Many-body localization
Arijeet Pal
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Professor David A. Huse
September 2012
c© Copyright by Arijeet Pal, 2012.
All rights reserved.
Abstract
A system of interacting degrees of freedom in the presence of disorder hosts a variety
of fascinating phenomena. Disorder itself has led to the striking pheonomena of local-
ization of classical waves and non-interacting quantum mechanical particles. There
are even phase transitons (like the glass transition) which are driven largely due the
effects of disorder. The work in this dissertation primarily addresses the interplay of
interactions and disorder for the fate of ergodicity in classical and quantum systems.
We specifically question the assumption of ergodicity in a generic, isolated spin-system
with interactions and disorder in the absence of coupling to an external heat bath. Our
results predict the existence of a novel phase transition at finite temperature (even
at ‘infinite’ temperature) in the quantum regime driven by the strength of disorder.
At relatively low disorder in the ergodic phase, an isolated quantum system can serve
as its own heat bath allowing any subsystem to thermalize. While at strong disorder
due to the localization of excitations, the isolated system fails to serve as a heat bath.
In the limit of infinite system size, there is a quantum phase transition between the
two phases with the critical point showing infinite-randomness like scaling properties.
Based on our conventional understanding, the low frequency dynamics of quantum
systems at finite temperature are often describable in terms of an effective classical
model. With this motivation in mind, we also studied the dynamics of an interacting,
disordered classical spin-model. Our results exclude the possibility of many-body
localization in classical systems. A classical many-body system at strong enough dis-
order becomes chaotic under the dynamics of its own hamiltonian thus converging to
thermal equilibrium at long times. Hence, many-body localization is a macroscopic
quantum phenomenon at extensive energies without a classical counterpart.
iii
Acknowledgements
A PhD dissertation is the culmination of five formative years of one’s life. It means
much more than just the hundred odd pages of words and figures which are part of
its final form. There are many stories told and lessons taught which can only be
read in between the lines. And countless people are a part of this experience which
can hardly be captured in this section. First and foremost, I would like to thank
my adviser, Professor Huse. Any words of appreciation will fall short of his actual
contribution to the work and my experience in graduate school. I am sure his acumen
as a physicist has been appreciated by many but as his student, I can vouch that he
is also a wonderful teacher of subtle and intricate ideas. The extreme care with which
he chooses his words is a rare quality to find. Starting from the summer of 2006 as
an undergraduate till now, he has introduced me to the art of research and taken
me through its various rigmaroles quite seamlessly. Without his boundless patience
and constant encouragement, my introduction to a career in physics would be an
entirely different experience. I also appreciate that he introduced me to the beautiful
problem of many-body localization at a very early stage in graduate school. Exploring
the cracks and corners of this problem with him has indeed been a great learning and
enriching experience.
I would also like to thank Professor Sondhi for fascilitating my first sojourn to
Princeton as a summer student. Over the years his advice on matters of importance
have been invaluable. Without his support my move into Princeton, and now as
I leave the place, would be quite a different story. I hope to continue working on
problems we have identified and look forward to further scientific interactions. The
experience of working in Professor Hasan’s lab as a beginning graduate student gave
a really good perception of the nature of research in Condensed matter physics. I
would like to thank him for giving me the chance to work on topological insulators for
my experimental project when the field was in its infancy. I also appreciate the advice
iv
and support he provided on taking the next step outside graduate school. I would
also like to thank Vadim for his support and useful suggestions not just on physics
but academic-life in general. I always eagerly looked forward to his trips from New
York and the interesting discussions they led to. I hope to continue this discourse in
the future.
At a personal level, graduate school has given me some great friends to cherish
for the years to come. The many hours spent in Jadwin would have seemed much
longer without discussions with Hans, Miro, BingKan, Anand, Anushya, Bo, Chris,
Charles, Sid, Hyungwon on physics and other random thoughts. Then there was also
the life outside Jadwin. Sharing the sentiments of winning and losing on the soccer
field with Pablo, John, Eduardo, Pegor and many others can hardly be replicated
outside the sports field. Finding the right tennis partners in Richard, Bo and Hans
helped me fulfil my childhood desire to play the sport. I also spent 4 memorable
years in 3V Magie with Darren and Ketra who were my partners in crime on many
occassions from Bollywood choreography at Dbar to cooking meals for friends. It was
always comforting to know that on occassions when I needed a ‘break ’ from physics,
I was only a walk away from interesting conversations over lunch or coffee with Rohit
D, Rohit L, Anna, Radha, Vinay, Franziska, Rotem and Udi. Last but definitely
not the least, my impressions on Princeton will be far from complete without Sare’s
companionship and her always being there when I needed.
Then there are the people outside Princeton who had as much influence. On this
occassion, I would also like to thank my mentors in Boys’ High School, Dr. Aditi
Mukhopadhyaya and St.Stephen’s College, Dr. Bikram Phookun. They channeled
my youthful exuberance and gave form to my random ideas. The late-night, light-
hearted conversations with Ranit, one of my closest friends from yesteryears, gave a
lot more perspective on ‘life’ than we had expected! I literally cannot describe in
words the contribution of my parents, Sripati and Paulina and, brother and sister-
v
in-law, Shubhojit and Manpreet. Without their efforts, reaching this stage of my life
would not just be impossible but inconceivable. Had it not been for the train journey
from Guwahati to Allahabad, I would very well be telling a different story.
vi
To my parents.
vii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction 1
1.1 Scaling theory for Anderson transition . . . . . . . . . . . . . . . . . 6
1.2 Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Ergodicity (Thermalization) . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Classical Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Berry’s conjecture (Quantum Chaology) . . . . . . . . . . . . 15
1.3.3 Eigenstate Thermalization Hypothesis . . . . . . . . . . . . . 16
1.4 Disorder and Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Variable range hopping . . . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Fermi glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.3 Many-Body Localization: Basko, Aleiner, Altshuler (BAA) . . 23
1.5 Possible signatures in experiments . . . . . . . . . . . . . . . . . . . . 27
1.6 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 The quantum many-body localization 32
2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
viii
2.2 Single-site observable . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Transport of conserved quantities . . . . . . . . . . . . . . . . . . . . 41
2.4 Energy-level statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Spatial correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.7 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Energy transport in disordered classical spin chains 68
3.1 Classical many-body localization? . . . . . . . . . . . . . . . . . . . . 68
3.2 Model, trajectories and transport . . . . . . . . . . . . . . . . . . . . 72
3.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.3 Finite-size and finite-time effects . . . . . . . . . . . . . . . . 77
3.3 Results: Macroscopic diffusion . . . . . . . . . . . . . . . . . . . . . . 79
3.3.1 Current autocorrelations . . . . . . . . . . . . . . . . . . . . . 79
3.3.2 DC conductivity: extrapolations and fits . . . . . . . . . . . . 80
3.4 Further explorations and outlook . . . . . . . . . . . . . . . . . . . . 86
3.5 Finite size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6 Chaos amplification of round-off errors . . . . . . . . . . . . . . . . . 89
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4 Conclusion and Future outlook 93
4.1 Question of Universality . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Topological order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Bibliography 98
ix
List of Tables
2.1 Properties of the ergodic and localized phases . . . . . . . . . . . . . 36
3.1 Estimates of the D.C. conductivity κ . . . . . . . . . . . . . . . . . . 82
x
List of Figures
1.1 Disorder in phosphorus doped silicon . . . . . . . . . . . . . . . . . . 2
1.2 ESR measurements of p-doped silicon . . . . . . . . . . . . . . . . . . 3
1.3 A typical diagrammatic term in the locator expansion . . . . . . . . . 6
1.4 Scaling function for single-particle localization . . . . . . . . . . . . . 9
1.5 Non-equilibrium initial conditions . . . . . . . . . . . . . . . . . . . . 12
1.6 Chaotic and regular trajectories . . . . . . . . . . . . . . . . . . . . . 14
1.7 Variable-range hopping between localized energy-levels . . . . . . . . 19
1.8 BAA phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9 Probability distribution of the relaxation rate (Γ) . . . . . . . . . . . 25
1.10 Density profile of Anderson-localized condensate . . . . . . . . . . . . 28
1.11 Schematic diagram of coupled SC qubits in microwave resonator . . . 29
1.12 I-V characteristic of InOx thin film . . . . . . . . . . . . . . . . . . . 30
2.1 Phase diagram of many-body localization transition . . . . . . . . . . 35
2.2 Decoupled precessing spins . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Difference of m(n)iα between adjacent eigenstates vs L . . . . . . . . . . 39
2.4 Probability distribution of m(n)iα . . . . . . . . . . . . . . . . . . . . . 40
2.5 Probability distribution of |m(n)iα −m
(n+1)iα | . . . . . . . . . . . . . . . 41
2.6 Dynamic part of initial spin-polarization . . . . . . . . . . . . . . . . 43
2.7 Probability distribution of r(n) . . . . . . . . . . . . . . . . . . . . . . 45
2.8 Ratio of adjacent energy gaps . . . . . . . . . . . . . . . . . . . . . . 48
xi
2.9 Spin-spin correlation (Czznα) in energy eigenstates . . . . . . . . . . . . 49
2.10 Measure of anti-correlation at long distances . . . . . . . . . . . . . . 51
2.11 Probability distributions of lnCzznα(i, i+ L/2) . . . . . . . . . . . . . . 52
2.12 Scaled width of the distribution of lnCzznα(i, i+ L/2) . . . . . . . . . . 53
2.13 Level spacing and ET in the ergodic and localized phases . . . . . . . 55
2.14 Contribution to the dynamic fraction from adjacent eigenstates (P (n)α ) 56
2.15 Probability distribution of P (n)α . . . . . . . . . . . . . . . . . . . . . 58
2.16 System + bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.17 Entanglement entropy of energy eigenstates vs L/2 . . . . . . . . . . 65
2.18 Entanglement spectrum of energy eigenstates . . . . . . . . . . . . . . 66
3.1 Diffusion constant vs relative spin-spin interaction strength . . . . . . 71
3.2 Short time behavior of current autocorrelation C(t) . . . . . . . . . . 80
3.3 Current autocorrelations on medium time scales . . . . . . . . . . . . 81
3.4 Long-time tail of the current autocorrelation function . . . . . . . . . 82
3.5 Estimation of the exponent of long-time tails . . . . . . . . . . . . . . 83
3.6 Long time tails in terms of η(t) ≡ κL(t)− κL(2t) . . . . . . . . . . . . 84
3.7 Variation of the D.C. conductivity κ(t) vs t . . . . . . . . . . . . . . . 85
3.8 Rescaled κ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.9 Long time limit of κ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.10 Finite-size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.11 Round-off effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
xii
Chapter 1
Introduction
Understanding the role of disorder in natural phenomena has been one of the most
puzzling questions in physical sciences which has received its requisite attention only
in the past few decades. Given the ubiquitous nature of disorder, it is important to
understand if disorder fundamentally changes our predictions which are usually based
on idealized and clean theoretical models. Although disorder is present on all scales,
it is interesting to note, its significance for current empirical observations is probably
most pronounced in condensed matter physics. In condensed matter physics itself,
this paradigm was brought to the forefront by the seminal work of P. W. Anderson
(1958) [1], where he was able to show that the quantum mechanical wavefunction of a
non-interacting particle is exponentially localized at all energies for sufficiently strong
but finite disorder. As the localized states do not carry currents over macroscopic
length scales hence, this had dramatic consequences for the transport properties of
a material. Thus, a complete description of transport in solid-state systems requires
taking into consideration effects due to disorder on an equal footing. At the time
of Anderson’s paper, this result was a paradigm shift from the conventional way of
1
thinking. It took some time before the community grew to realize the significance of
this work. Neville Mott and David Thouless were probably some of the first people to
understand the impact of this work and found its connection to physical realizations
of metal-insulator transitions.
