln(t) = 〈Szl (t)S
zn(t)〉 − 〈Sz
l (t)〉〈Szn(t)〉. (6.40)
For our initial states, (6.40) is zero at time t = 0; however, a ording to [174℄, after
a ertain time whi h depends on |l − n|, it starts to hange signi� antly.
150
For instan e [195℄, su h a time an be hosen the �rst in�e tion point τ . Varying
|l− n| in Eq. (6.40), one an numeri ally evaluate τ and provide a predi tion for the
speed vmax
at whi h information spreads in the system determining the ratio
|l−n|2τ
. We
will all vmax
the light- one velo ity. A ording to [189, 190, 191, 211, 212℄, after the
quen h and in absen e of intera tions pairs of entangled quasi-parti le with opposite
momenta move ballisti ally with a group velo ity �xed by their dispersion relation.
Within this framework, one should expe t τ to be state-independent and vmax = vg
where vg > 0 is the maximum over the k's of Eq. (6.39). We will a tually show that
vg is rather an upper bound for the observed vmax
whi h an indeed be dependent on
the symmetries of the initial state. Finally observe that we do not expe t, in absen e
of intera tions, the light- one velo ity to be dependent on �nite size e�e ts as long as
L≫ |l − n|.
To study the time-evolution of the orrelation fun tion (6.40), �rst, we need to
analyze the propagators
Fln(t) = 〈ψ0|c†l (t)c†n(t)|ψ0〉, (6.41)
Cln(t) = 〈ψ0|cl(t)c†n(t)|ψ0〉, (6.42)
where |ψ0〉 is a state of the type introdu ed in Se . 6.4.1. If there is no ambiguity,
we will drop it from the expe tation values. From Eqs. (6.41)-(6.42) and the Wi k
theorem, whi h applies to our states [224℄, it follows
∆ln(t) = |Fln(t)|2 − |Cln(t)|2. (6.43)
Note that as L×L matri es whose matrix elements are given in Eqs. (6.41)-(6.42),
F and C satisfy F
T = −F and C
† = C.
151
Generi Quadrati Hamiltonian
T11, T12, T21 and T22 are omplex matri es.
T21 = (T12)∗
T22 = (T11)∗
(T11)T = T11
(T22)T = T22
T22T12 +T21T22 = 0
Table 6.1: Properties of the four L × L blo ks of the matrix T for a quadrati
Hamiltonian (6.37). The notation is obvious and time dependen e is omitted here.
It is straightforward to show that for a fermioni model with a quadrati Hamil-
tonian as in Eq. (6.37) one has,
(t)
†(t)
= e
−it
A B
-B -A
︸ ︷︷ ︸
T (t)
(0)
†(0)
(6.44)
where = {c1, c2, ..., cL} and
† = {c†1, c†2, ..., c†L} are ve tors of length L; i.e. T is a
2L × 2L matrix. Exploiting the properties of the four L × L blo ks of the matrix T
olle ted in Tab. ??, Eq. (6.44) an be written as,
(t)
†(t)
=
T
∗22(t) T12(t)
T
∗12(t) T22(t)
(0)
†(0)
. (6.45)
Finally, after some easy algebra, expli it expression for the time-evolved matri es
F and C an be omputed and read respe tively (time dependen e is dropped from
the T 's)
F(t) =T∗12F
†(0)T†12 +T
∗12C(0)T22 +T22T
†12
−T22CT (0)T†
12 +T22F(0)T22, (6.46)
152
C(t) =T∗22F
†(0)T†12 +T
∗22C(0)T22 +T12T
†12
−T12CT (0)T†
12 +T12F(0)T22. (6.47)
Eqs. (6.46)-(6.47) are valid in prin iple for any free fermioni systems with Hamilto-
nian (6.37); however, they have mu h simpler forms in the XY hain for our parti ular
hoi e of initial states as we now dis uss.
Evolution of the orrelation fun tions in the XY hain
In the periodi XY hain, it turns out [A,B] = 0; these matri es are indeed trivially
diagonalized by the unitary transformation with elements Ulk = 1√Le−ilφk
and φk
given, for N = 1, below Eq. (6.39). Consequently, the four blo ks of T are mutually
ommuting and this leads to further simpli� ations. In parti ular, it is easy to verify
that
T11 = cos[t√
A
2 −B
2]− iA√
A
2 −B
2sin[t
√
A
2 −B
2], (6.48)
T12 =−iB
√
A
2 −B
2sin[t
√
A
2 −B
2]. (6.49)
The other two blo ks an be found observing that T22 = T
∗11 and T21 = −T12. The
eigenvalues of the matri es A and B are
λAk = 2J(−h + cosφk), (6.50)
λBk = −2iJγ sin φk. (6.51)
From Eqs. (6.50)-(6.51) and omparing with Eq. (6.38), it follows εk =√
(λAk )2 − (λBk )
2.
Re alling then Eqs.(6.48)-(6.49) we �nally obtain
λT11
k = [λT22
k ]∗ = cos(tεk)− iλAkǫk
sin(tεk) (6.52)
λT12k = −λT21
k = −iλBk
εksin(tεk). (6.53)
153
The time-evolved matri es F(t) and C(t) an now be al ulated for the lass of repre-
sentative initial states |ψ0〉 introdu ed in Se . 6.4.1. A trivial ase is |ψ0〉 = | ↓↓ . . . ↓↓〉,
i.e. a state without fermions; here F(0) = 0 and C(0) = 1. Then using unitarity of
the matrix T we obtain
[F(t)]ln =1
L
L∑
k=1
λT22k λT12
k e−iφk(l−n)(6.54)
[C(t)]ln = δln −1
L
L∑
k=1
(λT12
k )2e−iφk(l−n). (6.55)
A ording to Eqs. (6.52)-(6.55), the time evolution of ∆ln(t) in Eq. (6.40) is des ribed
by the ballisti spreading of quasi-parti les with dispersion relation εk as in (6.38).
As it an be also easily he ked numeri ally, orrelations spread at the maximum
group velo ity vg obtained from (6.39), in agreement with a standard quasi-parti le
interpretation.
For the initial states labelled by (1/p, 1), the matrix C(0) has elements
[C(0)]ln = δln
[
1− 1
p
p−1∑
j=0
e−2πi ljp
]
, (6.56)
f whereas F(0) = 0. From the expressions in Eqs. (6.46)-(6.47) and inserting the
unitary matrix U that diagonalize simultaneously all four blo ks of T we derive
[F(t)]ln =1
L
L∑
k=1
λT ∗12
k λT22k e−iφk(l−n)
− 1
Lp
p−1∑
j=0
e−2πinj
p
L∑
k=1
λT ∗12
k λT22
−Ljp+ke−iφk(l−n)
+1
Lp
p−1∑
j=0
e−2πinj
p
L∑
k=1
λT22k λT12
−Ljp+ke−iφk(l−n), (6.57)
154
and
[C(t)]ln =1
L
L∑
k=1
|λT22
k |2e−iφk(l−n)
− 1
Lp
p−1∑
j=0
e−2πinj
p
L∑
k=1
λT ∗22
k λT22
−Ljp+ke−iφk(l−n)
+1
Lp
p−1∑
j=0
e−2πinj
p
L∑
k=1
λT12k λT12
−Ljp+ke−iφk(l−n). (6.58)
In the next subse tions, we will pass to study Eqs. (6.57)-(6.58) in details. Similar
al ulations an be also arried out for the initial states labelled by (s/p, s) where
C(0) has matrix elements
[C(0)]ln = δln
[
1− 1
p
p−1∑
j=0
s−1∑
l′=0
e−2πi (l+l′)jp
]
; (6.59)
we will also brie�y investigate su h a possibility.
Evolution of the orrelation fun tions in the XX hain
We onsider preliminary the simple ase of the XX hain (γ = h = 0 and J = −1)
where the fermion number is onserved and B (and F) vanishes. Then the matrix C
for the initial states labelled by (1/p, 1) an be rewritten as
Cln = δln −1
Lp
p−1∑
j=0
e−2πinj
p
L∑
k=1
e−i
[
φk(l−n)+
(
εk−ε−
Ljp +k
)
t
]
, (6.60)
where εk = −2 cosφk and φk = 2πkL
(k = 1, . . . , L). Noti e also that vg = 2. For the
sake of determining the light- one velo ity, the last exponential in Eq. (6.60) implies
that
εe�k,j(p) ≡1
2
(
εk − ε−Ljp+k
)
, (6.61)
an be interpreted as an e�e tive dispersion relation. The e�e tive dispersion origi-
nates from the dis rete translational symmetry of p latti e sites of the initial state
155
that allows quasi-parti les to be produ ed with momenta shifted by multiples of 2π/p.
An e�e tive group velo ity an be de�ned from Eq. (6.61) as
ve�k,j(p) = sin φk − sin
(
φk −2πj
p
)
; (6.62)
whose maximum value (over j and k) is then the light- one velo ity
vmax
= 2 when p is even, (6.63)
2 cos
(π
2p
)
when p is odd. (6.64)
Note that the e�e tive maximum group velo ity o urs when j = p2and j = p−1
2for
even and odd p, respe tively. In parti ular, from Eq. (6.63) follows that the a tual
light- one velo ity is state-dependent and annot be faster than the maximum group
velo ity vg. The predi tion in Eq. (6.63) is of ourse in agreement with a numeri al
estimation of vmax, obtained from the in�e tion points of the orrelations ∆ln; see
Fig. 6.12 for examples with states labelled by (1/2, 1), (1/3, 1) and (1/2, 2). It is
also interesting to observe that the absolute minimum visible in the se ond and third
panel in Fig. 6.12 is a �nite size e�e t, indeed the envelope of |∆ln(t)| is monotoni ally
de reasing in the limit L→ ∞ after rea hing the �rst maximum.
A similar analysis for the initial states (s/p, s), shows that
Cln = δln −1
Lp
p−1∑
j=0
Ajs e− 2πinj
p
L∑
k=1
e−i[φk(l−n)+2tεe�k,j (p)](6.65)
where Ajs =∑s−1
q=0 e− 2πijq
p. It is lear that as long as Ajs 6= 0 for the values of j
orresponding to the e�e tive maximum group velo ity, Eq. (6.63) remains valid.
However, one an verify that Ajs is a tually zero in some ases. For example, for the
state labelled by (1/2, 2), we have A22 = 0 and therefore the light- one velo ity is
obtained from the maximum over of Eq. (6.62) at j = 1 and p = 4; namely vmax =√2.
156
14 16 18 20 22 24 26t
-0.012
0
-0.006
0
∆ ln(t
)
-0.011
0
40 44 48 52 56 60|l-n|
10
12
14
16
18
20
22
τ
(1/2 , 1)
(1/3 , 1)
(1/2 , 2)
Figure 6.12: ∆ln(t) for the XX hain with di�erent initial states. The verti al lines
are indi ating analyti al values of the time T . Here we have L = 144 and |l−n| = 60.
Evolution of the orrelation fun tions for arbitrary values of the
parameters
The results of the previous subse tion an be extended to arbitrary values of the
parameters in the XY hain. Consider the on�gurations (1/p, 1). Ea h term in the
sum over j in Eqs. (6.57)-(6.58) is asso iated to a time propagation with an e�e tive
dispersion
ǫe�k,j,±(p) =1
2
(
εk ± εLjp−k
)
, j = 0, . . . , p− 1, (6.66)
where it is understood that for j = 0 we get ba k Eq. (6.38). Moreover, also the
�rst line in Eqs. (6.57)-(6.58) ontains a state-independent ontribution whose time
evolution expands over the usual dispersion. Therefore as long as B 6= 0, one should
expe t a state-independent light- one velo ity vmax = vg for all the fun tions (6.41)-
(6.42). However, interestingly for p = 2, the state-independent terms are an elled
exa tly by the j = 0 ontributions of the sums in Eqs. (6.57)-(6.58). The latter
157
observation follows from
λT ∗22
k λT22
−k + (λT12
k )2 = 1, (6.67)
that is a onsequen e of the unitarity of the matrix T . For arbitrary values of the
parameters γ and h, the e�e tive maximum group velo ity (i.e. the light- one velo ity)
an be then obtained as
vmax
= maxj 6=0,±,k
dεe�k,j 6=0,±(p)
dk. (6.68)
A tually if p = 2, only j = 1 is allowed in Eq. (6.68), however as we will dis uss,
in some ases, the same result applies to p > 2. In the Ising hain (γ = 1) vmax is
obtained sele ting the negative sign in Eq. (6.66), and φk as lose as possible to
π2.