Anderson’s theoretical work at that time was motivated by experiments performed
in George Feher’s group at the Bell Laboratories [2–4]. They were particularly in-
terested in the phenomenon of spin relaxation in phosphorus doped silicon using
electron spin resonance techniques. The electronic wavefunction localized on a phos-
phorus atom in doped silicon has a Bohr radius of ∼ 20 Å. The electron in this state
felt the random environment of Si29 defects in Si28. The relaxation time of the spins
on these donor atoms was of the order of minutes as opposed to milliseconds which
was predicted by theoretical calculations based on Fermi Golden Rule taking into
account phonons and spin-spin interactions.
Figure 1.1: The electron on the donor phosphorus impurities are bound in a hydro-genic wavefunction with a large Bohr radius. The Si environment is slightly impuredue to the presence of Si29. (Figure from [5])
2
At low dopant concentration, the electron spin resonance signal is inhomoge-
neously broadened while as the dopant concentration increased the signal is homoge-
neously broadened signifying the localization transition.
Figure 1.2: Electron spin resonance signal for different phosporus concentration acrossthe Anderson transition. (a) In the localized phase (d) In the delocalized phase. (Plotfrom [3])
Anderson was considering energy transport in this spin system and conceptualized
it as a collection of interacting spins in a disordered environment. This is in general
an interacting (nonlinear) problem. In order to capture the essential physics he made
the “ linear ” approximation which made the problem tractable compared to the fully
3
interacting case. The hamiltonian for this simplified model can be written as
H =∑
i
Eini +∑
i,j
Vij c†i cj + h.c. (1.1)
where c†i is the creation operator on a localized state at site i. It is important
to note that when this hamiltonian is expressed in terms of spin operators through
the Jordan-Wigner transformation, it amounts to neglecting the Szi Szj term in the
hamiltonian. In terms of annhilation-creation operators this implies that the model
is linear and there are no interactions between the occupied states on the sites. In a
clean system with uniform nearest neighbor hopping, this hamiltonian is diagonalized
by Bloch states.
ψ(~k) =1√N
∑
i
ei~k·~ri|i〉 (1.2)
A perturbation theory setup in three dimensions or higher using the bloch states
as the unperturbed part while treating the disorder as a perturbation though qual-
itatively changes the transport from ballistic to diffusive but the states still remain
delocalized. While if the perturbation theory is performed in the localized limit
where the unperturbed states are eigenstates of H0 =∑
iEini while the hopping is
a perturbation (technically referred to as the locator expansion), the localized states
remain stable in the presence of the hopping terms at strong disorder. In one or two
dimensions all states are localized for arbitrarily small disorder (with time-reversal
symmetry and without spin-orbit coupling).
Let |i〉 be a localized state at site i. A general single-particle state can be repre-
sented as
ψ =∑
i
ai|i〉
4
The dynamics of the amplitudes are governed by the Schrodinger time-evolution.
iaj = Ejaj +∑
k 6=j
Vjkak
If we initialize our system with the particle localized at site i, the question of
localization is related to the t → ∞ limit of the probability amplitude ai. If the
system is localized at that energy the probability at site i remains finite while in the
delocalized state it diffuses and decays to zero. This can be formulated in terms of
the Green’s function
Gij(t) = 〈i|eiHtc†je−iHtci|i〉
Thus, the problem becomes amenable to a perturbative treatment where the basis
states are the eigenstates of H0 while the hopping terms are treated perturbatively.
This gives rise to many nuances. Firstly, the unperturbed energies are randomly dis-
tributed. So to maintain conservation of energy while hopping becomes a probabilistic
statement.
Secondly, the perturbation theory may have resonances which are due to almost
degenerate states with relatively large tunneling between them. These resonances
could affect the convergence of the perturbation theory. This can be addressed by
renormalizing the bare energy levels of the resonating sites self consistently which
mitigates the divergence. Thus, by taking into account terms at all orders of the
perturbation theory, for weak enough hopping and infinite system size, the initial
state has an infinite life time with probability one.
The distinction between the localized and delocalized states at this single-particle
level may require taking into consideration the probability distributions rather than
the averages of the chosen observable (For example, |Gij(t)|2 averaged over disorder
realizations does capture the transition between the localized and diffusive phases
while one needs to evaluate the probability distribution of ImGij(ω) to probe the
5
Figure 1.3: A typical path in the perturbation theory where the particle hops fromone site to the next and performs a quantum coherent random walk. Hopping backand forth between lattice sites 4 and 5 represents resonant tunneling. Figure from [1]
transition). Averaging over the myriad realizations of disorder or over the various
lattice sites washes away the effects of localization and results in a finite decay time.
This is reasonable from a physical point of view as any real system has one particular
realization of disorder.
1.1 Scaling theory for Anderson transition
Since Anderson’s perturbative approach to localization there have been many other
ways of addressing the problem which have uncovered the richness of the problem.
A scaling theory of the single-particle localization transition crucially depends on the
6
idea of Thouless energy. It is a measure of the shift of the eigenenergies (∆E) for
a finite-size system due to changing the boundary conditions from periodic to anti-
periodic. Intuitively speaking, a change in boundary conditions does not appreciably
affect the energy of an eigenstate exponentially localized in the bulk. Hence, the shift
in energy is only exponentially small in system size L.
∆Elocalized ∼ e−L/ξ (1.3)
where ξ is the localization length. On the other hand in the delocalized part of the
spectrum, a change in the boundary condition completely changes the state and the
shift in energy is comparable to the inverse of the diffusion time across the finite-size
sample. In a clean system the change in the boundary conditions on the Hamiltonian
can also be conceived of as a density modulation of wavenumber ∼ πL. Hence, the
Thouless energy in the diffusive phase is inverse of the decay time of the mode and
obeys the following relation.
∆Ediffusive =~
tdiff=L2
D(1.4)
where D is the diffusion constant. Thus, the ratio of the energy shift to the energy
spacing (δW ) is a useful measure of localization proposed by Edwards and Thouless in
1972 [6, 7]. And this was an important ingredient for proposing the scaling theory of
the transition later in the decade by the Gang of Four [8]. The average level-spacing
for single-particle states in a finite system scales as a power-law in the middle of the
band and is given by
δW =
(
dE
dnL−d
)
(1.5)
where d is the dimension of the space and dndE
is the density of states per unit
volume. In order to develop a scaling theory, the eigenstates of a system of linear
7
dimension aL has to be expressible as an admixture of states of ad sub-systems of
linear dimension L. The energy levels within the various subsystems are mixed and
broadened due to tunneling matrix elements at the boundary between adjacent sub-
systems. The crucial insight from Thouless’ work was that the physical quantity
which behaves universally (in the RG sense) is the conductance G defined in units
of e2/~ and not the conductivity (σ). Also, the dimensionless conductance can be
expressed as a universal function parametrized by a single parameter ∆EδW
.
g =G
e2/~(1.6)
As we combine ad blocks of linear dimension L to form a larger block of size aL,
the dimensionless conductance can be expressed as a one parameter scaling function
which satisfies the following renormalization group equation
d ln g(L)
d lnL= β(g(L)) (1.7)
For large conductance (weak disorder) the system must obey Ohm’s law for weak
scattering providing the system with finite conductivity. Hence,
G(L) = σLd−2 (1.8)
Therefore, for g → ∞, β → d− 2. In the other limit of small conductance (strong
disorder), the leading order behaviour at long distances is
g = g0e−αL (1.9)
and β → ln(g/g0). Assuming the beta function is continuous and doesn’t have
singularities, the behaviour of the system can be represented as in Fig. 1.4.
8
Figure 1.4: For d > 2 there is a critical value of conductance gc above which under theRG flow the conductance flows to infinity implying the system behaves as a metal.While for d = 1, 2 for any value of initial conductance g0 the conductance at scale Lrenormalizes to 0 and the system is localized. Figure from [8]
1.2 Random Matrix Theory
Freeman Dyson and Eugene Wigner had studied the spectral properties of random
matrices in the 50s and 60s in an attempt to describe spectral properties of complex
nuclei [9–13]. It was found that when the elements of matrices satisying certain global
symmetries (e.g. orthogonality, unitarity, symplectic) are chosen from a Gaussian
distribution, the eigenvalue distribution of the matrix has universal characteristics1.
1Riemann hypothesis: It is conjectured that ζ function:
ζ(z) =∞∑
n=1
=1
nz=
∏
p∈set of primes
1
1− p−z(1.10)
has zeros lying on the line z = 1
2+ iEi where Ei is real. Interestingly, the statistical fluctuations of
Eis behave like the eigenvalues of a random Hermitian matrix. This has been numerically verifiedfor a large number of zeros of the function. Thus, it might be possible that this abstract mathemat-ical problem is related the quantum chaotic behaviour of a physical system without time-reversalsymmetry.
9
Specifically, behaviour of the level spacing (δ) distribution close to zero is only de-
pendent on the global symmetry of the ensemble of matrices. The disappearance of
the weight of the probability distribution at zero is a signature of spectral rigidity.
An intuitive understanding of this effect can be developed in terms of eigenvalues of
2× 2 matrices.
H11 H12
H∗12 H22
where Hij is the matrix element between two adjacent states in energy. The off-
diagonal part is due to a perturbation coupling the states. The eigenvalues of the
matrix are
E± =1
2
(
H11 +H22 ±√
(H11 −H22)2 + 4|H12|2)
(1.11)
For an orthogonal matrix, H12 is purely real. Therefore, only 2 parameters need
to be tuned for the perturbed energies to be degenerate i.e., H11 = H22 and H12 = 0.
While for a unitary matrix, the number of such real parameters to be tuned is 3.
Thus, close to the s = 0, the level spacing distribution behaves as ∼ sd−1 where
s = Ei+1 −Ei and d is the number of free parameters to be adjusted.
The distribution of level spacing for the Gaussian orthogonal ensemble is approx-
imately given by
PGOE(s) ≈π
2
s
δ2exp
(
−π4
(s
δ
)2)
(1.12)
where δ = 〈s〉 and the angular brackets 〈. . . 〉 imply ensemble averaging. In the
case, where the matrix is sparse which is what physical Hamiltonians correspond
to, the level spacing distribution captures the effects of localization. In the regime
of extendend states, the spectrum of the Hamiltonian experiences level repulsion
10
and shares the same universal properties as the GOE ensemble. The delocalized
eigenstates have finite matrix elements due to the disorder which provides it the
spectral rigidity. While at strong disorder when the states are localized, the off-
diagonal terms are exponentially suppressed in L as two adjacent states in energy
are typically localized far apart in space. This amounts to energy eigenvalues being
completely random without any correlations between them. Hence, the level spacing
has a Poisson distribution in the localized phase.
PPoisson(s) =1
δexp
(
−sδ
)
(1.13)
This argument is rather general and doesn’t assume if the Hamiltonian is that of
a single-particle or many-particles. As long as the eigenstates are localized in real-
space, the off-diagonal matrix elements for the disorder potential will be exponentially
suppressed in the localized phase. In the many-body case the matrix elements Hij
are evaluated between states in Fock space which would be eigenstates of the clean
Hamiltonian.
1.3 Ergodicity (Thermalization)
A classic textbook example used to motivate the idea of ergodicity is a collection
of atoms in a box. The atoms begin from an arbitrary initial state where they are
manifestly out of equilibrium (for example, either localized in a part of the box or
all atoms moving in one direction). How does this gas of atoms reach a steady
state describe by thermodynamic quantities like pressure and temperature whose
statistical fluctuations are governed by equilibrium statistical mechanics? Given the
generality of this phenomena it is quite striking how nascent our understanding is of
this phenomena not just in the quantum but, arguably, to some extent in the classical
realm as well. Though in the first instance the quantum and classical world seem
11
Figure 1.5: Particles in a box start from an initial non-equilibrium distribution
disparate. But if we believe that the description of phenomenon is always quantum
at the microscopic level and the classical description is valid only in a coarse-grained
sense at the macroscopic scale, a complete description of thermalization must have
elements of the classical as well as quantum.
The relevance of localization for many quantum interacting degrees of freedom
to thermalize in the absence of coupling to a heat bath was though not directly ad-
dressed but was indeed recognized in Anderson’s 1958 paper. Ever since, the connec-
tion between localization and ergodicity in a many-body quantum system is relatively
unexplored. How does an isolated system reach a state of thermal equilibrium from
a generic initial condition? Under what conditions can a system serve as its own
heat-bath? This is a question of fundamental importance not just for quantum me-
chanical but also classical systems. The equations of motion governing the dynamics
of observables are time reversal invariant and yet at long times in a statistical sense
an arrow of time emerges. Hence, at long times effective equations of motion become
Markovian. What permits the existence of such a solution to the dynamical equations
remains a question of broad interest relevant to many fields in physics.
12
There are some differences between classical and quantum systems in their theo-
retical treatment which a priori is not clear if they are relevant. Nonetheless, let me
highlight them for the sake of completeness. For a classical system Hamiltonian dy-
namics is completely deterministic. Therefore, if we measure a specific observable at
a fixed time for a fixed initial condition, there will be no fluctuations in this quantity.
Hence, in order to have a reasonable definition which results in a distribution for the
observed quantity, one either needs to average over an ensemble of initial conditions
or perform an average in time. While for a quantum system, even if we start from
the same initial state, quantum dynamics inherently allows for fluctuations. Thus,
several measurements of a specific observable generates a distribution and if the final
state is indeed thermalized, this distribution function should coincide with the Gibb’s
measure. Also, for classical systems phase space is continuous even for a finite system.
The equivalent concept in a quantum system is the Hilbert space spanned by its basis
states. Even for a many paricle system, this space (Fock space) is discrete for any
finite system and the notion of distance (geometry) in this space is very different from
the classical phase space.
1.3.1 Classical Chaos
Classical degrees of freedom in the presence of strong enough non-linearities is ex-
pected to exhibit chaos where at long times the trajectory of a classical system
uniformly visits all points of the phase space on the constant energy surface. The
Kolmogorov-Arnol’d-Moser (KAM) theorem addresses this issue to some extent. A
classical integrable system can be represented in terms action-angle co-ordinates Ii−θi(i = 1, . . . , d) where the action variable is conserved and the angle variable oscillates
at a frequency. Ii is the integral of motion. A particular set of conserved actions Ii
13
defines a d−dimensional torus in angle space (θi).
∂Ii∂t
= 0 (1.14)
θi = 2πωit (1.15)
A simple example of a classically integrable system is a freely propagating parti-
cle in a rectangular box. In this case the integrals of motion are the two orthogonal
components of linear momentum. According to the KAM theorem, for small enough
perturbations from the integrable system, the dynamics of the system still preserves
most/some (depending on the nature of the integrability breaking term) of the in-
variant tori. Hence, the system is not fully chaotic even though some of the action
variables do cease to be conserved i.e., corresponding trajectories become stochastic
at long times. The system is fully ergodic when the total energy is the only integral
of motion.
Figure 1.6: (a) Chaotic trajectory of a particle in a stadium (b) Regular orbit in anintegrable system (Figure from [14])
14
1.3.2 Berry’s conjecture (Quantum Chaology)
Due to the linearity of the Schrödinger equation, the definition of chaos for a quan-
tum mechanical system is subtle. It is not analogous to classical chaos which implies
exponential sensitivity to initial conditions. Some of the subtleties also arise from the
~ → 0 limit of quantum mechanics. This limit in a sense is singular. As opposed to the
case of special relativity where the classical newtonian regime can be reached pertur-
batively in orders of (v/c)2, there is no such correspondence where classical mechanics
can be developed from quantum mechanics perturbatively in ~. Therefore, Michael
Berry defines quantum chaology as “the study of semiclassical, but non-classical, be-
haviour characteristic of systems whose classical motion exhibits chaos". Hence, the
question of quantum chaos is well-posed for the highly excited states of a hamiltonian
which in its classical limit behaves chaotically. One of the nonclassical measures of
quantum chaos is in terms of the statistics of the spectrum of the Hamiltonian for a
bounded system. For a hamiltonian exhibiting quantum chaos the spectrum exhibits
level repulsion while an integrable system has Poissonian statistics. This distinction
is exactly like the difference between the extended and localized phases of a single
particle Anderson model.
The other measure concerns the properties of the wavefunctions of the highly
excited states where the system behaves semiclassically (~ → 0) [15]. For a classically
chaotic system, Berry conjectured that the energy eigenstates when expressed as
a linear combination of the basis states, the amplitudes behave as gaussian random
functions of the quantum number corrsponding to the basis states [16]. For a concrete
example, let us consider the case of a gas of hard spheres of radius a in a box of linear
dimension 2L. The phase space of the classical system is known to be fully chaotic.
In this case, the natural basis states are the momentum eigenstates
Φ~P (~X) = exp(i ~P · ~X) (1.16)
15
where ~P = (~p1, . . . , ~pN) and ~X = (~x1, . . . , ~xN) are the momenta and positions
of the N hard spheres. The energy eigenstates Ψn can be expressed as a linear
combination of Φ~P (~X) where the wavefunctions vanish outside the domain D. The
domain is defined as
D = (~x1, . . . , ~xN) : −L ≤ xµi ≤ L; |~xi − ~xj | > 2a (1.17)
and
Ψn( ~X) =∑
~P
Cn, ~PΦ~P (~X) (1.18)
The momenta are also constrained by the total energy condition. In the case of
the hard-sphere gas
En =
N∑
i=1
~p2i2m
(1.19)
In the limit of large N and L with the density held fixed, Berry’s conjecture is
equivalent to assuming that Cn, ~P is an uncorrelated gaussian random variable in ~P
only to be limited by the energy of the eigenstate. Also, Cn, ~P and Cm,~P for two differ-
ent eigenstates (n 6= m) are also completely uncorrelated. Hence, a typical eigenstate
at the chosen energy satisfies the statistical properties of a Gaussian ensemble. This
property of the Berry’s conjecture forms the basis of Eigenstate Thermalization hy-
pothesis (ETH).