The expli it value is
vmax =2Jh√1 + h2
. (6.69)
Similarly, in the regions of the parameters h = 1 and γ < γ∗, one �nds
vmax =2J
√
1 + γ2; (6.70)
where γ∗ is the solution of γ∗ = 1√1+(γ∗)2
. However, there is not a losed formula
for vmax; see the �rst and se ond panel in Fig. 6.17 for the numeri al estimations. In
Fig. 6.13, we plot ∆ln(t) for di�erent values of γ and h for quen hes from the initial
on�guration (1/2, 1). The light- one velo ities estimated from the in�e tion points
are in agreement with Eqs. (6.69)-(6.70) and more generally with Eq. (6.68).
For p > 2, the determination of vmax is more involved. Let us fo us on the the
riti al Ising hain. Sin e the �rst terms in Eqs. (6.57)-(6.58) are now not an elled
expli itly we should expe t that the C and F matrix elements will propagate with
velo ity vg. This is indeed orre t, see for instan e the red and blue urves in the two
panels of Fig. 6.14 with initial states p = 3 and p = 4. However unexpe tedly, when
158
0 2 4 6 8 10 12t
-0.007
0
0.007
-0.012
0
0.012
∆ ln(t
)
-0.012
0
0.012
15 20 25 30|l-n|
4
6
8
10
12
τ
( γ = 1 , h = 0.75 )
( γ = 0.5 , h = 1 )
( γ = 0.5 , h = 0.5 )
Figure 6.13: ∆ln(t) for the XY hain with various γ and h parameters with the
initial state (12, 1). The verti al lines are indi ating analyti al values of the time τ .
Here we have L = 96 and |l − n| = 12.
ombined into ∆ln, the two signals almost exa tly an el around the �rst maximum,
leaving a light- one velo ity slower than vg. The latter an be still al ulated as in
ase p = 2 from Eq. (6.68). See the green urve in Fig. 6.14 for an illustration. This
unexpe ted result holds also for any h ≤ 1 (ferromagneti phase) as we support
analyti ally in the Appendix 6.5.7. For h > 1 (paramagneti phase), Appendix 6.5.7
shows that su h a an ellation does not happen, therefore vmax = vg. This asymmetry
in the light- one velo ity between the two phases is hard to spot numeri ally, sin e
the state-independent term is of order 1/h2 and the di�eren e between Eq. (6.68) and
vg drops to zero fast as h in reases.
We studied the orrelation fun tion ∆ln for several di�erent values of the param-
eters γ and h, and for the initial states (1p, 1) with p > 2 we found numeri ally vmax
always to be given by Eq. (6.68) or vg. However a lear pattern does not emerge from
the numeri al analysis. For the on�guration (1/2, 2) with p = 4 again the terms that
159
are independent from the initial states an el out expli itly. For instan e in the Ising
hain the light- one velo ity is
vmax
=
√2Jh
√
1−√2h + h2
; (6.71)
a result that an be veri�ed numeri ally from the in�e tion points. Similar arguments
are also valid for all the on�gurations with (s/p, s).
10 12 14 16 18 20 22 24 26 28 30t
-0.006
-0.003
0
0.003
0.006| F
ln |
2
| Cln
|2
∆ln
( 1/3 , 1)
10 12 14 16 18 20 22 24 26 28 30t
-0.004
0
0.004
0.008| F
ln |
2
| Cln
|2
∆ln
( 1/4 , 1)
Figure 6.14: |Cln|2, |Fln|2, and ∆ln(t) for the XY hain with γ = 1 and h = 1. Theverti al lines in Maroon and Magneta, are indi ating analyti al values of the time τ al ulated from the maximum group velo ity and maximum e�e tive group velo ity,
respe tively. Here we have L = 144 and |l − n| = 60.
6.4.3 Airy s aling of orrelation fun tions at the edges of the light-
one
To understand analyti ally the behaviour of the fermioni propagators is more on-
venient to onsider the in�nite volume limit L→ ∞. We will now re- ast the results
in Se . 6.4.2 dire tly in Fourier spa e and obtain an approximation of orrelation
fun tions near the boundary of the light- one involving an Airy fun tion.
We onsider then fermioni operators ci and c†i with {c†i , cj} = δij , {ci, cj} =
{c†i , c†j} = 0 de�ned for any i, j ∈ Z. The onvention for the Fourier transforms are
cj =
∫ π
−π
dk√2π
eikjc(k), c(k) =1√2π
∑
j∈Ze−ikjcj, (6.72)
160
from whi h follows that {c†(k), c(k′)} = δ(k−k′) and {c(k), c(k′)} = {c†(k), c†(k′)} =
0. To be de�nite we only onsider initial on�gurations labelled by the pair (1/p, 1).
In XY hain, the time evolved operators c(−k, t) and c†(k, t) are linearly related (see
Eqs. (6.90)-(6.92)) to the orrespondent operators at time zero. In parti ular one has
c†(k, t)
c(−k, t)
=
T11(k, t) T12(k, t)
T21(k, t) T22(k, t)
c†(k)
c(−k)
(6.73)
where the matrix elements are
T11(k, t) = cos(ε(k)t) + i cos(θ(k)) sin(ε(k)t) (6.74)
T12(k, t) = sin(ε(k)t) sin(θ(k)). (6.75)
and T11(k, t) = [T ∗22(k, t)], T21(k, t) = −T12(k, t). From unitarity it follows more-
over |T11|2 + T 212 = 1 and T ∗
11T12 + T ∗21T22 = 0. The Bogolubov angle is θ(k)/2, see
Appendix 6.5.7 for expli it expressions. Fermioni orrelation fun tions are double
integrals in Fourier spa e. For instan e let us denote by fαa fermioni operator with
f+ ≡ c and f− ≡ c†, from the de�nition of the Fourier transform (6.72) we have
〈fαl (t)f
βn (t)〉 =
∫ ∫dkdk′
2πeiαkl+iβk′n〈fα(k, t)fβ(k′, t)〉, (6.76)
with integrals in the domain k, k′ ∈ [−π, π]. The time evolution of the matrix ele-
ment in (6.76) is obtained from Eq. (6.73) as a linear ombination of four matrix
elements of the same type at time t = 0. Among those four the only non-trivial on
the lass of initial states we are onsidering is g(k, k′) = 〈c†(k)c(k′)〉. Noti e indeed
that 〈c†(k)c†(k′)〉 = 〈c†(k)c†(k′)〉 = 0 and 〈c(k)c†(k′)〉 an be obtained from the anti-
ommutation relations. For our initial state (1p, 1) the fun tion g is given by
g(k, k′) =1
2π
∑
n∈Zeinp(k−k′) =
1
p
p−1∑
j=0
δ
(
k − k′ − 2πj
p
)
. (6.77)
161
It is then straightforward to derive integral representations for the orrelators Fln(t)
and Cln(t) that we write as
Fln(t) =
p−1∑
j=0
F jln(t) and Cln(t) =
p−1∑
j=0
Cjln(t). (6.78)
Expli it expressions are in Eqs. (6.93)-(6.96). Ea h integral F jln(t) and C
jln(t) des ribes
a time propagation with a velo ity that an be derived from the e�e tive dispersion
relation
εe�j,±(k, p) =1
2
[
ε(k)± ε
(
−k + 2πj
p
)]
, j = 0, . . . , p− 1, (6.79)
that is Eq. (6.66) in the limit L → ∞. As in Se . 6.4.2, the predi ition of the light-
one velo ity follows from al ulating the maximum e�e tive group velo ities obtained
from Eq. (6.79). In parti ular, it an be easily veri�ed (see Eqs. (6.93) and (6.95) in
parti ular) that for p = 2, the maximum e�e tive group velo ity is always smaller
than vg obtained from Eq. (6.39).
As shown in Fig. 6.14, the numeri s indi ates that a an ellation of the fastest
j = 0 ontributions in Eq. (6.79) appears also for p 6= 2 in the riti al Ising hain.
This observation extends to the whole ferromagneti phase h ≤ 1. It an be under-
stood analyti ally omparing the behaviours near their in�e tion points of |F 0ln(t)| and
|C0ln(t)| in (6.78) and showing that they are the same. At the edge of the light- one
those fun tions an be approximated by an Airy fun tion with in reasing a ura y as
t→ ∞. The te hni al details are presented in Appendix 6.5.7: we refer to Eqs. (6.106)-
(6.107) for the ferromagneti phase h < 1; Eqs. (6.109)-(6.110) for the paramagneti
phase h > 1 and Eq. (6.113) for the quantum riti al point h = 1.
Here, we demonstrate the validity of the Airy-approximation of free-fermioni
orrelation fun tions, through the neatest example of a quen h from the Néel state
(p = 2) in the riti al Ising hain with J = 1/2. In this ase the integrals F 0ln(t) and
C0ln(t) vanish for any time. For t → ∞ and x/t �nite, being x ≡ l − n > 0, using the
162
te hniques des ribed in the Appendix 6.5.7 we obtain
(4t)2/3∆ln(t) ≃ −Ai2(−X), t≫ 1. (6.80)
In (6.80), X = 2vmax
t−x[−te′′′(kmax)]1/3
∈ R, kmax = π/2 and a ording to (6.79)
e(k) ≡ εe�1,−(k, 2) =1
2[ε(k)− ε(−k + π)] (6.81)
vmax
= maxk∈[−π,π]
de(k)
dk=
1√2. (6.82)
The value of vmax is of ourse the same as in Eq. (6.68) taking J = 1/2. A omparison
of the stationary phase approximation in Eq. (6.80) with a numeri al evaluation of
the orrelation fun tion ∆ln(t) is presented in Fig. ??. It is �nally worth to remark
that for X = 0 (i.e. t = |l−n|2v
max
), Eq. (6.80) is within the 10% from the exa t value
already for t = 50. Noti e also that Ai
′′(0) = 0 whi h explains why for large times
the in�e tion point of the signal is lose to
|l−n|2v
max
.
Figure 6.15: The ontinuous blue urve is the fun tion −Ai2(−X). The points rep-resent numeri al evaluationof (4t)2/3∆ln(t) for a quen h from the Neel state in the
riti al Ising hain. Here |l − n| = 2ve�maxt− [−te′′′(kmax)]1/3X and t = 500
163
6.4.4 Evolution of the entanglement entropy
In this �nal se tion before the on lusions, we present numeri al results for the time
evolution of the entanglement entropy of a subsystem of size l for the di�erent initial
states dis ussed in Se . 6.4.1. Based on a quasi-parti le pi ture [211, 212℄, we expe t
the entanglement entropy, denoted by S(t), to grow linearly in time up to τs whi h is
approximately
l2vg
. After τs, whi h we will all the saturation time, the entanglement
entropy onverges [213℄ to the von Neumann entropy of the stationary state [217℄. In
the numeri s τs is obtained as the earliest time where the se ond derivative of the
signal hanges.