1.3.3 Eigenstate Thermalization Hypothesis
Assuming that highly excited energy eigenstates satisfy Berry’s conjecture, what can
one say about approach to thermal equilibrium of an isolated quantum system? Let
us start the system in some pure quantum state (ψ(0)) with a well defined average
energy (E) with small fluctuations (∆ ≪ E; this implies that the energy eigenstates
16
contributing to the initial state are within an energy window ∼ ∆ - energy window in a
microcanonical ensemble). For an out-of-equilibrium initial condition the co-efficients
αn have a very detailed and specific arrangement.
ψ(0) =∑
n
αnΨn (1.20)
E =∑
n
|αn|2En (1.21)
∆2 =∑
n
|αn|2(En − E)2 (1.22)
Eigenstate thermalization hypothesis [17, 18] states that the expectation value of
local observables O at long times equilibrates to the microcanonical average. This
equilibrium average can be well represented by just the expectation value in a typical
eigenstate within the microcanonical energy span.
limt→∞
〈ψ(t)|O|ψ(t)〉 =∑
|En−E|≤∆〈Ψn|O|Ψn〉N∆
= typ〈Ψn|O|Ψn〉typ (1.23)
where N∆ is the number of states in the energy window. For any finite t the
expectation value is
〈O〉t =∑
n
|αn|2Onn +∑
n 6=m
α∗mαne
−i(En−Em)tOnm (1.24)
Onm is the matrix element of the operator between eigenstates n and m. The
off-diagonal terms gives rise to dephasing on time evolution. For an ergodic system,
decoherence occurs possibly for two reasons. For generic initial conditions, the com-
plex phases are randomized over time ∼ ~/∆. But for the decoherence of finely tuned
non-equilibrium initial conditions, Onm also tend to zero exponentially in system size.
The decay of the off-diagonal matrix element can be argued based on Berry’s con-
jecture but this behaviour has not been rigorously shown. Once the second term has
17
decayed to zero, the diagonal term survies which still depends on the intial conditions
αn.
It is important to remember some of the limitations of ETH. Intuitively, it must
depend on the time scales of dynamics. In the case of a few degrees of freedom at
equilibrium, the relevant time scale is the time needed to diffusively relax an excitation
in a finite system (τdiff ) while the mean level spacing (δ) governs the time scale of
the fast dynamics [19–22]. Hence, the limit in which ETH is valid is δ ≪ ~/τdiff i.e.,
diffusive relaxation in a finite system occurs at a much shorter time scale compared
to δ. This is exactly the condition which is violated for localized states and results
in the breakdown of ETH. The semiclassical limit also implicitly assumes that the
states under study are highly excited (The mean level spacing is small between high
energy states). Hence, one should expect ETH to break down at low energies for finite
systems. For instance, it is evident that the ground state will not satisfy ETH because
there is a lot of structure in the wavefunction which gives the state its special status
(Also entanglement entropy (to be discussed later) doesn’t satisfy a volume law). It
remains to be explored if the breakdown of ETH indeed means non-thermalization or
is there another mechanism which can still result in thermalization at lower energies.
This suggests that for energies close to the ground state the excitations can only
thermalize by coupling to an external heat bath.
1.4 Disorder and Interactions
Understanding the effects of disorder combined with interactions is a major challenge
in condensed matter physics. Part of the difficulty lies in the fact that there are
few theoretical tools which allow the treatment of disorder and interactions on an
equal footing. The robustness of the Anderson insulator to interactions has perplexed
physicists from the early days of localization. On those lines, Mott had posed the
18
question - What is the result of coupling a single-particle localized insulator to an
external heat bath?
1.4.1 Variable range hopping
Figure 1.7
In the limit of strong enough disorder where all the single particle states near
the fermi-level are localized, two adjacent states in energy are localized far apart
in space. While a heat bath by definition has delocalized excitations for excitation
energies arbitrarily close to zero. In essence the intuitive picture suggests that the
localized states can exchange energy with the heat bath to hop over long distances
from one localized state to another state close in energy thus resulting in conduction
as shown in Fig. 1.7. Assuming that we are dealing with fermions, therefore at low
temperatures there is a well-defined fermi level with long-lived excitations restricted
only close to EF . Lets consider the transport due to the tunneling between two states
with energies E1 > EF and E2 < EF and their localization centers separated by
19
distance R. The probability to produce excitations of order ∼ E1 − E2 = ǫ in the
heat bath goes as exp(−ǫ/kBT ). On the other hand the tunneling matrix element
decays as ∼ exp(−2R/ξ) where ξ is the localization length of the states. Hence, at
leading order the conductivity at low temperatures behaves as
σ(T ) ∼ exp(−2R
ξ− ǫ
kBT) (1.25)
The typical separation between the states is given by
Rtyp =
(
dn(EF )
dE(E1 − E2)
)− 1d
(1.26)
where d is the dimension of the space. Hence, the two terms in the exponential
of Eq. 1.25 have competing dependence on ǫ. Mott argued that the conductivity
will be dominated by states where the tunneling and activation are optimal. Thus,
maximizing over ǫ one gets the result
ǫoptimal ∼ (kBT )d
1+d (1.27)
This gives the conductivity of the system at low temperatures to be
σvariable(T ) = σ0(T ) exp
(
−(
T0T
)1
1+d
)
(1.28)
σ0 and T0 depends on the details of the model. σ0 has weak dependence (power-
law) on T . If one had expected just naively that the transport in the presence of
a bath would be due to activation across the mobility edge, conductivity would be
given as
σactivation(T ) = σ′0 exp
(
−E1 −EckBT
)
(1.29)
20
Ec is the mobility edge of the sample. Mott’s variable range hopping argument
predicts a different exponent for the power in the exponential from the transport
just due to activation across the mobility edge. The difference is more conspicuous
in higher dimensions. The variation from variable range hopping conductivity at
low T to activated transport at high T has been experimentally observed in doped
semiconductors.
1.4.2 Fermi glass
An Anderson insulator without any coupling to a heat bath has zero D.C. conductiv-
ity at zero temperature (At finite temperature if the entire spectrum is localized D.C.
conductivity is still zero). At the same time Mott’s result of finite hopping conductiv-
ity in the presence of the bath beckons the question if electron-electron interactions
can play a similar role in the absence of an external heat bath. Can the electrons
serve as their own heat bath? This was recognized to be an important issue in order
to completely understand transport phenomenon of electrons in semiconductors. An
early work [23, 24] attempted to address this problem using a perturbative analysis
much on the lines of Landau’s Fermi liquid theory. Albeit, the breakdown of transla-
tional invariance due to disorder introduces complications. Let us consider the case
in which the fermi level is below the mobility edge for the single-particle problem.
Hence, all the low energy-excitations are exponentially localized. Much like Ander-
son’s locator expansion predicting single-particle localization, the important quantity
to probe localization in the presence interaction is the behaviour of the imaginary
part of self energy (Im(Σ(ω))) of the Green’s function.
G(ω) =1
ω − H0 − Σ(ω)(1.30)
21
H0 is the non-interacting disordered hamiltonian. One of the crucial ingredients to
setup a perturbative calculation is the basis in which it is performed. It was realized
that working in the basis of single-particle localized states (from now on denoted by
|α〉) helps to keep the perturbative expansion relatively clean. The feature which
makes it particularly useful is that Im(Σ(ω)) tends to zero for ω → µ.
limω→µ
Im(Σαα′(ω)) = 0 (1.31)
In this sense, local excitations have infinite lifetime close to the fermi-level. For
the purposes of decay of excitations due to inelastic scattering, Re(Σ(ω)) acts only
to renormalize the disorder potential. For strong enough disorder this has negligible
effect on the dynamics. In the α-basis, if Im(Σ(ω)) is a continuous function of ω, it
is a signature of decay of the single-particle excitations. On the other hand if it is
finite only at a discrete set of points in ω (pure-point spectrum: dense set of points
with measure zero), it implies that the excitations do not decay via single or many-
particle processes. The behaviour of the self energy has clear features which can be
understood at 1st order in perturbation theory.
In the single-particle channel at frequency ω, if the interactions are short ranged
two states α and α′ can have significant tunneling only if they are localized within a
finite distance off each other. But this imposes energy restrictions as two states nearby
in space are far apart in energy. In this channel both conditions are satisfied only
for a finite number of states and the probability of such an occurence tends to zero.
Hence, they only contribute as poles to the imaginary part of self energy. A similar
argument for the many-particle channel taking into account the available phase space
volume for scattering also contributes only isolated poles to the Im(Σ(ω)). At the 1st
order in perturbation theory the low ω part of spectral support is discrete and the
22
state are bound. This hints that Anderson insulator with zero D.C conductivity is
stable in the presence of short-range interactions.
This treatment of the problem has a few limitations. Firstly, being perturbative
in nature it cannot discount non-perturbative affects at strong interactions. Secondly,
a priori it is not clear if higher-order terms in the perturbative expansion converge
to the same conclusion. The work of Basko, Aleiner and Altshuler [25] addresses at
least one of these issues.
1.4.3 Many-Body Localization: Basko, Aleiner, Altshuler
(BAA)
Following the work of Fleishman, Licciardello and Anderson, there were many efforts
to resolve the question of localization in the presence of interaction but only an
inconclusive picture emerged. But a compelling evidence in favour of many-body
localization was reported by BAA based on a rigourous perturbative treatment where
they summed up Feynman diagrams up to all orders. They made a striking claim that
localization persists upto a finite temperature (or energies of O(N) as temperature is
ill-defined in the many-body insulator). There is a phase transition from the insulating
to the conducting phase.
The treatment of the problem shared many features with Anderson’s locator ex-
pansion. In this case one works in the limit where all single-particle eigenstates are
localized. This is true in d = 1, 2 for arbitrarily weak disorder while for d > 2 above
a critical disorder strength. There is no single-particle mobility edge as the spectrum
is bounded (in the tight-binding limit). The pertubation theory is in the basis of
occupied single-particle states.
|Ψα〉 = |nα0 , . . . , nα2N〉 (1.32)
23
Figure 1.8: Below a critical temperature Tc the D.C. condustivity is strictly equalto zero. At high temperatures, the system becomes ergodic and the delocalized andhas finite conductivity. λ is the strength of the interaction and δζ is the mean levelspacing with the localization volume ζd. (Figure from [25])
nαi(= 0, 1 for spinless fermions) is the occupation number of the eigenstate with
energy Eαiand localized around site ~rαi
with localization length ξ. Ψα is a state in
Fock space corresponding to the occupation numbers. The Hamiltonian in this basis
is expressed as
H =∑
α
ǫαc†αcα +
1
2
∑
αβγµ
Vαβγµc†αc
†β cγ cµ (1.33)
The matrix element Vαβγµ is restricted in energy and space. Due to the exponential
localization the matrix elements are chosen to be finite only for states satifying
|~rα − ~rβ| . ξ
|~rβ − ~rγ | . ξ
...
24
Also, the matrix elements are neglected for states separated in energy by more
than the typical single-particle level spacing within the localization volume (δξ).
|ǫα − ǫγ |, |ǫβ − ǫµ| . δξ
|ǫα − ǫµ|, |ǫβ − ǫγ | . δξ
In this terminology the interaction term generates hops in the Fock space of many-
body states. It plays the same role as tunneling played in the single-particle Anderson
problem. The Anderson problem is studied in fixed dimension d in the limit L→ ∞
while the way this problem is conceived it is the study of localization in the very
high-dimensional Fock space (d → ∞). BAA studied the statistics of the imaginary
part of the single-particle self-energy which governs the quasiparticle relaxation for
a finite size system. The limit L → ∞ is taken at the end of the calculation to be
discussed later.
Γα(ω) = Im(Σα(ω)) (1.34)
Figure 1.9: (a) In the delocalized phase (dashed line) Γ(ǫ) is a continuous function ofenergy. While in the localized phase (solid line), the delta function is smeared out dueto the dissipation added by hand (finite η; at the end the limit η → 0 is taken). (b)The probability distribution of Γ in the loclaized (solid line) and the ergodic phase(dashed line). (Figure from [25])
25
Since, Γ varies from sample-to-sample, a naïve average over disorder realizations
cannot distinguish between the two phases. For a single realization of disorder, in the
delocalized phase Γ is expected to be a smooth function of ǫ in the limit L→ ∞ as the
excitation decays into the continuum. This results in a gaussian probability distribu-
tion for Γ peaked around the mean value. While in the localized phase the spectrum
is expected to be a discrete point spectrum. Hence, the probability distribution is a
delta function at zero. This kind of a singular distribution is difficult to analyse in
a theoretical calculation. Thus, a method originally employed by Anderson for the
single-particle problem serves to be useful. The self-energy is analytically continued
to small imaginary values of ω (Im(ω) = η). We’ll take the η → 0 at the end of the
calculation. Physically, it is as if the system is coupled to an external bath. This
procedure leaves the delocalized phase unaffected. But in the localized case it has the
effect of broadening the δ-function peaks in Γ into Lorentzians thus giving the states
a finite lifetime. In this case, the distribution function develops a tail and the peak
shifts to η from zero as shown in Fig. 1.9 (b). It is important to note that before
taking the limit η → 0 one has to send the system volume to ∞, first. This limiting
procedure has to be treated carefully. η shouldn’t tend to zero faster than the mean
level spacing: η > exp(−Ld). In this case the order of limits are not interchangeable
as for any finite closed system the spectrum is always a sequence of delta functions.
The spectral weight for a single eigenstate (even for an infinite system) in the local-
ized phase is finite only at a discrete set of points but, it is a different set of points
for different eigenstates. For an arbitrary initial state which is a linear combination
of many eigenstates, this procedure must still produce a pure-point spectrum for the
system to be localized.
limη→0
limL→∞
P (Γ > 0) =
finite for a metal
0 for an insulator(1.35)
26
As shown in Fig. 1.9 (b), the probability of Γ > η behaves as ∼ η in the many-
body localized phase. Hence, the probability for any finite decay rate in the insulating
phase tends to zero as the coupling to the bath is switched off. While in the delocalized
phase, the decay rate stays finite in this limit.
1.5 Possible signatures in experiments
Manifestations of single-particle localization have been measured in early transport
experiments in doped semiconductors. As a matter of fact some of the theoretical work
was born out of attempts to understand impurity band conduction of electrons and
holes in doped semiconductors. This was verified by careful transport experiments at
low temperatures. Since, quantum coherence of the wavefunction over large distances
plays a crucial role in localization, its effects are only measureable at low temperatures.
A direct measurement of exponential localization of the single-particle wavefunction
eluded experiments until recently. Since, it is primarily a wave phenomena the first
direct observations were using light or classical photons [26, 27]. Because of the non-
iteracting nature of photons, this is truly an observation of Anderson localization.
In material systems, any description of localization is incomplete without taking
into account electron-elctron or electron-phonon interactions. Since, these are usually
not within an experimentalist’s control in real materials, a direct observation of the
localized wavefuntion remained illusive. With the advent cold atomic systems, where
the strength of the interactions is a finely tunable experimental knob via a Feshbach
resonance, Anderson localization was directly imaged in a system of bosonic atoms
[28, 29]. These experiments were performed in the limit of negligible interactions
due to the low density of the cloud. The disorder potential is realized by an optical
speckle pattern whose strength can be controlled by the intesity of the laser beam.
27
Figure 1.10: Atomic density profile of the BEC cloud. The condensate wavefunctionis exponentially localized with the tails fitted to an exponential. The inset of figure(d) shows that in the absence of random potential (VR = 0) the rms width of cloudgrows linearly while when it is switched on (VR 6= 0) it stops growing after some time.Plot from [28]
These experiments are particularly promising to study phenomena pertaining to
many-body localization not just due to the tunability of parameters but also, the lack
of an external heat bath makes the system extremely isolated to a very good approx-
imation. In real materials even though phonons are often neglected in a calculation
the assumption of thermal equilibrium pre-supposes the existence of a heat bath at
low temperatures. Due to the lack of a physically dynamic lattice in cold atoms or
other degrees of freedom which can serve as a bath in an obvious way, this assumption
may not be a bad approximation for realistic experiments. Thus, the possibility of
28
observing a signature of the many-body localized insulator may not be a far-fetched
one.
Figure 1.11: Superconducting (SC) circuits as qubits: (a) A transimission line res-onator with an array of SC qubits (b) The basis building block for the array - Super-conducting quantum interference device (SQUID). It consists of 2 superconductingislands connected by a tunnel junction. (Figure from [30])
There are other experimental setups which are being developed to emulate phe-
nomena in materials. Most of them are being conceptualized as possible platforms
for quantum information processing. One such system is that of superconducting cir-
cuits in transmission line resonators [30]. In this system, the photons inside the cavity
are prepared to interact with each other by coupling via the superconducting qubits.
Therefore, the dressed photons behave as effective particles with on-site interaction
29
and hopping on a lattice. This can be used to study quantum many-body physics of
photons [31, 32].
The other physical system where the effects of many-body localization may be
relevant for experiments is the problem of disordered superconductivity in two dimen-
sions. Experiments performed on InOx and TiN thin films have given some intriguing
results. These thin films undergo a superconducting transition at low temperatures.
At low temperatures in the presence of a magnetic field I-V characteristics shows
highly non-linear behaviour on applying a D.C. bias voltage [33–36] on the insulating
side.
Figure 1.12: InOx thin film showing a jump in I-V characteristic in the insulatingphase for T = 0.01K. (Figure from [34])
This was explained by invoking the idea of electron overheating. On applying
a voltage, the electrons are excited to a higher temperature than the phonon bath
(Tel > Tph) as the phonon and electron degrees of freedom are decoupled from each
other [36, 37]. Thus, the resistance of the sample is well-behaved in terms of Tel
(assuming the electrons are thermalized) and the apparent non-linearity is due to
the overheating of the electrons compared to phonons (Tph). The jump in the I-V
30
characteristic is reflecting the bistability of the electronic system where on applying
a strong enough voltage the electronic system goes to the metastable state with
the higher Tel. This phenomena hints that under suitable conditions the electronic
degrees of freedom can be decoupled from the phononic heat bath. Thus, making the
realization of a many-body localized insulator more feasible.
1.6 Thesis outline
In Chapter 2, I will be discussing the numerical treatment of localization of the
excited states in the presence of interactions and disorder. We specifically search for
signatures of localization at infinite temperature. In our case the relative strength of
disorder is the only tunable parameter. I will explain in detail the various measures
(motivated by ideas mentioned in the introduction) we used to probe the physics of
many-body localization. Continuing from there I will highlight some of our results
showing the existence and distictions between the ergodic and insulating phases.
This will lead to throwing some light on the properties of the critical point separating
these two phases. Chapter 3 will explore the possibility of realizing the phenomena
of many-body localization in an effective classical model with disorder. I will discuss
the numerical method employed to study the dynamics of the model and results on
energy transport. Phase transitions at finite temperature are mostly described by
effective classical model. Since, the many-localization transition is also at nonzero
temperature in this work we explore if a classical model can capture the transition.
The concluding chapter will discuss the overall picture of this interesting transition
that our work has realized. Also, discuss the prospects for future work on this problem
and other open questions related to it.
31
Chapter 2
The quantum many-body localization
As originally proposed in Anderson’s seminal paper [1], an isolated quantum system of
many locally interacting degrees of freedom with quenched disorder may be localized,
and thus generically fail to approach local thermal equilibrium, even in the limits of
long time and large systems, and for energy densities well above the system’s ground
state. In the same paper, Anderson also treated the localization of a single particle-
like quantum degree of freedom, and it is this single-particle localization, without
interactions, that has received most of the attention in the half-century since then.