One an al ulate the entanglement entropy from the orrelation fun tions as
follows [263℄
S = −Tr
[1 + Γ
2ln
(1 + Γ
2
)]
, (6.83)
where Γ is a 2l × 2l blo k matrix whi h an be written as
Γmn =
⟨(axmaym
)
(axn ayn)
⟩
− δmn12×2
=
〈axmaxn〉 − δmn 〈axmayn〉
〈aymaxn〉 〈aymayn〉 − δmn
. (6.84)
Here axm = c†m + cm and aym = i(cm − c†m) and the indexes m,n belong to the one-
dimensional subsystem of size l. One an easily �nd all the di�erent elements of the
matrix Γ as,
Γ11 = F + F † +C −CT , (6.85)
Γ12 = i(1−C −CT − F + F †), (6.86)
Γ21 = −i(1−C −CT + F − F †), (6.87)
Γ22 = −F − F † +C −CT . (6.88)
164
At this stage we have all the ingredients to build the matrix Γ and onsequently to
al ulate the entanglement entropy from Eq. (6.83). In fa t, one an al ulate the
entanglement from
S = −l∑
j=1
[1 + νj
2ln
(1 + νj
2
)
+1− νj
2ln
(1− νj
2
)]
, (6.89)
where νj's are the positive eigenvalues of the matrix Γ. It is tempting to onje ture
that the saturation time ould be given by
l2vmax
, with vmax is the light- one velo ity
extra ted from the orrelation fun tions, dis ussed in Se . 6.4.2. It is indeed evident
from the numeri s that, as long as p > 1 and γ 6= 0, vg does not ne essarily play a
role into the time evolution of the entanglement entropy. See in parti ular Fig. 6.16
and ompare the blue urve, orresponding to S(t) for p = 1 with the green that
refers instead to p = 3; here γ = 2 and h = 1.5. It is lear that starting from an
initial state with p 6= 1, the saturation time an in rease. Noti e also that for p = 1,
vmax = vg whereas for p = 3, vmax is obtained from Eq. (6.68) that is already non-
trivial. However, one should be areful in drawing on lusions based on the numeri s.
Consider for instan e the ase p = 2 with γ = 2 and h = 1.5. A ording to the
analysis in Se . 6.4.2, the light- one velo ity vmax is given by Eq. (6.68), sin e the
fastest terms travelling with velo ity vg are zero. However the entanglement entropy
displays a saturation time larger than
l2vmax
as it an be learly seen in Fig. 6.16 (red
urve). At present we do not have an analyti al understanding of this e�e t.
6.4.5 Con lusions
In this paper we have analyzed the in�uen e of the initial state on the maximum
speed at whi h orrelations an propagate, a ording to the Lieb-Robinson bound.
We investigated the XY hain and global quen hes from a lass of initial states that
are fa torized in the lo al z- omponent of the spin and have a rystalline stru ture.
165
0 10 20 30 40 50 60t
0
10
20
30
40
50
60
70
S(t)
( 0 , 1 )( 1/2 , 1 )( 1/3 , 1 )
Figure 6.16: ( olor online) The evolution of the entanglement entropy in the XY
hain with J = 1 and system size L = 600 and l = 72. Entanglement entropy as a
fun tion of time at γ = 2 and h = 1.5. The verti al dashed lines are in orresponden eof t = l
2vmax, being vmax the light- one velo ity for the orrelation fun tions obtained
in Se . 6.4.2. Comparing the blue (p = 1) and green (p = 3) urves is evident thee�e t of the initial state on the saturation times. The red urve orresponds to p = 2and shows instead that in su h a ase the saturation time is learly larger than
l2vmax
.
We demonstrated expli itly that momentum onservation in the rystal leads to a
state-dependent light- one velo ity vmax that rules how fast orrelations spread. We
have given, and he ked numeri ally, analyti al predi tions for the light- one velo ities
for several values of the parameters γ, h and p. We also dis ussed an approximation
of fermioni orrelations fun tions in in�nite volume that shows, in agreement with
previous results in [192℄, that the behaviour at the light- one edge an be hara terized
by integer powers of t−1/3. In parti ular this is the ase when the light- one velo ity
is a maximum with vanishing se ond derivative (and non-zero third derivative) of the
e�e tive dispersion. The degree of universality of the t−1/3-s aling and in parti ular
its dependen e on the initial state, however, have been not lari�ed yet [228, 229, 230,
231, 232℄.
166
We have then studied numeri ally the evolution of the entanglement entropy and
showed that the hoi e of the initial state a�e ts also the saturation time. When om-
pleting this paper, a preprint[233℄ appeared that analyzes entanglement dynami s in
the XX hain (γ = 0) for the lass of initial states here labelled as (1/p, 1). In parti -
ular an interesting semi lassi al interpretation in terms of entangled p-plets of quasi-
parti les is proposed. Our al ulations in Se . 6.4.2 for the light- one velo ity in the
XX hain are in agreement with su h a quasi-parti le pi ture. It will be important to
investigate how this an be adapted to determine the light- one velo ity vmax and the
linear growth of the entanglement entropy also for γ 6= 0. Our analysis suggests that
these observables are not easily predi table on the whole parameter spa e, therefore
a generalized quasi-parti le pi ture will be likely initial state dependent. Finally, it
will be relevant to study the e�e t of the initial state on �nite size e�e ts.
6.4.6 Additional details
Bogolubov rotation. The Bogolubov rotation in the XY hain is
b†(k)
b(−k)
=
R(k)︷ ︸︸ ︷
cos(θ(k)/2) −i sin(θ(k)/2)
−i sin(θ(k)/2) cos(θ(k)/2)
c†(k)
c(−k)
(6.90)
with cos θ(k) = 2J(cos(k)−h/J)ε(k)
, sin θ(k) = 2Jγ sin(k)ε(k)
and ε(k) = 2J√
(cos(k)− h/J)2 + γ2 sin2(k).
The Bogolubov fermions b(k) and b†(−k) have simple time evolution
b†(k, t)
b(−k, t)
=
U(k,t)︷ ︸︸ ︷
eiε(k)t 0
0 e−iε(k)t
b†(k)
b(−k),
(6.91)
and therefore we obtain Eq. (6.73) of Se . 6.4.3 as
c†(k, t)
c(−k, t)
=
T (k,t)︷ ︸︸ ︷
R†(k)U(k, t)R(k)
c†(k)
c(−k)
. (6.92)
167
Fermioni orrelators. Expli it expressions for the fun tions F jln(t) and C
jln(t) are
F 0ln(t) =
∫dk
2πe−ik(l−n)
[
T12(k, t)T11(−k, t)+
1
p(T11(k, t)T12(−k, t)− T12(k, t)T11(−k, t))
]
, (6.93)
F j 6=0ln (t) =
e−2πinj
p
p
∫dk
2πe−ik(l−n)
[
T11(k, t)×
×T12(
−k + 2πj
p, t)
− T12(k, t)T11
(
−k + 2πj
p, t)]
; (6.94)
and analogously
C0ln(t) =
∫dk
2πeik(l−n)
[
T22(−k, t)T11(k, t)(
1− 1
p
)
+
1
pT12(k, t)
2
]
, (6.95)
Cj 6=0ln (t) =
e−2πinj
p
p
∫dk
2πeik(l−n)
[
T21(−k, t)×
×T12(
k +2πj
p, t)
−T22(−k, t)T11(
k +2πj
p, t)]
. (6.96)
Noti e that, onsistently with the dis ussion in Se . 6.4.2, the integrals in Eqs. (6.93)
and (6.95) are vanishing for p = 2.
Behaviour at the in lination point. Consider an integral of the form
I(x, t) =
∫ π
−π
dk
2πH(k)e2itε(k)−ikx
(6.97)
where we assume x > 0 and H(k) a fun tion with support k ∈ [−π, π]. As it is
dis ussed for instan e in [226, 227℄, for large x, t with their ratio �xed we an approx-
imate (6.97) by an Airy fun tion. At the boundary of the light- one, the solution kmax
of the stationary phase equation
ε′(kmax) =x
2t, (6.98)
168
satis�es by de�nition ε′′(k
max
) = 0. A remarkable ex eption in presen e of intera tions
is ontained in [231℄. At k = kmax, ε′(k) has a maximum if t > 0 and a minimum if
t < 0. Therefore in a Taylor expansion near k = kmax of the phase in (6.97) we need to
keep the third order term. If H(kmax) 6= 0 we then obtain the following approximation
of the integral I(x, t) near the boundary of the light- one
I(x, t) ≃ e2itε(kmax)−ikmaxxH(kmax)
[−tε′′′(kmax)]1/3Ai(−X), (6.99)
being X = 2ε′(kmax)t−x
[−tε′′′(kmax)]1/3. It should be noti ed that
1. Eq. (6.99) is determined in the limit x, t → ∞ but it gives a fairly good approx-
imation of the integral as long as
|2ε′(kmax)t− x| ≪ [−tε′′′(kmax)]1/3. (6.100)
2. The in lination point onsidered in the main text orresponds to X = 0. It
follows that I(x, t) an be approximated at the in lination point t = τ as
I(x(τ), τ) ≃ e2iτε(kmax)−ikmaxx(τ) H(kmax)
[−τε′′′(kmax)]1/3Ai(0), (6.101)
with x(τ) = 2τε′(kmax) as in (6.98).
Example: Ising hain (J = 1/2; γ = 1). The fun tions F 0ln(t) and C0
x(t) an be
written as
F 0ln(t) = H0 +
∑
σ=±
∫ π
−π
dk
2πHσ(k)e
2iσε(k)t−ik(l−n), (6.102)
C0ln(t) = K0 +
∑
σ=±
∫ π
−π
dk
2πKσ(k)e
2iσε(k)t−ik(l−n); (6.103)
where H± and K± are obtained expanding the integrands in (6.93) and (6.95) and
we also used C0ln(t) = C0
nl(t). The onstant terms H0 and K0 an be also determined
169
expli itly from the residue theorem and they are given by
H0 = K0 = −hx−2(h2 − 1)(p− 2)
8p, h ≤ 1; (6.104)
H0 = −K0 = −h−x+2(h2 − 1)(p− 2)
8p, h > 1, (6.105)
where we de�ned x ≡ (l − n) ≥ 2. They are then vanishing at h = 1 or are exponen-
tially small with the distan e x when h 6= 1. Sin e we will onsider only large values
of the in lination point, it turns out that x ≫ 1 also (see below Eq. (6.101)).
We fo us �rst on the ase h < 1. Here we have kmax = arccos(h), ε′(kmax) ≡ vg =
h, ε′′′(kmax) = −h; using (6.99) we obtain for x → ∞ the approximations at the
in lination point τ
F 0ln(τ) ≃
2H+(kmax) cos[x(τ)φ(h)]
[x(τ)/2]1/3Ai(0) (6.106)
C0ln(τ) ≃
2K+(kmax) cos[x(τ)φ(h)]
[x(τ)/2]1/3Ai(0). (6.107)
where φ(h) =√
1/h2 − 1− arccos(h) and x(τ) is de�ned below (6.101). It turns out
that H+(kmax) = H−(−kmax) = −iK+(kmax) = −iK−(−kmax) and
H+(kmax) = −ip− 2
4p, (6.108)
Therefore at the in lination point |F 0ln|2 and |C0
ln|2 have the same approximation in
terms of an Airy fun tion. Although the argument applies for large t, we believe
that it justi�es the di� ulty in the numeri s to spot the the fastest group velo ity
ontribution when h < 1.
We now pass to dis uss the ase h > 1. Here we have kmax = arccos(1/h),
ε′(kmax) ≡ vg = 1, ε′′′(kmax) = −1. A similar expansion gives the approximations
F 0ln(τ) ≃
eix(τ)φ(h−1)H+(kmax) + e−ix(τ)φ(h−1)H−(kmax)
[x(τ)/2]1/3Ai(0) (6.109)
C0ln(τ) ≃
2K+(kmax) cos[x(τ)φ(h−1)]
[x(τ)/2]1/3Ai(0), (6.110)
170
where now K+(kmax) = K−(kmax) and
H±(±kmax) = −i(h∓√h2 − 1)(p− 2)
4h2p(6.111)
K+(kmax) =p− 2
4h2p. (6.112)
There is no more a an ellation of the two terms as for h < 1. The absen e of a
peak at τ = l−n2
(l ≫ n) in the observable ∆ln(t) might be however understood as
a ombination of two e�e ts. For large h ≫ 1, the values at the in lination point of
|F 0ln|2 and |C0
ln|2 are suppressed by a fa tor 1/h2 and 1/h4 respe tively. On the other
hand, as h→ 1, an exa t an ellation between |F 0ln| and |C0
ln| must happen.