Much more recently, Basko, et al. [25] have presented a very thorough study of
many-body localization with interactions at nonzero temperature, and the topic is
now receiving more attention; see e.g. [38–48].
Many-body localization at nonzero temperature is a quantum phase transition that
is of very fundamental interest to both many-body quantum physics and statistical
mechanics: it is a quantum “glass transition” where equilibrium quantum statistical
mechanics breaks down. In the localized phase the system fails to thermally equili-
brate. These fundamental questions about the dynamics of isolated quantum many-
32
body systems are now relevant to experiments, since such systems can be produced
and studied with strongly-interacting ultracold atoms [49]. And they may become rel-
evant for certain systems designed for quantum information processing [50, 51]. Also,
many-body localization may be underlying some highly nonlinear low-temperature
current-voltage characteristics measured in certain thin films [37].
2.1 The model
Many-body localization appears to occur for a wide variety of particle, spin or q-bit
models. Anderson’s original proposal was for a spin system [1]; the specific simple
model we study here is also a spin model, namely the Heisenberg spin-1/2 chain with
random fields along the z-direction [40]:
H =L∑
i=1
[hiSzi + J ~Si · ~Si+1] , (2.1)
where the static random fields hi are independent random variables at each site
i, each with a probability distribution that is uniform in [−h, h]. Except when stated
otherwise, we take J = 1. The chains are of length L with periodic boundary con-
ditions. This is one of the simpler models that shows a many-body localization
transition. Since we will be studying the system’s behavior by exact diagonalization,
working with this one-dimensional model that has only two states per site allows us
to probe longer length scales than would be possible for models on higher-dimensional
lattices or with more states per site. We present evidence that at infinite temper-
ature, β = 1/T = 0, and in the thermodynamic limit, L → ∞, the many-body
localization transition at h = hc ∼= 3.5 ± 1.0 does occur in this model. The usual
arguments that forbid phase transitions at nonzero temperature in one dimension do
not apply here, since they rely on equilibrium statistical mechanics, which is exactly
what is failing at the localization transition. We also present indications that this
33
phase transition might be in an infinite-randomness universality class with an infinite
dynamical critical exponent z → ∞.
Our model has two global conservation laws: total energy, which is conserved for
any isolated quantum system with a time-independent Hamiltonian; and total Sz.
The latter conservation law is not essential for localization, and its presence may
affect the universality class of the phase transition. For convenience, we restrict our
attention to states with zero total Sz.
For simplicity, we consider infinite temperature, where all states are equally prob-
able (and where the sign of the interaction J does not matter). The many-body
localization transition also occurs at finite temperature; by working at infinite tem-
perature we remove one parameter from the problem, and use all the eigenstates from
the exact diagonalization (within the zero total Sz sector) of each realization of our
Hamiltonian. We see no reason to expect that the nature of the localization transition
differs between infinite and finite nonzero temperature (with an extensive amount of
energy in the system),although it is certainly different at strictly zero temperature
[52]. It is important to emphasize that temperature is not a well-defined macroscopic
observable in the many-body localized phase. In cases, where the isolated system
doesn’t thermalize to a mixed state with a Gibb’s distribution at finite temperature
one should consider the parameter being tuned as the energy density. The the "tem-
perature” T can be defined as the temperature that would give that energy density
at thermal equilibrium. Note that this is a quantum phase transition that occurs at
nonzero (even infinite) temperature. Like the more familiar ground-state quantum
phase transitions, this transition is a sharp change in the properties of the many-
body eigenstates of the Hamiltonian, as we discuss below. But unlike ground-state
phase transitions, the many-body localization transition at nonzero temperature ap-
pears to be only a dynamical phase transition that is invisible in the equilibrium
thermodynamics [39].
34
The model we chose to study has a finite band-width. An infinite temperature
limit of such a system is studied by considering states at high energy densities i.e.
eigenstates in the middle of the band. We weigh the observables evaluated from these
states with equal probability in order to study their thermal expectation values. A
practical benifit of working in this limit is the utilization of all the data we acquire
from the full diagonalization of the Hamiltonian which is the most computer time-
consuming part of the calculation.
There are many distinctions between the localized phase at large random field
h > hc and the delocalized phase at h < hc. We call the latter the “ergodic” phase,
although precisely how ergodic it is remains to be fully determined [53]. The dis-
tinctions between the two phases all are due to differences in the properties of the
many-body eigenstates of the Hamiltonian, which of course enter in determining the
dynamics of the isolated system.
Figure 2.1: The phase diagram as a function of relative interaction strength h/J atT = ∞. The critical point is (h/J)c ≈ 3.5. For h < hc the system is ergodic whilefor h > hc, it is many-body localized.
In the ergodic phase (h < hc), the many-body eigenstates are thermal [17, 18,
54, 55], so the isolated quantum system can relax to thermal equilibrium under the
35
Ergodic phase Many-body localized phase• An infinite system is a heat bath • An infinite system is not a heat bath• Many-body eigenenergies obey GOElevel statistics
• Many-body eigenenergies have Pois-son level statistics
• System achieves local thermal equi-librium
• Doesn’t thermally equilibrate- quan-tum glass
• Finite D.C. transport of energy andother globally conserved quantities
• D.C. transport is zero
• Extensive entanglement in eigen-states
• “Area-law” entanglement in eigen-states
• Eigenstate thermalization is true • Eigenstate thermalization is false
Table 2.1: Properties of the ergodic and localized phases
dynamics due to its Hamiltonian. In the thermodynamic limit (L→ ∞), the system
thus successfully serves as its own heat bath in the ergodic phase. In a thermal eigen-
state, the reduced density operator of a finite subsystem converges to the equilibrium
thermal distribution for L → ∞. Thus the entanglement entropy between a finite
subsystem and the remainder of the system is, for L → ∞, the thermal equilibrium
entropy of the subsystem. At nonzero temperature, this entanglement entropy is
extensive, proportional to the number of degrees of freedom in the subsystem.
In the many-body localized phase (h > hc), on the other hand, the many-body
eigenstates are not thermal [25]: the “Eigenstate Thermalization Hypothesis” [17,
18, 54, 55] is false in the localized phase. Thus in the localized phase, the isolated
quantum system does not relax to thermal equilibrium under the dynamics of its
Hamiltonian. The infinite system fails to be a heat bath that can equilibrate itself.
It is a “glass” whose local configurations at all times are set by the initial conditions.
Here the eigenstates do not have extensive entanglement, making them accessible to
DMRG-like numerical techniques [40, 56]. A limit of the localized phase that is simple
is J = 0 with h > 0.
H =L∑
i=1
hiSzi (2.2)
36
Figure 2.2: In the limit J = 0 randomly oriented spins precess around their localmagnetic field hi with frequency Ωi =
hi~
Here the spins do not interact, all that happens dynamically is local Larmor
precession of the spins about their local random fields. No transport of energy or
spin happens, and the many-body eigenstates are simply product states with each
spin either “up” or “down”.
Any initial condition can be written as a density matrix in terms of the many-body
eigenstates of the Hamiltonian as ρ =∑
mn ρmn|m〉〈n|. The eigenstates have different
energies, so as time progresses the off-diagonal density matrix elements m 6= n dephase
from the particular phase relations of the initial condition, while the diagonal elements
ρnn do not change. In the ergodic phase for L→ ∞ all the eigenstates are thermal so
this dephasing brings any finite subsystem to thermal equilibrium. But in the localized
phase the eigenstates are all locally different and athermal, so local information about
the initial condition is also stored in the diagonal density matrix elements, and it is
the permanence of this information that in general prevents the isolated quantum
system from relaxing to thermal equilibrium in the localized phase.
Our goals in this work are (i) to present results in the ergodic and localized phases
that are consistent with the expectations discussed above, and (ii), more importantly,
to examine some of the properties of the many-body eigenstates of our finite-size
37
systems in the vicinity of the localization transition to try to learn about the nature
of this phase transition. Although the many-body localization transition has been
discussed by a few authors, there does not appear to be any proposals for the nature
(the universality class) of this phase transition or for its finite-size scaling properties,
other than some very recent initial ideas in Ref. [45]. It is our purpose here to
investigate these questions, extending the previous work of Oganesyan and Huse [39],
who looked at the many-body energy-level statistics of a related one-dimensional
model. Since the many-body eigenstates have extensive entanglement on the ergodic
side of the transition, it may be that exact diagonalization (or methods of similar
computational “cost” [45]) is the only numerical method that will be able to access
the properties of the eigenstates on both sides of the transition.
2.2 Single-site observable
As a first simple measure to probe how thermal the many-body eigenstates appear
to be, we have looked at the local expectation value of the z component of the spin.
m(n)iα = 〈n|Szi |n〉α (2.3)
at site i in sample α in eigenstate n. If the system of spins does thermalize, the
equilibrium properties of a single spin coupled to the rest of the spins has thermal
behaviour. For each site in each sample we compare this for eigenstates that are
adjacent in energy, showing the mean value of the difference: [|m(n)iα − m
(n+1)iα |] for
various L and h in Fig. 2.3, where the eigenstates are labeled with n in order of their
energy. The square brackets denote an average over states, samples and sites. The
number of samples used in the data shown in this work ranges from 104 for L = 8, to
50 for L = 16 and some values of h.
38
7 8 9 10 11 12 13 14 15 16 17−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
L
ln [
|miα (
n) −
miα (
n+
1) |]
8.0
5.0
3.6
2.7
2.0
1.0
0.6
Figure 2.3: The natural logarithm of the mean difference between the local mag-netizations in adjacent eigenstates (see text). The values of the random field h areindicated in the legend. In the ergodic phase (small h) where the eigenstates are ther-mal these differences vanish exponentially in L as L is increased, while they remainlarge in the localized phase (large h).
In our figures we show one-standard-deviation error bars. The error bars are
evaluated after a sample-specific average is taken over the different eigenstates and
sites for a particular realization of disorder. Here and in all the data in this work
we restrict our attention to the many-body eigenstates that are in the middle one-
third of the energy-ordered list of states for their sample. Thus we look only at
high energy states and avoid states that represent low temperature. In this energy
range, the difference in energy density between adjacent states n and (n + 1) is of
order√L2−L and thus exponentially small in L as L is increased. If the eigenstates
are thermal then adjacent eigenstates represent temperatures that differ only by this
exponentially small amount, so the expectation value of Szi should be the same in
39
these two states for L → ∞. From Fig. 2.3, one can see that the differences do
indeed appear to be decreasing exponentially with increasing L in the ergodic phase
at small h, as expected. In the localized phase at large h, on the other hand, the
differences between adjacent eigenstates remain large as L is increased, confirming
that these many-body eigenstates are not thermal.
Figure 2.4: The probability distribution of local magnetization for L = 14 andh = 0.6, 3.0, 4.0 and 10.0.
In the two phases the probabability distribution of m(n)iα has distinctive behaviour.
At infinite temperature, in equilibrium we expect m(n)iα ≈ 0 which manifests itself as a
peak in the probability distribution around zero as shown in Fig. 2.4. With increasing
strength of disorder, the probability distribution becomes bimodal and peaked around
+12
and −12. The tendency of the eigenstates to have the maximal z-component of
spin at a single site suggests that the energy eigenstates are approximate product
states i.e. |n〉 ∼ | ↑↓ · · · ↑↓↓〉 at strong but finite disorder. In the localized phase, we
40
find the emergence of an effective spin-12
degree of freedom (two-level system) which
is the dressed form of the original spin due to finite interaction strengths.
Figure 2.5: The probability distribution of difference of local magnetization betweenadjacent eigenstates for disorder strengths h = 0.6, 2.0, 3.0 and 6.0. The differentsystems sizes are marked in the legend. For low disorder in the ergodic phase, thedifference distribution becomes more sharply peaked around zero. While in the local-ized phase the distribution develops a bump close to 1. At strong disorder the finitesize effects are negligible.
2.3 Transport of conserved quantities
Thermalization requires the transport of energy. In the present model with conserved
total Sz, it also requires the transport of spin. To study spin transport on the scale
41
of the sample size L, we consider the relaxation of an initially inhomogeneous spin
density:
M1 =∑
j
Szj exp (i2πj/L) (2.4)
is the longest wavelength Fourier mode of the spin density. Consider an initial
condition that is at infinite temperature, but with a small modulation of the spin
density in this mode, so the initial density matrix is
ρ0 =eǫM
†1
Z≈ (1 + ǫM †
1)/Z (2.5)
where ǫ is infinitesimal, and Z is the partition function. The time-evolution of the
initial magnetization is give by
〈M1〉t = Tr(ρtM1) (2.6)
= Tr(e−iHtρ0eiHtM1) (2.7)
=ǫ
Z
∑
n,m
|〈n|M1|m〉|2ei(En−Em)t (2.8)
The initial spin polarization of this mode is then
〈M1〉0 =∑
n
〈n|ρ0M1|n〉 =ǫ
Z
∑
n
〈n|M †1M1|n〉 . (2.9)
If we consider a time average over long times, then the long-time averaged density
matrix ρ∞ is diagonal in the basis of the eigenstates of the Hamiltonian, since a
generic finite-size system has no degeneracies and the off-diagonal matrix elements of
ρ each time-average to zero. As a result, the long-time average of the spin polarization
in this mode is
42
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.50.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
[fα (
n) ]
h
8
10
12
14
16
Figure 2.6: The fraction of the initial spin polarization that is dynamic (see text).The sample size L is indicated in the legend. In the ergodic phase (small h) thepolarization decays substantially under the dynamics, while in the localized phase(large h) the decay is small, and this distinction gets sharper as L increases.
〈M1〉∞ =ǫ
Z
∑
n
〈n|M †1 |n〉〈n|M1|n〉 . (2.10)
Thus for each many-body eigenstate in each sample we can quantify how much it
contributes to the initial and to the long-time averaged polarization. We then define
the fraction of the contribution to the initial polarization that is dynamic and thus
decays away (on average) at long time, as
f (n)α = 1− 〈n|M †
1 |n〉〈n|M1|n〉〈n|M †
1M1|n〉. (2.11)
In the ergodic phase, the system does thermalize, so the initial polarization does
relax away and f (n)α → 1 for L→ ∞. In the localized phase, on the other hand, there
43
is no long-distance spin transport, so f(n)α → 0 for L → ∞. In Fig. 2.6 we show
the mean values of f for each L vs. h. They show the expected behavior in the two
phases (trending with increasing L towards either 1 or 0), and the phase transition
is indicated by the crossover between large and small f that occurs more and more
abruptly as L is increased.
2.4 Energy-level statistics
A spectral distinction between the many-body localized and the ergodic phases is
based on the statistics of energy eigenvalues. The spectral rigidity in the ergodic
phase is reflected in the level repulsion. This repulsion between eigenvalues can be
understood heuristically from the point of view of second-order perturbation theory
where any local perturbation from the Hamiltonian in the ergodic phase leads to
the increase in gap between adjacent eigenvalues for a finite size system. While in
the many-body localized phase a local perturbation has exponentially small overlap
between two adjacent many-body eigenstates thus producing negligible repulsion be-
tween the levels. This picture has been theoretically and numerically substantiated
for single-particle localization [57].
A qualitatively similar finite-size scaling plot to Fig. 2.6 also indicating the phase
transition is obtained by examining the many-body eigenenergy spacings as was done
in Ref. [39], and is shown as Fig. 2.8. We consider the level spacings δ(n)α = |E(n)α −
E(n−1)α |, where E(n)
α is the many-body eigenenergy of eigenstate n in sample α. Then
we obtain the ratio of adjacent gaps as r(n)α = minδ(n)α , δ(n+1)α /maxδ(n)α , δ
(n+1)α , and
average this ratio over states and samples at each h and L. A choice of a two-gap
quantity was made as opposed to the single gap distribution. A single gap distribution
(p(s) = [〈δ(s− δn〈δn〉
)〉]) requires an appropriate definition of the mean gap 〈δn〉. The
mean gap has significant variations over the range of the spectrum. It is indeed
44
Figure 2.7: The probability distribution of r(n) for L = 16 and h = 1.0, 3.6, 4.0 and6.0. For h < hc the spectrum’s finite level-repulsion can be seen as a peak at finiter(n) in the distribution. While for large disorder h > hc, the distribution is peaked atzero.
45
exponentially small in system size in the middle of the band but towards the edge of
the band the gap decreases as a power law in L. Such variations significantly affect
the numerical estimates of the single-gap distribution. Hence, looking at a two-gap
quantity alleviates this concern. Also, a particular realization of the random potential
for small system sizes is comprised of just a few numbers but the energy eigenvalues
are exponential large in number leading to significant correlation between them even
in the localized phase where they are expected to be uncorrelated. It is assumed that
such effects decay as L → ∞.
In the ergodic phase, the energy spectrum has GOE (Gaussian orthogonal en-
semble) level statistics and the average value of r converges to [r]GOE ∼= 0.53 for
L → ∞. This can be verified by looking at an ensemble of large random matrices
and numerically estimating [r]GOE. While in the localized phase the level statistics
are uncorrelated and Poissonian. The distribution of r in the localized phase can be
derived by calculating⟨
δ(
r − minδ(n),δ(n+1)
maxδ(n),δ(n+1)
)⟩
PP (r) =⟨
δ
(
r − minδ(n), δ(n+1)maxδ(n), δ(n+1)
)
⟩
=1
s2
∫ ∞
0
d δn
∫ ∞
0
d δn+1δ
(
r − minδ(n), δ(n+1)maxδ(n), δ(n+1)
)
exp
(
−δn
s
)
exp
(
−δn+1
s
)
s is the mean level spacing. On splitting the integral into two parts, first part has
δ(n) > δ(n+1) while for the second part δ(n) < δ(n+1).
46
PP (r) =1
s2
∫ ∞
0
d δn
∫ δn
0
d δn+1δ
(
r − δn+1
δn
)
exp
(
−δn
s
)
exp
(
−δn+1
s
)
+1
s2
∫ ∞
0
d δn+1
∫ δn+1
0
d δnδ
(
r − δnδn+1
)
exp
(
−δn
s
)
exp
(
−δn+1
s
)
=2
s2
∫ ∞
0
d δn
∫ δn
0
d δn+1δ
(
r − δn+1
δn
)
exp
(
−δn
s
)
exp
(
−δn+1
s
)
Since r ≤ 1 by definition and the peak of the δ-funtion is at rδn therefore, the
integral over δn+1 picks up this value.
PP (r) =2
s2
∫ ∞
0
d δn
[
δn exp
(
−(1 + r)δn
s
)
]
(2.12)
On evaluating the above integral the distribution of r is given by Pp = 2(1+r)2
and
[r]p → 2 ln 2− 1 ∼= 0.39. Note that our model is integrable at h = 0, so will not show
GOE level statistics in that limit, and this effect is showing up for our smallest L and
lowest h in Fig. 2.8.
The crossings of the curves for different values of L in Figs. 2.6 and 2.8 give
estimates of the location hc of the phase transition. Both plots show these estimates
“drifting” towards larger h as L is increased, with the crossings at the largest L being
slightly above h = 3. In both cases this “drifting” is also towards the localized phase,
suggesting the behavior at the phase transition is, by these measures, more like the
localized phase than it is like the ergodic phase. This drift towards the localized
phase could also be due to the choice of the observable. Due to the lack of a finite-
size scaling theory for the transition, our choice of r could have contributions from
irrelevant operators in the RG sense whose finite size effects have a slower decay with
increasing system size. Thus, resulting in the drift of the critical point. This issue
can be be possibly addressed either by changing the observable or the hamiltonian
47
0.5 2.5 4.5 6.5 8.5 10.5 12.50.38
0.4
0.42
0.44
0.46
0.48
0.5
0.52
0.54
[rα (
n) ]
h
8
10
12
14
16
Figure 2.8: The ratio of adjacent energy gaps (defined in the text). The sample size Lis indicated in the legend. In the ergodic phase, the system has GOE level statistics,while in the localized phase the level statistics are Poisson.
which may reverse the direction of the drift and/or reduce the size of the finite-size
effect from the irrelevant operator.