For h = 1, the two distin t stationary points ±kmax a tually merge at k = 0 and
the asymptoti expansion involve only one term. One has vg = 1 and ε′′′(0+) = −1/4,
leading to the �nal result
F 0ln(τ) ≃ −i p− 2
2p[x(τ)]1/3Ai(0), C0
ln(τ) = −iF 0ln(τ), (6.113)
that ones again shows the an ellation of the fastest parti le ontribution for arbitrary
values of p. Analogous approximations near the orrespondent in lination points ould
be al ulated for F jln(t) and Cj
ln(t) with the same te hnique. In the main text we
estimate, for instan e, ∆ln(t) for p = 2 and h = 1.
171
Figure 6.17: Up(left)ve�max(2) and ve�
max(1) for di�erent values of a and h. Up(right)d = ve�max(1)− ve�max(2) for di�erent values of a and h. Down(left)ve�max(3) and v
e�
max(1)for di�erent values of a and h. Down(right) d = veffmax(1)− veffmax(3) for di�erent valuesof a and h..
172
6.5 Formation probabilities and Shannon information and their time
evolution after quantum quen h in the transverse-field XY
hain
Studying orrelation fun tions in many-body systems has been onsidered one of the
main topi s in statisti al me hani s and ondensed matter physi s for many years.
Although, for a long-time the main quantities of interest were the orrelation fun tions
of lo al observables the re ent interest in al ulating non-lo al quantities, espe ially
the entanglement entropy, has made signi� ant hanges. One of the main reasons for
this interest (at least in 1 + 1 dimensional riti al systems) is that by al ulating
some of the non-lo al observables one an derive the entral harge of the system
without referring to the velo ity of sound, for the ase of entanglement entropy see
[234℄. Another non-lo al quantity whi h has been studied for many years with Bethe
ansatz te hniques and some other methods is emptiness formation probability [145,
146, 147, 235, 236, 237, 238℄. In the ase of spin hains, it is the probability of �nding
a sequen e of up spins in the system (note that almost all of the studies in this
regard on entrate on the σzbases). These studies show that this probability, with
respe t to the sequen e size, de reases like a Gaussian in the ase of systems with
U(1) symmetry [236℄ and exponentially in other ases [145, 146, 238℄. In the riti al
ases, the Gaussian and exponential are a ompanied with a power-law de ay with a
universal exponent. In a re ent development [147℄ it was shown that for those riti al
systems without U(1) symmetry this universal exponent is dependent on the entral
harge of the system. The argument is based on onne ting the on�guration of all
spins up to some sort of boundary onformal �eld theory. One should noti e that the
This hapter is reprinted from K Naja�, MA Rajabpour, Formation probabilities and
Shannon information and their time evolution after quantum quen h in the transverse-�eld
XY hain, Physi al Review B 93, 12:125139, Copyright(2013)
173
argument works just for those bases that an be onne ted to boundary onformal
�eld theory.
In apparently onne ted studies re ently many authors investigated the Shannon
information of quantum systems in di�erent systems [239, 240, 241, 242, 243, 244,
245, 246, 247, 248, 249℄. The Shannon information is de�ned as follows: Consider the
normalized ground state eigenfun tion of a quantum spin hain Hamiltonian |ψG〉 =∑
I aI |I〉, expressed in a parti ular lo al bases |I〉 = |i1, i2, · · ·〉, where i1, i2, · · · are
the eigenvalues of some lo al operators de�ned on the latti e sites. The Shannon
entropy is de�ned as
Sh = −∑
I
pI ln pI , (6.114)
where pI = |aI |2 is the probability of �nding the system in the parti ular on�guration
given by |I〉. As it is quite lear this quantity is bases dependent and to al ulate it
one needs in prin iple to know the probability of o urren e of all the on�gurations.
The number of all on�gurations in reases exponentially with the size of the system
and that makes analyti al and numeri al al ulations of this quantity quite di� ult.
Note that the emptiness formation probability is just one of the whole possible on-
�gurations. The Shannon information of the system hanges like a volume law with
respe t to the size of the system so in prin iple one does not expe t to extra t any
universal information by studying the leading term. The universal quantities should
ome from the subleading terms. To study subleading terms, it is useful to de�ne yet
another quantity alled Shannon mutual information. By onsidering lo al bases it is
always possible to de ompose the on�gurations as a ombination of the on�gura-
tions inside and outside of a subregion A as |I〉 = |IAIA〉. Then one an de�ne the
marginal probabilities as pIA =∑
IApIAIA
and pIA =∑
IApIAIA
for the subregion A
174
and its omplement A. Then the Shannon mutual information is
I(A, A) = Sh(A) + Sh(A)− Sh(A ∪ A), (6.115)
where Sh(A) and Sh(A) are the Shannon information of the subregions A and A.
From now on instead of using pIAIAwe will use just pI . Sin e in the above quantity the
volume part of the Shannon entropy disappears Shannon mutual information provides
a useful te hnique to study the subleading terms. Note that one an similarly de�ne
the above quantity also for two regions A and B, i.e. I(A,B) that are not ne essarily
omplement of ea h other. The Shannon mutual information has been studied in
many lassi al [250, 251, 252, 253℄ and quantum systems [254, 255, 256, 257, 258℄.
Numeri al studies on variety of di�erent periodi quantum riti al spin hains show
that for parti ular bases (so alled onformal bases) we have [255, 257, 258℄
I(A, A) = β ln[L
πsin(
πl
L)], (6.116)
where L and l are the total size and subsystem size respe tively and β is very lose
to
c4, with c the entral harge of the system. Note that if one takes an arbitrary base
(non- onformal bases) the oe� ient β is nothing to do with the entral harge. It is
worth mentioning that based on [257℄ the onformal bases are those bases that an
be onne ted to some sort of boundary onformal �eld theory in the sense of [147℄.
For example in the transverse �eld Ising model the σxand σz
bases are the onformal
bases. Note that if one onsiders the Shannon information of the subsystem we will
have Sh(A) = αl + β ln[Lπsin(πl
L)]. Sin e to extra t the Shannon information one
needs to use all the probabilities the only way to onsider all of them in the numeri al
al ulations is exa t diagonalization. This makes the numeri al al ulation for large
sizes very di� ult. The results of the arti les [255, 257, 258℄ are all for periodi systems
with L = 30 whenever there are spin one-half system and smaller sizes for systems
175
with bigger spins. In a re ent work [256℄ the author was able to onsider an in�nite
system and study the Shannon information of the subsystem up to the size l = 40.
It was on luded that for the XX hain β is
c4with c = 1 but for the Ising model
although it is very lose to
c4with c = 1
2the results are suggesting that probably β is
not exa tly onne ted to the entral harge. Noti e that all of the above al ulations
are done by onsidering periodi boundary onditions for the onne ted regions A
and A. It is not lear how the equation (6.116) might hange if one onsiders open
boundary onditions. Finally it is worth mentioning that some of the the above results
have re ently been extended to dis onne ted regions in [259℄.
Motivated by the studies of emptiness formation probability and Shannon infor-
mation of the subsystem we study here some related quantities. First of all, as it is
natural one might be interested in studying the s aling limit of some other on�gu-
rations with respe t to the size of the subsystem. For example, onsider an antiferro-
magneti on�guration in the Ising model or any other on�guration with a pattern.
It is very important to know that these on�gurations are also �owing to some sort
of boundary onformal �eld theory or not. This study will learly also help to under-
stand the nature of the Shannon mutual information. In addition, this kind of study
also is very useful in the al ulations of post-measurement entanglement entropy and
lo alizable entanglement entropy [260, 261℄. Having the above motivations in mind,
we study formation probabilities , Shannon information and their evolution after a
quantum quen h in the quantum XY hain in the σzbases.
The outline of the paper is as follows: In the next se tion we will �rst de�ne our
system of interest, i.e. XY hain and then we will provide a method to al ulate the
probability of any on�guration in the free fermioni systems. In se tion three we
will list all the known analyti al results regarding emptiness formation probability
for in�nite systems and also �nite systems with periodi and open boundary ondi-
176
tions. Expli it distinguishment is made between riti al Ising model and XX hain
with U(1) symmetry. In se tion IV, we will de�ne many di�erent on�gurations with
spe ially de�ned pattern and al ulate their orresponding probabilities numeri ally.
Here again, we dis uss Ising and XX universalities separately for in�nite systems and
for the systems with boundary. We also dis uss on�gurations that do not have any
pattern. In se tion V, we study the Shannon information in the transverse �eld riti al
Ising model and also riti al XX hain. We will lassify the on�gurations based on
their magnetization and show that in prin iple just a small part of the on�gurations
an have a �nite ontribution to Shannon information in the s aling limit. Se tion
VI is devoted to the evolution of formation probabilities and Shannon information
after a quantum quen h. We prepare the system in a parti ular state and then we
let it evolve with another Hamiltonian and study the time evolution of the formation
probabilities and espe ially Shannon information. Finally, the last se tion is about
our on lusions and possible future works.
6.5.1 Formation probabilities from redu ed density matrix
In this se tion, we �rst de�ne the system of interest and after that using the redu ed
density matrix of this system we will �nd a very e� ient method to al ulate formation
probabilities for systems that an be mapped to free fermions. The Hamiltonian of
the XY- hain is as follows:
H = −L∑
j=1
[
(1 + a
2)σx
j σxj+1 + (
1− a
2)σy
j σyj+1 + hσz
j
]
. (6.117)
After using Jordan-Wigner transformation, i.e. cj =∏
m<jσzm
σxj −iσy
j
2and N =
∏L
m=1σzm = ±1 with c†L+1 = 0 and c†L+1 = N c†1 for open and periodi boundary
177
onditions respe tively the Hamiltonian will have the following form:
H =L∑
j=1
[
(c†jcj+1 + ac†jc†j+1 + h.c.)− h(2c†jcj − 1)
]
+N (c†Lc1 + ac†Lc†1 + h.c.). (6.118)
The above Hamiltonian has a very ri h phase spa e with di�erent riti al regions [262℄.
In �gure 1 we show di�erent riti al regions of the system. To al ulate probability of
formations for di�erent patterns we �rst write the redu ed density matrix of a blo k
of spins D by using blo k Green matri es. Following [263, 264℄, we �rst de�ne the
operators
ai = c†i + ci, bi = c†i − ci. (6.119)
The blo k Green matrix is de�ned as
Gij = tr[ρDbiaj ]. (6.120)
The elements of the Green matrix an be al ulated following [265℄ and we will men-
tion their expli it form for open and periodi boundary onditions later.
To al ulate the redu ed density matrix after partial measurement we need to �rst
de�ne fermioni oherent states. They an be de�ned as follows
|ξ >= |ξ1, ξ2, ..., ξN >= e−∑N
i=1 ξic†i |0 >, (6.121)
where ξi's are Grassmann numbers following the properties: ξnξm + ξmξn = 0 and
ξ2n = ξ2m = 0. Then it is easy to show that
ci|ξ >= −ξi|ξ > . (6.122)
Using the oherent states (6.121) the redu ed density matrix has the following form
[264℄
ρD(ξ, ξ′) = < ξ|ρD|ξ′ >
= det1
2(1−G)e
12(ξ∗−ξ′)TF (ξ∗+ξ′), (6.123)
178
where F = (G+1)(1−G)−1. One an use the above formula to extra t the formation
probability for arbitrary on�guration in the σz bases as follows: �rst of all to extra t
the probability of parti ular on�guration we need to look to the diagonal elements
of ρD(ξ, ξ′). When the spin in the σz
dire tion is up in the fermioni representation
it an be understood as the la k of a fermion in that site whi h in the language of
oherent states means that the orresponding ξ is zero in the equation (6.123). After
putting some of the ξ's equal to zero one will have a new redu ed density matrix in the
oherent state basis with this onstraint that some of the spins are �xed to be up. In
other words in the equation (6.123) instead of F we will have F whi h is a sub-matrix
of the matrix F . The elements of the new redu ed density matrix will be ρD(η,η′),
where we put the ξ's orresponding to the sites �lled with fermions equal to η. To
extra t the probability of formation one just needs to integrate over all the η's that
orrespond to the down spins. In other words after using formulas of the Grassmann
Gaussian integrals the formation probability will have the following formula
P (Cn) = det[1
2(1−G)]MCn
F , (6.124)
whereMCnF is the minor of the matrix F orresponding to the on�guration Cn. Noti e
that we just need prin ipal minors of the matrix F . Sin e the sum of all the prin ipal
minors of the matrix F is equal to det(1 + F ) the normalization is ensured. We have
(lk
)number of rank-k minors for matrix F with size l. Summing over the number of all
prin ipal minors, one an obtain 2l whi h is the number of all possible on�gurations.
Con�gurations with the same minor rank have the same number of up spins and
by in reasing k we a tually in rease the number of down spins in the orresponding
on�guration. For example k = 0(l) is the ase with all spins up (down) and sometimes
it is alled emptiness formation. The above formula gives a very e� ient way to
al ulate the formation probabilities in numeri al approa h. Sin e, as we will show, all
179
of these probabilities are exponentially small with respe t to the size of the subsystem
it is mu h easier to work with the logarithm of them and de�ne logarithmi formation
probabilities
Π(Cn) = − lnP (Cn). (6.125)
All of the al ulations done in this paper are based on the formula (6.124) and are
performed using Mathemati a. In the next se tions, we will study the formation
probabilities for di�erent on�gurations with the rystal order (pattern formation
probabilities) with respe t to the size of the subsystem for some parti ular riti al
regions of the system.
6.5.2 Emptiness formation probability: known results
Before presenting our results, we �rst review here the well-known fa ts regarding
emptiness formation probability (k = 0 and l). For reasons that will be lear in
the next se tion, we will all these two on�gurations x = 0 and 1 respe tively. Using
Fisher-Hartwig theorem the emptiness formation probability was already exhaustively
studied in Ref. [145, 146℄. The results were generalized to arbitrary onformal riti al
systems in [147℄. Due to the U(1) symmetry there is an important di�eren e between
the XX riti al line and the Ising riti al point. For this reason, we report the results
regarding these two possibilities separately.
Criti al Ising point
We list here the results regarding the logarithmi emptiness formation probability at
the riti al Ising point (a = h = 1).
180
Con�guration Cx=0, i.e. (| ↑, ↑, ... ↑>): This on�guration orresponds to k = 0 and
has the highest probability and using the equation (6.124) one an easily show that
P (x = 0) = det[1
2(1−G)] (6.126)
The above formula is valid independent of the boundary onditions and the size of
the system. For the in�nite system at the riti al Ising point the G matrix has the
following form:
Gij = − 1
π(i− j + 1/2). (6.127)
Sin e the above matrix is a Toeplitz matrix using Fisher-Hartwig onje ture in [? ℄
it was shown that the logarithmi probability for this on�guration hanges with the
subsystem size as follows:
Π(x = 0) = αl + β ln l + γln l
l+O(1), (6.128)
where β = 116
= 0.0625, and α and γ are some non-universal numbers. At the riti al
point of the Ising model these numbers are known α = ln 2 − 2C/π = 0.11002, with
C the Catalan onstant and γ = − 132π
= −0.00994. The ln llterm is the result of the
paper [147℄. Using onformal �eld theory te hniques in [147℄ it was argued that for
generi riti al systems the oe� ient of the logarithm should be β = c8, where c is
the entral harge of the riti al system. For the Ising universality lass c = 12.
When the size of the total system is �nite L depending on the form of the boundary
onditions, periodi or open; G has the following two forms
GPij = − 1
L sin(π(i−j+1/2)L
), (6.129)
GOij = − 1
2L+ 1
( 1
sin(π(i−j+1/2)2L+1
)+
1
sin(π(i+j+1/2)2L+1
)
)
.
(6.130)
181
Noti e that for L→ ∞ the �rst equation redu es to (6.127) and the se ond equation
will give the result for a semi-in�nite hain. The results for emptiness formation
probabilities for the above two ases are [147℄
ΠP (x = 0) = αl + β ln[L
πsin
πl
L] + γπ cot(
πl
L)ln l
L+O(1), (6.131)
ΠO(x = 0) = αl + βo ln[4L
π
tan2 πl2L
sin πlL
] + γoπ2− cos(πl
l)
sin πll
ln l
L+O(1), (6.132)
where β = c8and βo = − c
16. It is also onje tured that γ = − c
8πaand γo = c
32πa. The
above two equations are derived using boundary onformal �eld theory te hniques
and in prin iple they are valid in the Ising ase be ause x = 0 on�guration is related
to the free onformal boundary ondition in the onformal Ising model.
Con�guration Cx=1, i.e. | ↓, ↓, ... ↓>: This ase, whi h is also studied in [145, 146℄
and [147℄, orresponds to k = l and has the lowest probability. One an easily show
that
P (x = 1) = det[1
2(1−G)] detF = det[
1
2(1 +G)]. (6.133)
For the in�nite system it follows similar formula as (6.128) with also an extra ν (−1)l√l
term, in other words,
Π(x = 1) = αl + β ln l + ν(−1)l√
l+ γ
ln l
l+O(1), (6.134)
where β = c8. At the riti al Ising point α = ln 2+2C/π = 1.27626 and ν = −0.21505
and γ is unknown. The os illating ν term is mathemati ally explained by using gen-
eralized Fisher-Hartwig onje ture in [145, 146℄. To the best of our knowledge its
presen e at the riti al point has not been understood by physi al arguments [266℄.
Our numeri al results in the next se tion will show that the term is present whenever
182
the parity of the number of down spins hanges with the subsystem size. The term is
very important to be onsidered in numeri al al ulations to get reliable results for β
whi h is the universal and the most interesting term.
When the size of the system is �nite depending on the type of the boundary
onditions boundary hanging operators an play an important role. The following
formulas are presented in [147℄:
ΠP (x = 1) = αl + β ln[L
πsin
πl
L] + ..., (6.135)
ΠO(x = 1) = αl + βo1 ln[
L
πsin
πl
L] + βo
2 ln[L
πtan
πl
2L] + ...,
(6.136)
where β = c8, βo
1 = c16
and βo2 = 4h− c
8with h = 1
16being the onformal weight of the
boundary hanging operator. The dots are the subleading terms.
XX riti al line
The riti al XX hain a = 0 has U(1) symmetry whi h as it is already dis ussed
extensively in the literature is the main reason for having Gaussian de aying emptiness
formation probability [147, 236℄. Sin e in this model < c†ic†j >=< cicj >= 0 the
equation (6.123) has simpler form
ρD(ξ, ξ′) = det(1− C)eξ
∗Fξ′(6.137)
where Cij =< c†icj > and F = C(1− C)−1. Finally we have
P (Cn) = det[1− C]MCnF (6.138)
The form of the C matri es in the periodi and open ases are [267℄:
CPij =
nf
πδij + (1− δij)
sin(nf(i− j))
L sin(π(i−j)L
), (6.139)
COij =
(1
2− (
L
2(L+ 1)−n′f
π))
δij + (1− δij)1
2(L+ 1)
(sin(n′f (i− j))
sin(π(i−j)2L+2
)−
sin(n′f (i+ j))
sin(π(i+j)2L+2
)
)
,
183
where nf = πL
(
2⌈ L2π
arccos(−h)⌉ − 1)
is the Fermi momentum and n′f = π
2(L+1)
(
1 +
2⌊ (L+1)π
arccos(−h))⌋)
with ⌈x⌉(⌊x⌋) as the losest integer larger (smaller) than x.
The all spins up and down on�gurations do not lead to onformal boundary
onditions and so none of the equations that we mentioned in the last subse tion are
valid. However, using Widom onje ture it is already known that, see for example
[238℄, the probabilities for both Cx=0 and Cx=1 show Gaussian behavior. For systems
with U(1) symmetry one expe ts the following behavior for logarithmi emptiness
formation probability [236℄:
Π(x = 0) = α2l2 + αl + β ln l +O(1), (6.140)
where β = 14for riti al XX hain.
6.5.3 Logarithmi pattern formation probabilities
In this se tion, we study the logarithmi pattern formation probability de�ned as
Π(C) = − lnP (C) with respe t to the size of the subsystem. The easiest on�gurations
to study are those that have some kind of rystal stru ture. Although everything is
already known and he ked numeri ally for the emptiness formation probabilities we
will also report the results on erning these ases as ben hmarks. Here we introdu e
the on�gurations that we studied numeri ally. None of these on�gurations have
been onsidered before in the literature.
We study here the on�gurations with k = l2, l3, l4, ..., l
10with rystal pattern and
we all them on�gurations x = 12, 13, 14, ..., 1
10and for some ranks we will study the
two most basi ases. For example we will study
Con�gurations with k = l2:
184
a (| ↓, ↑, ↓, ↑, ... >)
b (| ↓, ↓, ↑, ↑, ↓, ↓, ↑, ↑, ... >)
Con�gurations with k = l3:
a (| ↑, ↑, ↓, ↑, ↑, ↓, ↑, ↑, ... >)
b (| ↑, ↑, ↑, ↑, ↓, ↓, ↑, ↑, ↑, ↑, ↓, ↓, ... >)
All the on�gurations with the same k belong to the ases with an equal rank of the
minor in the equation (6.124). Note that in all of the up oming numeri al al ulations
in every step we in rease the size of the subsystem with a number whi h is devidable to
the length of the base of the orresponding on�guration. For some parti ular k's the
a and b on�gurations di�er by the parity e�e t. For example, in k = l2a depending
on l = 4i or l = 4i − 2 with i = 1, 2, ... the subsystem has even or odd number of
down spins. This means that the parity of the number of down spins hanges with
the subsystem size for this on�guration. However, for k = l2b this is not the ase
be ause in order to have "perfe t rystal" in the subsystem we need to onsider a
subsystem with l = 4i with i = 1, 2, ... whi h has always even number of down spins
inside. Be ause of this di�eren e in parity e�e t for k = l2a and k = l
2b we expe t
di�erent subleading behavior for these two ases. Finally noti e that one an simply
de�ne on�gurations like k = l2c and k = l
3c by simply taking bigger bases for the
rystals. For example, k = l2c an be understood as a on�guration with the base:
three down spins and then three up spins.
185
Transverse-field Ising hain
Using the equation (6.124) and (6.127) we �rst studied the rystal on�gurations
introdu ed in the previous subse tion for the ase of in�nite hain. To al ulate the
formation probability for every on�guration we �rst use the matrix G introdu ed
in (6.127) to �nd the matrix F . Then for every on�guration we use an appropriate
minor to al ulate the orresponding probability in (6.124). For example, in the ase
of k = l2a this an be done by just �nding a minor of F whi h an be derived by
al ulating the determinant of a submatrix F obtained from F by removing every
other row and olumn. The results for α and β (the most interesting quantities in
this study) are shown in the Table 6.2. Based on the numeri al results one an derive
the following on lusions regarding rystal on�gurations:
1. All the rystal on�gurations follow either the equation (6.128) or (6.134) with
β = 116.
2. Whenever the parity of the number of down spins in a on�guration hanges
with respe t to the size of the subsystem we have the os illating term
1√l. For
example, in the ase of k = l2a we have the subleading term
(−1)l2√
lbut
1√l
orre tion is absent in k = l2b. It appears again for k = l
3a in the form of
(−1)l3√
l.
Generalization to other on�gurations is strightforward. .
3. Although in some ases α for bigger x is smaller than α with smaller x in average
α in reases with x.
We then studied the same on�gurations for the periodi boundary ondition. In
the �gure 6.18, it is shown that all of the on�gurations follow the formula (6.131)
. The ase of the open boundary ondition is more tri ky and depending on the
186
Con�guration α βx = 0 0.110025 0.062498x = 1 1.276267 0.062465x = 1
2(a) 0.984708 0.062462
x = 12(b) 0.755726 0.062496
x = 13(a) 0.818715 0.062468
x = 13(b) 0.542109 0.062491
x = 14(a) 0.710620 0.062481
x = 14(b) 0.434286 0.062524
x = 15(a) 0.634016 0.062495
x = 16(a) 0.576551 0.062509
x = 17(a) 0.531651 0.062523
x = 18(a) 0.495482 0.062537
x = 19(a) 0.465643 0.062549
x = 110(a) 0.440555 0.062562
Table 6.2: Fitting parameters for the logarithmi formation probabilities of di�erent
rystal on�gurations of the riti al Ising hain dis ussed in the text. All the data were
extra ted by �tting the data in the range l ∈ (2000, 2500) to αl+β ln l+γ ln ll+ δ 1
l+η
for those ases that do not show parity e�e t and to αl+β ln l+γ ln ll+ν (−1)m√
l+δ 1
l+η
(with suitable m) for those ases that show parity e�e t [268℄.
0 500 1000 1500 2000l
-0.2
0
0.2
0.4
0.6
Π(l
,L)-
α l
CFTx=0x=Lx=1/2(a)x=1/2(b)x=1/3(a)x=1/3(b)
Figure 6.18: Π(l, L)−αl for periodi system with total length L = 2000 with respe tto l for di�erent on�gurations. The dashed lines are the results expe ted from CFT,
i.e.
116ln[L
πsin πl
L] + η.
187
0 500 1000 1500 2000l
0
0.5
1
1.5
2
2.5
3
Π(l
,L)
- α
l
CFTx=1x=1/2(a)x=1/3(a)x=1/4(a)
(a)
0 500 1000 1500 2000l
-1
-0.8
-0.6
-0.4
-0.2
0
Π(l
,L)
- α
l
CFTx=0x=1/2(b)x=1/3(b)x=1/4(b)
(b)
Figure 6.19: Π(l, L)−αl for open system with total length L = 2000 with respe t tol for di�erent on�gurations. a) on�gurations without boundary hanging operators
and b) on�gurations with boundary hanging operators. The dashed lines are the
results expe ted from CFT.
on�guration we have two possibilities: When the parity of the number of down spins
is independent of the size of the subsystem (for example in the on�gurations x = 12b
and
13b) we have the formula (6.132) but when we have the possibility of having odd
or even number of down spins in a on�gurations (for example x = 12a, 1
3a) we have
the formula (6.136). The results are shown in the Figure 6.19. This behavior ould
be anti ipated based on the di�eren e between on�gurations k = 0 and k = l that
we dis ussed before. It looks like that the boundary hanging operator plays a role
whenever there is the parity e�e t in the on�guration. Looking to the problem in
the language of Eu lidean two dimensional lassi al system one an argue that in the
ase of open boundary ondition we have a strip with a slit on it [147℄. However,
the boundary onditions on the boundary of strip an be di�erent from the boundary
ondition on the slit, onsequently, one needs to onsider boundary hanging operator
on the point where the boundary ondition hanges. However, in general it is not lear
188
whi h on�gurations lead to di�erent boundary onditions on the slit and on the
boundary of the strip. Our numeri al results indeed give a hint that depending on the
bahavior of the parity of the number of down spins in a on�guration the onformal
boundary ondition on the slit an be di�erent. In the next two subse tions, we
will �rst omment on the validity of the above results in other ases su h as non-
rystal on�gurations. Then we will also indi ate the possible universal behavior of
our results.
Logarithmi formation probability of non- rystal onfigurations
The number of rystal on�gurations is mu h smaller than the number of the whole
on�gurations. In fa t, the number of rystal on�gurations grows polynomially with
the subsystem size but the number of whole on�gurations grows exponentially. How-
ever, numeri ally it is very simple to he k the formula for many on�gurations that
have a small deviation from the rystal states. For example, one an onsider the ase
k = 1 with all spins up ex ept one and al ulate the logarithmi formation proba-
bility using the equation (6.124). It is lear that one does not expe t the result be any
di�erent from the equation (6.128) and indeed numeri al results on�rm this expe -
tation. The important on lusion of this numeri al exer ise is that there are many
on�gurations " lose" to rystal on�gurations that indeed follow either the equation
(6.128) or equation (6.134) with all having the same β's but di�erent α's.
The above results strongly suggest that all of the rystal and non- rystal on-
�gurations dis ussed in this se tion are �owing to some sort of onformal boundary
onditions in the s aling limit.
189
Universality
To he k that the above results are the properties of the Ising universality lass we
also studied the riti al XY- hain whi h has also entral harge c = 12. The Green
matrix, in this ase, is given by
Gij =
∫ π
0
dφ
π
(cosφ− 1) cos[(i− j)φ]− a sinφ sin[(i− j)φ]
√
(1− cosφ)2 + a2 sin2 φ.
Our numeri al results depi ted in the Figure 4 show that the oe� ient of the
logarithmi term is a universal quantity whi h means that it has a �xed value on
the riti al XY-line. The oe� ient of the linear term hanges by varying a whi h
indi ates its non-universal nature.
0.5 0.6 0.7 0.8 0.9 1a
0.058
0.06
0.062
0.064
0.066
0.068
β
x=1/2(a)x=1/3(a)0.0625
Figure 6.20: The oe� ient of the logarithmi term in (6.128) for two on�gurations
x = 12a and x = 1
3a for di�erent values of a. The dashed line is the CFT result. The
size of the largest subsystem was l = 500 and all the results were extra ted by �tting
the data to αl+β ln l+γ ln ll+ν (−1)m√
l+δ 1
l+η with suitablem in the range l ∈ (100, 500).
Estimated errors in the numeri s are in the order of the size of the markers.
XX hain
We repeated the al ulations of the last se tion for also riti al XX hain. The entral
harge of the system is c = 1. The results of logarithmi formation probabilities for
190
di�erent magneti �eld h are shown in the Table 6.3 and Table 6.4. Based on the
numeri al al ulations we on lude the followings:
1. The on�gurations with x =nf
πfollow the equation (6.128) with β = 1
8. This
means that in the s aling limit most probably all of these on�gurations �ow
to some sort of bounday onformal onditions. Note that as far as there is no
boundary hanging operator in the system the equation (6.128) is valid for any
CFT independent of its stru ture.
2. All the other on�gurations follow the equation (6.140) with β whi h is di�erent
for di�erent on�gurations.
As we mentioned earlier XX hain has a U(1) symmetry whi h means that the number
of parti les is onserved. The only on�gurations that respe t this symmetry in the
subsystem level are the on�gurations with x =nf
π. Any inje tion of the parti les
into the subsystem hanges drasti ally the formation probability. In the ase of x =
0 this phenomena is already explained in [147℄ based on ar ti phenomena in the
dimer model. It is quite natural to expe t that similar stru ture is valid for all the
on�gurations with x 6= nf
π. Note that based on our results the oe� ient of the ln is
nf -dependent and stri tly speaking is not a universal quantity.
nf Con�guration α βx = 1/2(a) 0.3465735 0.124998
π/2 x = 1/2(b) 0.5198604 0.124997x = 1/2(c) 0.7127780 0.124597
π/3 x = 1/3(a) 0.3662041 0.124987π/4 x = 1/4(a) 0.3432345 0.125024
Table 6.3: Fitting parameters for the logarithmi formation probability of antifer-
romagneti on�gurations with di�erent �lling fa tors. All the results were extra ted
by �tting the data in the range l ∈ (100, 300) to αl + β ln l + γ ln ll+ δ 1
l+ η.
191
Con�guration α2 α βx = 0, 1 0.346573 0.000000 0.250054x = 1/3(a) 0.035191 0.366228 0.524293x = 1/3(b) 0.035188 0.597663 1.578683x = 1/4(a) 0.080911 0.346599 0.829767x = 1/4(b) 0.080910 0.587114 2.744492x = 1/5(a) 0.118119 0.321924 1.144949x = 1/6(a) 0.147178 0.298674 1.465399x = 1/7(a) 0.170072 0.278052 1.788319x = 1/8(a) 0.188433 0.260021 2.112155x = 1/9(a) 0.203427 0.244263 2.436061x = 1/10(a) 0.230432 0.230432 2.759481
Table 6.4: Fitting parameters for di�erent on�gurations with x < 12in the XX hain
with nf = π2. All the data were extra ted by �tting the data in the range l ∈ (100, 300)
to α2l2 + αl + β ln l + η.
We also studied the �nite size e�e t in this model. The results of the numeri al
al ulations of periodi boundary ondition for x = 12are shown in the Figure 6.21.
It is shown that all of the on�gurations with x = 12follow the formula (6.131) with
c = 1.
We also repeated the same al ulations for open boundary onditions. We �rst
performed the al ulations for semi-in�nite open hain and �tted the results to (6.128)
and extra ted the β. The oe� ient of the logarithm not only depends on the rank
of the on�guration but also to the on�guration itself. It also hanges with nf . We
were not able to �nd any universal feature in this ase.
The above results suggest that most probably all of the rystal on�gurations with
x =nf
πin the periodi boundary ondition �ow to a boundary onformal �eld theory.
In the language of Luttinger liquid, the orresponding boundary ondition should be
192
0 50 100 150 200 250 300l
0.4
0.8
1.2
1.6
Π(l
,L)
- α
l
CFTx=1/2(a)x=1/2(b)
Figure 6.21: Π(l, L)−αl for periodi system with total length L = 300 with respe t
to l for di�erent on�gurations for riti al XX- hain with nf = π2. The dashed lines
are the results expe ted from CFT.
the Diri hlet boundary ondition [147℄. The ase of the open boundary ondition is
intriguing and we leave it as an open problem.
6.5.4 Shannon information of a subsystem
In this se tion, we study Shannon information of a subsystem in transverse-�eld
Ising model and XX hain. For both models, the Shannon information is already
al ulated in [256℄ up to the size l = 40 whi h it seems to be the urrent limit for
lassi al omputers. The reason that we are interested in revisiting this quantity is to
have a more detailed study of the ontribution of di�erent on�gurations. This will
give an interesting insight regarding the possible s aling limit for this quantity.
193
Criti al Ising
In the last se tion, we studied many di�erent on�gurations in the riti al Ising model
and we found that all of them follow PCn = eαnl
lc8
with c = 12. The natural expe tation
is that if we plug this formula in the de�nition of the Shannon information we get
Sh(l) = αl +c
8ln l + ... (6.141)
where the dots are the subleading terms. The above formula is onsistent with [255℄.
However, one should be areful that although there are a lot of rystal on�gurations
(polynomial number of them) and " lose" to rystal on�gurations that are onne ted
to the entral harge it is absolutely not lear what is going to happen in the s aling
limit. For example we repeated the al ulations of [256℄ and realized that extra tion
of the oe� ient of the ln in the above equation is indeed very di� ult, see appendix.
Here we show where one should look for the most important on�gurations. After a
bit of inspe tion and numeri al he k, one an see that the on�guration with the
highest probability is the x = 0. Although the proof of the above statment doesn't
look straightforward one an understand it qualitatively by starting from the ground
state of the Ising model with h → ∞ and approa hing to the riti al point h = 1.
The ground state of the Ising model with h → ∞ is made of a on�guration with
all spins up. When we de rease the transverse magneti �eld the other on�gurations
start to appear in the ground state. Although the amplitude of the on�guration with
all spins up de reases by de reasing h it still remains always bigger than the other
on�gurations. Another way to look at this phenomena is by looking at the variation
of the expe tation value of the Hamiltonian H = 〈C|H |C〉 for di�erent on�gurations
C. It is easy to see that H is minimum for the on�guration C with all spins up. This
simply means that most probably when one onstru t the ground state of the Ising
model using the variational te hniques this on�guration playes the most important
194
role. The least important on�guration is x = 1 with the lowest probability. This an
also be understood with the same heuristi argument as above. For every rank k of
the minors, as we dis ussed, we have
(lk
)number of on�gurations whi h means that
for every x for large l we have ef(x)l with f(x) = −x ln x− (1−x) ln(1−x) number of
on�gurations. It is obvious that the number of on�gurations in every rank should
be high enough to ompensate the exponential de rease of probabilities. We realized
that in every rank the on�gurations a has the lowest probabilities. One an again
understand this fa t using the variational argument. In this ase it is mu h better to
make �rst the anoni al transformation: σx → −σzand σz → σx
in the Hamiltonian
of the rti al Ising model. Then one an simply argue that H is big if there are a
lot of domain walls, i.e. 〈C|σzjσ
zj+1 |C〉 = −1 in the system whi h is the ase for the
on�gurations a. Other important on�gurations are the on�gurations whi h divide
the subsystem to two onne ted regions with in one part all the spins are up and in
the other part all the spins are down. These on�gurations are interesting be ause
they have the biggest probabilities among all the on�gurations orresponding to their
minor rank. Note that in this ase we have just one domain wall. It is not di� ult
to see that the probability of all of these on�gurations de ay exponentially with the
following oe� ient α:
αmin(x) =4C
πx+ ln 2− 2C
π, (6.142)
where C is the Catalan onstant. In the two extreme points, we re over the pre-
vious results. We also he ked the validity of the above formula numeri ally. Having
the biggest and smallest probabilities for every rank, we an now easily read the
most important ranks. In Figure 6.22, we depi ted the αmax and αmin for di�erent
on�gurations. We also depi ted the graph of the number of on�gurations in every
rank. The Figure learly show that the on�gurations with x > 12 an not have any
195
signi� ant ontribution in the s aling limit be ause the number of on�gurations is
not enough to ompensate the exponential de ay of the probabilities. A similar story
seems to be valid also for the values of x lose to zero. The reason is that the number
of on�gurations with small x is su h low that an not ompensate exponential de ay
of the probabilities in this region to have a signi� ant ontribution in the Shannon
information. Just the region between the points that the two lines ut ea h other
will most likely survive in the s aling limit. The numeri al results indeed prove our
0 0.2 0.4 0.6 0.8 1x
0
0.5
1
1.5
2
f(x)α
minα
max
Figure 6.22: Values of f(x), αmin and αmax with respe t to x. The red urve is the
fun tion f(x) = −x log x − (1 − x) ln(1 − x) and the blue line is the linear fun tion
(6.142). The separated points are the αmax regarding the on�gurations dis ussed in
the text.
expe tation. In Figure 6.23, we depi ted the ontribution of every rank Shk(l) in
Shannon information for two di�erent sizes. As it is quite lear the most important
ontributions ome from 0 < x < 12. The ontribution of the on�gurations with
x > 12is exponentially small. This means that ignoring a lot of on�gurations will
produ e a very small amount of error in the �nal result of Shannon information. To
quantify this argument we al ulated the amount of error in the evaluation of the
Shannon information if we just keep the on�gurations with ranks up to xm. Suppose
Sh(l, xm) is the ontribution of the on�gurations with all ranks equal or smaller
196
than xm. Then the error of trun ation an be al ulated by E(xm, l) = Sh(l,1)−Sh(l,xm)Sh(l,1)
.
Interestingly we found that the logarithm of the error fun tion is a linear fun tion of
l, see Figure 6.24. In other words
ln E(xm, l) = −λ(xm)l + δ(xm), (6.143)
where λ(xm) is equal to zero and in�nity for xm = 0 and xm = 1 respe tively. λ(xm)
for the other values are shown in the inset of the Figure 6.24. The above formula
shows that one an al ulate Shannon information with a ontrollable a ura y by
ignoring non-important on�gurations. Although the above trun ation method help
to al ulate the Shannon information with good a ura y (espe ially the oe� ient
of the linear term α) it is still not good enough to al ulate the oe� ient of the
logarithm with ontrollable pre ision.
0 5 10 15 20 25 30k
0
0.5
1
1.5
2
2.5
3
Shk(l
)
l=14l=26
Figure 6.23: (The ontributions of di�erent ranks k in the Shannon information for
two sizes l = 14 and 26.
XX hain
The Shannon information of the subsystem in the XX- hain is already dis ussed
in [256℄ and based on numeri al results it is on luded that the equation (6.142)
is valid with β = 18whi h is onsistent with the onje ture in [255℄. Here we just
197
0 5 10 15 20 25 30l
-6
-5
-4
-3
-2
-1
0
lnε(
x m,l)
xm
=1/2
xm
=1/3
xm
=1/4
0 0.1 0.2 0.3 0.4 0.5x
m
0
0.05
0.1
0.15
0.2
0.25
λ(x m
)
Figure 6.24: The error E(xm, l) in the evaluation of Shannon information oming
from the trun ation at the rank k = xml. Inset: −λ(xm) with respe t to xm.
omment on the ontribution of di�erent ranks whi h shows very di�erent behavior
from the transverse �eld Ising hain ase. First of all, as we dis ussed in the previous
se tion when the external �eld is zero the only on�gurations that de ay exponentially
are those that respe t the half �lling stru ture of the total system. The rest of the
on�gurations s ale like a Gaussian whi h simply indi ates that their ontribution is
very small in the Shannon information. This is simply be ause the number of these
on�gurations s ale just exponentially. Based on this simple fa t one an anti ipate
that the only on�gurations that an survive in the s aling limit are those with k = l2.
Numeri al results depi ted in the Figure 6.25 indeed support this idea. Although the
k = l2is only one among l possible minor ranks the number of on�gurations with
this rank is highest with respe t to the others whi h an be one of the reasons that
one an obtain a good estimate for the oe� ient of the logarithm in (6.142) with
relatively modest sizes.
198
0 6 12 18 24k
0
2
4
6
8
Shk(l
)
l=12l=24
Figure 6.25: The ontributions of di�erent ranks k in the Shannon information for
two sizes l = 12 and 24 in the XX- hain.
6.5.5 Evolution of Shannon and mutual information after global
quantum quen h
Inspired by experimental motivations, the �eld of quantum non-equilibrium systems
has enjoyed a huge boost in the re ent de ade [269℄. One of the interesting dire tions
in this �eld is the study of information propagation after quantum quen h, see for
example [270, 271, 272, 273, 274, 275℄. Based on semi lassi al arguments and also
using Lieb-Robinson bound it is shown [270, 272℄ that in one dimensional integrable
system one an understand the evolution of entanglement entropy of a subsystem
based on quasi-parti le pi ture [270℄. The argument is as follows: after the quen h,
there is an extensive ex ess in energy whi h appears as quasiparti les that propagate
in time. The quasi-parti les emitted from nearby points are entangled and they are
responsible for the linear in rease of the entanglement entropy of a subsystem with
respe t to the rest. In this se tion, we �rst study the time evolution of formation prob-
abilities and subsequently Shannon and mutual information after a quantum quen h.
199
One an onsider this se tion as a omplement to the other studies of information
propagation after quantum quen h. To keep the dis ussion as simple as possible, we
will on entrate on the most simple ase of XX- hain or free fermions. Following [276℄
onsider the Hamiltonian
H = −1
2
+∞∑
m=−∞tm(c
†mcm+1 + c†m+1cm). (6.144)
The time evolution of the orrelation fun tions in the half �lling are given as
0 5 10 15 20 25Time
0
90
180
270x=025.30
8.75
10x=1/2(a)10.25
8
12
16
20x=1/2(b)13.58
Figure 6.26: (The evolution of logarithmi formation probability of di�erent on�g-
urations with respe t to time t after quantum quen h. The size of the subsystem is
taken l = 20.
Cmn(t) = in−m∑
jl
ij−lJm−j(t)Jn−l(t)Cjl(0), (6.145)
where, J is the Bessel fun tion of the �rst kind. Here we onsider the dimerized initial
onditions with t2m = 1 and t2m+1 = 0. The dimerized nature of the initial state will
help later to onsider di�erent possibilities for the initial Shannon mutual information.
Then at time zero we hange the Hamiltonian to tm = 1 and let it evolve. The time
evolution of the orrelation matrix is given by [276℄
Cmn(t) =1
2
(
δm,n +1
2(δm+1,n + δm−1,n) + e−iπ
2(m+n) i(m− n)
2tJm−n(2t)
)
.(6.146)
200
To al ulate the time evolution of the probability of di�erent on�gurations one
0 2 4 6 8 10 12 14Time
6
9
12
15
13.2186.628l=20l=10
Figure 6.27: (The evolution of Shannon information of a subsystem with di�erent
sizes with respe t to time t after quantum quen h. The full lines are the equation
(6.147). The saturation points t∗ = l2are marked by verti al arrows.
just needs to use the above formula in (6.138). The results for few on�gurations
are shown in the Figure 6.26. Of ourse sin e the sum of all the probabilities should
be equal to one some of the probabilities in rease with time and some de rease. All
the probabilities hange rapidly up to time t∗ ≈ l2and after that saturate. One
an also simply al ulate the evolution of the Shannon information with the tools of
previous se tions. In Figure 6.27, we depi ted the evolution of Shannon information of
a subsystem with respe t to the time t. The numeri al results show an in rease in the
Shannon information up to time t∗ ≈ l2and then saturation. This is similar to what we
usually have in the study of the time evolution of von Neumann entanglement entropy
after quantum quen h [270℄. However, one should be areful that in ontrast to the
von Neumann entropy the Shannon information of the subsystem is not a measure
of orrelation between the two subsystems. In addition, the in rease in the Shannon
information of the subsystem is not linear as the evolution of the von Neumann
entanglement entropy. Our numeri al results indi ate that apart from a small regime
201
at the beginning the Shannon information in reases as
Sh(l) = altb − dt t < t∗ (6.147)
where b ≈ 0.15(2) and a and d are positive l independent quantities.
0 10 20 30 40 50 60 70Time
0
0.0012
0.0024d=60
0
0.0015
0.003 d=400
0.003
0.006 d=20
Figure 6.28: (The mutual information between a pair of dimers lo ated at distan e
d with respe t to time.
0 5 10 15 20Time
0
0.1
0.2
0.3
0.4
0.5
0.6
I(l,l
)
l=12(a)l=20(a)l=12(b)l=20(b)
Figure 6.29: The evolution of mutual information between two adja ent subsystems
in two di�erent ases: when at the boundary between the two subsystems there is
a dimer, ase a and when there is no dimer, ase b. In the �rst ase the mutual
information starts from a non-zero value but in the se ond ase it starts from zero.
To study the time evolution of orrelations, it is mu h better to study another
quantity, Shannon mutual information of two subsystems. To investigate this quantity
we �rst studied the time evolution of Shannon mutual information of a ouple of
202
dimers lo ated far from ea h other. The results depi ted in the Figure 6.28 show
that the Shannon mutual information of the dimers are zero up to time t∗ ≈ l2and
after that in reases rapidly and then again de ays slowly. This pi ture is onsistent
with the quasiparti le pi ture. The two regions are not orrelated up to time that
the quasiparti les emitted from the middle point rea h ea h dimer[277℄. However, the
similarity between the evolution of the von Neumann entropy and mutual Shannon
information ends here. To elaborate on that we onsider the mutual information of
two adja ent regions with sizes
l2. Be ause of the dimerized nature of the initial
state there are two possibilities for hoosing the subsystems: at time zero at the
boundary between the two subsystems there an be a dimer or not. In the �rst ase
at time zero the Shannon mutual information between the two subsystems is not
zero but in the se ond ase it is zero. In the se ond ase naturally one expe ts an
overall in rease in the mutual information but in the �rst ase a priory it is not
lear that the mutual information should in rease or de rease. In the Figure 6.29,
we have depi ted the results of the numeri s for the two adja ent subsystems for
di�erent sizes. The numeri al results show that for the un orrelated initial onditions
the Shannon mutual information �rst in reases rapidly and then it de ays and �nally
saturates at time t∗ = l2. In the orrelated ase, we have overall de ay in the mutual
information and �nally the saturation again at time t∗ = l2. This behavior is very
di�erent from the quantum mutual information of the same regions whi h for the
onsidered initial states �rst in reases linearly and then saturates at time t∗ = l2. The
interesting phenomena is that after a short initial regime that the Shannon mutual
information is initial state dependent the system enters to a regime that this quantity
is ompletely independent of initial state and it de reases "almost" linearly and then
saturates. This an be also easily seen from the equation (6.147), where we an simply
203
drive
I(l, l) = −dt t < t∗. (6.148)
The saturation regime is independent of the size of the subsystem, this is simply
be ause in the equlibrium regime the Shannon mutual information follows the area-
law [250℄ and so it is independent of the volume of the subsystems.
6.5.6 Con lusions
In this paper, we employed Grassmann numbers to write the probability of o ur-
ren e of di�erent on�gurations in free fermion systems with respe t to the minors of
a parti ular matrix. The formula gives a very e� ient method to study the s aling
properties of logarithmi formation probabilities in the riti al XY- hain. In parti -
ular, we showed that the logarithmi formation probabilities of rystal on�gurations
are given by the CFT formulas for the riti al transverse �eld Ising model. This is
he ked by studying the probabilities in the in�nite and �nite (periodi and open
boundary onditions) hain. In the ase of riti al XX- hain whi h has a U(1) sym-
metry just the on�gurations with x =nf
πfollow the CFT formulas. The rest of the
on�gurations de ay like a Gaussian and do not show mu h universal behavior. We
also studied the Shannon information of a subsystem in the transverse �eld Ising
model and XX- hain. In parti ular, for the Ising model, we showed that in the s aling
limit just the on�gurations with a high number of up spins ontribute to the s aling
of the Shannon information. In prin iple, if one onsiders all the on�gurations, with
our method one an not al ulate the Shannon information with lassi al omputers
for sizes bigger than l = 40 in a reasonable time. However, if one admits a ontrollable
error in the al ulation of Shannon information it is possible to hire the results of
se tion V to go to higher sizes. It would be very ni e to extend this aspe t of our
204
al ulations further to al ulate the universal quantities in the Shannon information
with higher a ura y. For example, one interesting dire tion an be �nding an expli it
formula for the sum of di�erent powers of prin ipal minors of a matrix. This kind of
formulas an be very useful to al ulate analyti ally or numeri ally the Rényi entropy
of the subsystem.
Finally, we also studied the evolution of formation probabilities after quantum
quen h in free fermion system. In this ase, we prepared the system in the dimer
on�guration and then we let it evolve with homogeneous Hamiltonian. The evolution
of Shannon information of a susbsytem shows a very similar behavior as the evolution
of entanglement entropy after a quantum quen h. Espe ially our al ulations show
that the saturation of the Shannon information of the subsystem o urs at the same
time as the entanglement entropy. This is probably not surprising be ause the t = t∗
is also the time that the redu ed density operator saturates.
It will be very ni e to extend our al ulations in few other dire tions. One dire tion
an be investigating the evolution of mutual information after lo al quantum quen hes
as it is done extensively in the studies of the entanglement entropy [278, 279, 280℄.
The other interesting dire tion an be al ulating the same quantities in other bases,
espe ially those bases that do not have any dire t onne tion to CFT.
6.5.7 Additional details: Shannon information for riti al transverse-
field Ising model
In this subse tion, we will provide more details regarding the Shannon information of
transverse riti al Ising hain and XX hain. The data regarding Shannon information
for a subsystem with length l up to l = 39 is listed in the Table A1. Having the
data, we he ked many di�erent fun tions with di�erent parameters to study the
oe� ient of the logarithm. Needless to say in reasing the possible parameters an
205
make a di�eren e in the �nal result. In [256℄ the results �tted to
Sh(l) = αl + β ln l +5∑
n=1
bnln
+ δ (6.149)
show that the best value is β = 0.060. This an be also he ked using the data provided
in the TableA1. It is worth mentioning that one an also get reasonable results using
the data up to l = 40 for some formation probabilities (not all) if we onsider extra
terms
∑5n=1
bnlnin the �tting pro edure. Although we found that the equation (6.149)
is the most stable �t with the least standard deviation based on our results in the
main text we found it is hard to ex lude the term
ln llbe ause it is present in all the
on�gurations studied there. If one in ludes this term and does not add the terms
∑5n=1
bnlnthe β oe� ient will be 0.0617. If one keeps all the terms
∑5n=1
bnlnthe result
will be β = 0.060. The �nal on lusion is that as far as one justi�es the presen e of
the terms
∑5n=1
bnln
in the Shannon information formula the best value for β with the
urrent available data is 0.060.
206
l Shannon l Shannon
1 0.473946633733778 21 9.094267377324401
2 0.925441055292197 22 9.520258511384927
3 1.367970612016317 23 9.946131954351737
4 1.805854593071358 24 10.37189747498959
5 2.240889870728481 25 10.79756367448224
6 2.674003797245196 26 11.22313816533366
7 3.105734740754158 27 11.64862771729621
8 3.536422963908594 28 12.07403837729498
9 3.966297046625437 29 12.49937556879184
10 4.395517906953372 30 12.92464417475445
11 4.824203084194648 31 13.34984860684562
12 5.252441034545332 32 13.77499286454422
13 5.473946633733777 33 14.21026702317442
14 6.107833679024358 34 14.62511508430369
15 6.535085171703405 35 15.05009939651494
16 6.962089515106671 36 15.47503630258492
17 7.388875612253789 37 15.89992835887542
18 7.815467577831834 38 16.32477792018708
19 8.241885740227190 39 16.74960160654153
20 8.668147394540807
Table A1: Shannon information al ulated for sizes l = 1, 2, ..., 39.
207
7
Con lusion
�Learn from yesterday, live for today, hope for tomorrow. The important
thing is not to stop questioning.�
� Albert Einstein
With the progressing development in experimental te hniques su h as pump-probe
spe tros opy, designing and manufa turing two-dimensional materials, the advent of
the highly tunable ultra old opti al latti es an array of ions and atoms, investigating
the dynami s of strongly orrelated systems have be ome one of the most a tive
and e�e tive parts of the ondensed matter physi s. Motivated by numbers of re ent
experiments [8, 16, 20, 25, 30℄, in this thesis, we have used various omputational
and exa t formalisms su h as DMFT, sum rules, nonequilibrium Green's fun tion and
quantum �eld theory to investigate a wide range of systems su h as high Tc super-
ondu tors, quantum ele troni devi es, mixtures of fermioni and bosoni atoms in
ultra old opti al latti es, quantum spin hains, and quantum simulators. Below, we
will provide a review of the main results of ea h proje t and brie�y mention some of
the remaining open questions for future dire tion.
In hapter 3, we �rst, derived a general formalism for the nth derivative of a time-
dependent operator in the Heisenberg representation and we employ this identity to
�nd the spe tral sum rules for retarded Green's fun tion up to the third moment
and onsequently, we used the results to obtain the �rst moment of the self-energy
208
in the normal state. These sum rules provide a powerful self- onsisten y he k for
the omputational or experimental results, and it may provide insight into the re ent
experiment of the e�e t of the pump on the weakening of the ele tron-phonon inter-
a tion. In fa t, our results in the atomi limit suggest that the hanges in spe tral
weight in di�erent time intervals ompensate ea h other and the integral over the
frequen y remains onstant as it is expe ted from the sum rules. Furthermore, our
results agree with a re ent omputational study in Ref. [68℄ whi h suggests that one
an dete t the weakening in tr-Arpes of the pump-probe experiment as a result of
the indu ed pump without any hange in the ele tron-phonon intera tion. In order
to investigate the pump e�e t in the super ondu ting state, the next step would be
generalizing the sum rules into the super ondu ting state. One possible approa h is
using the Nambu-Gorkov formalism in whi h the normal Green's fun tion does not
hange, but one needs to modify the sum rules for the anomalous Green's fun tion.
Furthermore, there has been a few studies re ently, indi ating an enhan ed transient
ele tron-phonon oupling in whi h the me hanism for su h enhan ement still remains
un lear [94℄. Re ently, the role of nonlinear ele tron-phonon oupling in the indu e-
ment of transient ele tron-phonon oupling has been suggested in Ref. [95℄. So, one
may examine the sum rules for Hubbard-Holstein model with an addition of extra
nonlinear ele tron-phonon oupling.
Then, we devoted hapter 4 to study the nonlinear response of a multilayer devi e
by obtaining the urrent-voltage pro�le. The devi e that we study onsists of a single
barrier with a number of metalli leads on both sides whi h are atta hed to the left
and right bulk. We use the urrent-voltage biasing previously proposed by Freeri ks
in Ref [17℄ to obtain the urrent and �lling a ross the devi e. Due to s attering in
barrier region, the urrent and �lling drop passing through the barrier. Then, in order
to sustain the urrent and �lling onserved, we apply a lo al ele tri �eld a ross the
209
barrier and furthermore, we have developed an optimization pro edure to obtain the
best values for the ele tri �eld. We report the results for single barrier plane after
applying an ele tri �eld. Our results show that in fa t, the optimization pro edure
improves the urrent and �lling onservation. Then, we report the urrent-voltage
pro�le for metalli U = 1 and insulating U = 4 phase. As a next step, one may add
more insulating planes in the barrier region and improve the results of the urrent
and �lling pro edure by taking into a ount more parameters.
In hapter 5, we address the problem of dete ting quantum ordering su h as anti-
ferromagneti ordering in the ultra old atoms as it requires a hieving very low tem-
peratures whi h are not a essible with urrent ooling te hniques. Based on the
DMFT solution of the Fali ov-Kimball model [18℄, we have proposed a new method
to enhan e the riti al temperature of quantum ordering by in reasing the degenera y
of the light parti le in the mixture of light-heavy mixtures. Our method suggests that
the enhan ement for N = 3 is 1.4Tc, whi h is lose to the urrent a hieved tempera-
ture, and for N = 4, the enhan ement will be ompletely a essible with the urrent
te hnique. We further, proposed new mixtures su h as Y b−Cs and Sr−Cs as some of
the promising mixtures with whi h one may dete t the enhan ement of the quantum
order.
Finally, in hapter 6, we study dynami al properties of the XY hain as one of the
most studied quantum spin hains. This model provides a strong platform both for
the theoreti al study of dynami s of quantum system out of equilibrium and pra ti al
point of view in whi h an be realized in trapped ion or other experimental realization
whi h play an important role in the simulation of quantum systems [25, 30℄. We
have studied di�erent dynami al properties su h as the probability of revivals, the
light one velo ity in whi h we analyti ally derive an expression that is dependent
on the initial state [32, 33℄. We have also studied the formation probability of the
210
so- alled rystal on�gurations and the Shannon information of the transverse Ising
hain [31℄. Sin e all of our al ulations have been performed in a omputational basis,
it has advantages that are very ompatible with urrent experimental setup su h as
trapped ions and a one-dimensional array of Rydberg atoms [25, 30℄. We have also
al ulated the post-measurement entanglement entropy and full ounting statisti s
whi h we refer the interested reader to Refs.[34, 35℄.
211
8
Publi ation List of Khadijeh Najafi
1. J. K. Freeri ks, K. Naja�, A. F. Kemper, and T. P. Devereaux, Nonequilibrium
sum rules for the Holstein model, in Femtose ond ele tron imaging and spe tros opy:
Pro eedings of the onferen e on femtose ond ele tron imaging and spe tros opy,
FEIS 2013, 2013 Key West, FL, USA, 2013.
2. K. Naja�, MM Ma±ka, K Dixon, PS Julienne, JK Freeri ks, Enhan ing quantum
order with fermions by in reasing spe ies degenera , Phys Rev A 96 (5), 053621.
3. K. Naja�, MA Rajabpour, Formation probabilities and Shannon information
and their time evolution after quantum quen h in the transverse-�eld XY hain,
Physi al Review B 93, 12:125139, 2016.
4. K. Naja�, MA Rajabpour, On the possibility of omplete revivals after quantum
quen hes to a riti al point, Physi al Review B 96, 1:014305, 2017.
5. K. Naja�, MA Rajabpour, J. Viti, Light- one velo ities after a global quen h
in a non-intera ting model,arXiv:1803.03856.
6. K. Naja�, MA Rajabpour, Entanglement entropy after sele tive measurements
in quantum hains, JHEP, 12:124, 2016.
7. K. Naja�, MA Rajabpour, Full ounting statisti s of the subsystem energy for
free fermions and quantum spin hains, Phys. Rev. B 96, 235109, 2017.
In preparation:
8. K. Naja�, J. A. Ja oby, and J. K. Freeri ks, Nonequilibrium sum rules for the
Holstein-Hubbard model.
212
9. K. Naja� and J. K. Freeri ks, Nonequilibrium urrent-voltage pro�le of a
strongly orrelated materials heterostru ture using non-equilibrium dynami al mean
�eld theory.
10. F. Yang, J. Cohn, K. Naja�, and J. K. Freeri ks, Keldysh-Eigenstate thermal-
ization quantum omputing.
11. K. Naja�, MA Rajabpour, J. Viti, Los hmidt e ho and revivals in the XY-
hain.
213
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