2.5 Spatial correlations
To further explore the finite-size scaling properties of the many-body localization
transition in our model, we next look at spin correlations on length scales of order
the length L of our samples. One of the simplest correlation functions within a
many-body eigenstate |n〉 of the Hamiltonian of sample α is
Czznα(i, j) = 〈n|Szi Szj |n〉α − 〈n|Szi |n〉α〈n|Szj |n〉α . (2.13)
48
0 1 2 3 4 5 6 7 8 9−13
−12
−11
−10
−9
−8
−7
−6
−5
−4
−3
[ln
|C
nα
zz(i,i+
d)|
]
d
810121416
h=0.6
h=3.6
h=6.0
Figure 2.9: The spin-spin correlations in the many-body eigenstates as a function ofthe distance d. The sample size L is indicated in the legend. The correlations decayexponentially with d in the localized phase (h = 6.0), while they are independent ofd at large d in the ergodic phase (h = 0.6). Intermediate behavior at h = 3.6, whichis near the localization transition, is also shown.
Due to periodic boundary conditions, the correlations are only shown up to d =
L/2. For d > L/2 the correlation is identically equal to the correlation at distance
L−d for each eigenstate |n〉. This particular correlation function has a sum-rule due to
the global Sz conservation which proves useful at evaluating the thermal expectation
value of the correlation at infinite temperature. For fixed i (j), the correlation function
summed over j (i) results in zero for every eigenstate and realization of disorder.
∑
j
Czznα(i, j) = 0 . (2.14)
In Fig. 2.9 we show the mean value [ln |Czznα(i, i+ d)|] as a function of the distance
d between the two spins for representative values of h in the two phases and near the
49
phase transition. Data are presented for various L. This correlation function behaves
very differently in the two phases:
In the ergodic phase, for large L this correlation function should approach its ther-
mal equilibrium value. For the states with zero total Sz that we look at, 〈n|Szi |n〉 ∼= 0
in the thermal eigenstates of the ergodic phase. The thermal correlation at infinite
temperature at large distances for finite system sizes is entirely constrained by the
sum-rule as the Boltzmann weight (e−βH) tends to 1. Correlation on the same site is:
Czznα(i, i) = 1/4. Therefore,
∑
j 6=i
Czznα(i, j) = −1
4. (2.15)
The conservation of total Sz does result in anti-correlations so that Czznα(i, j) ≈
−1/(4(L − 1)) for well-separated spins. These distant spins at sites i and j are
entangled and correlated: if spin i is flipped, that quantum of spin is delocalized and
may instead be at any of the other sites, including the most distant one. These long-
range correlations are apparent in Fig. 2.9 for h = 0.6, which is in the ergodic phase.
Note that at large distance the correlations in the ergodic phase become essentially
independent of d = |i− j| at large L and d, confirming that the spin flips are indeed
delocalized. Although we only plot the absolute value of the correlations, in fact these
correlations are almost all negative, as expected, in this large L ergodic regime.
In the localized phase, on the other hand, the eigenstates are not thermal and
〈n|Szi |n〉 remains nonzero for L → ∞. If spin i is flipped, within a single eigenstate
that quantum of spin remains localized near site i, with its amplitude for being at site
j falling off exponentially with the distance: Czznα(i, j) ∼ exp (−|i− j|/ξ), with ξ the
localization length. In the localized phase the typical correlation and entanglement
between two spins i and j thus fall off exponentially with the distance |i−j| (except for
|i− j| near L/2, due to the periodic boundary conditions). This behavior is apparent
50
in Fig. 2.9 for h = 6.0, which is in the localized phase and has a localization length
that is less than one lattice spacing. We note that in the localized phase, as well as
near the phase transition, the long distance spin correlations Czz are of apparently
random sign.
Figure 2.10: The excess fraction of states with anti-correlations at distance |i− j| =L/2. In the ergodic side the correlations are mostly negative while in the localizedcase positive and negative correlations are equally likely in which case the fractiontends to zero for larger system sizes.
The data of Figs. 2.3-2.10 show the existence of and some of the differences
between the ergodic and localized phases. We have also looked at entanglement
spectra of the eigenstates, which also support the robust existence of these two phases.
In addition to confirming the existence of these two distinct phases, we would like to
locate and characterize the many-body localization phase transition between them.
However, in the absence of a theory of this transition, the nature of the finite-size
scaling is uncertain, which makes it difficult to draw any strong conclusions from
51
these data with their modest range of L. In studies of ground-state quantum critical
points with quenched randomness, very broadly speaking, one first step is to classify
the transitions by whether they are governed (in a renormalization group treatment)
by fixed points with finite or infinite randomness [58–61]. In this real-space RG
technique, as the local high-energy terms of the Hamiltonian above the cutoff are
traced out, infinite randomness at the critical point results in broad distribution
functions for the coupling constants and of long distance correlations.
−30 −25 −20 −15 −10 −50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
φ=ln|Cnαzz (i,i+L/2)|
P(φ
)
0.6
1.8
2.7
3.6
5.0
6.0
8.0
Figure 2.11: The probability distributions of the natural logarithm of the long distancespin-spin correlation in the many-body eigenstates for sample size L = 16 and thevalues of the random field h is indicated in the legend.
To explore this question for our system, we next look at the probability distribu-
tions of the long distance spin correlations. For quantum-critical ground states gov-
erned by infinite-randomness fixed points, these probability distributions are found
to be very broad [58–60]. In particular, we look at
52
φ = ln |Czznα(i, i+ (L/2))| , (2.16)
whose probability distributions for L = 16 are displayed in Fig. 2.11 for various
values of h. Note the distributions are narrow, as expected, in the ergodic phase
and consistent with log-normal, as expected, in the localized phase. In between, in
the vicinity of the apparent phase transition, the distributions are quite broad and
asymmetric.
0.5 2.5 4.5 6.5 8.5 10.512.50.15
0.2
0.25
0.3
0.35
h
σL
810121416
Figure 2.12: The scaled width σ of the probability distribution of the logarithm of thelong-distance spin correlations (see text). The legend indicates the sample lengthsL. In the ergodic phase at small h and in the localized phase at large h, this widthdecreases with increasing L, while near the transition it increases. To produce theone-standard-deviation error bars shown, we have calculated the σ (see text) for eachsample by averaging only over sites and eigenstates within each sample, and thenused the sample-to-sample variations of σ to estimate the statistical errors. We havealso (data not shown) calculated σ by instead averaging φ and φ2 over all samples;this produces scaling behavior for σ that is qualitatively the same as shown here, butwith σ somewhat larger in the localized phase and near the phase transition.
53
To construct a dimensionless measure of how these distributions change shape as
L is increased, we divide φ by its mean, defining η = φ/[φ]. Then we quantify the
width of the probability distribution of η by the standard deviation σ =√
[η2]− 1.
This quantity is shown in Fig. 2.12 vs. h for the various values of L. By this measure,
in both the ergodic and localized phases the distributions become narrower as L is
increased, as can be seen in Fig. 2.12. This happens in the localized phase because
although the mean of −φ grows linearly in L, the standard deviation is expected
to grow only ∼√L. Over the small range of L that we can explore, σ is found
to decrease more slowly than the expected L−1/2 in the localized phase, but it does
indeed decrease.
This scaled width σL(h) of the probability distribution of φ as a function of the
random field h for each sample size L shows a maximum between the ergodic and
localized phases. In the vicinity of the phase transition, σ actually increases as L
is increased, suggesting that its critical value is nonzero, like for quantum-critical
ground states that are governed by an infinite randomness fixed point. This suggests
the possibility that this one-dimensional many-body localization transition might also
be in an infinite-randomness universality class. The peak in this plot is close to h = 4,
and is thus suggesting a slightly higher estimate of hc than the crossings in Figs. 2.6
and 2.8.
2.6 Dynamics
In the study of the spectral and localization properties of noninteracting particles in
finite samples (such as quantum dots), there are two very important energy scales:
the level spacing δ and the Thouless energy ET . The Thouless energy is ~ times the
rate of diffusive relaxation on the scale of the sample. The diffusive (nonlocalized or
ergodic) phase is where ET is larger than δ, and for d-dimensional samples with d ≥ 3,
54
the localization transition occurs when these two energy scales are comparable. Since
the single-particle level spacing in a d-dimensional system of linear size L behaves as
δ ∼ L−d and this sets the relaxation time at the localization transition, the dynamic
critical exponent for the single-particle localization transition is z = d.
Figure 2.13: Energy levels in the middle of the band for h = 1.0 and 8.0 of systemsize L = 16. In the localized phase, ET ≈ δ while in the ergodic phase it is many > δ
A possibility that we will now investigate is that the many-body localization tran-
sition also occurs when the Thouless energy is of order the many-body level spacing.
Since the many-body level spacing behaves as log δ ∼ −Ld, this corresponds to an
infinite dynamic critical exponent z → ∞. Note also that even for our model with
d = 1 this is a stronger divergence of the critical time scales than occurs at the known
infinite-randomness ground-state quantum critical points, where log δ ∼ −Lψ with
ψ ≤ 1/2.
55
0.5 2.5 4.5 6.5 8.50
0.05
0.1
0.15
0.2
0.25
[Pα (
n) ]
h
8
10
12
14
16
Figure 2.14: Contribution to the dynamic part of 〈M1〉 from matrix elements betweenadjacent energy states (see text). In the ergodic and localized phase the contributionis decreasing to zero with increasing sample size. The sample size L is indicated inthe legend. The maximum contribution from adjacent states is close to the criticalpoint.
It is important to note that the model (1) we study has two globally conserved
quantities; total energy and total Sz. Their respective transport times (and hence
their corresponding Thouless energy) in the ergodic phase may have different scaling
properties close to the critical point. By studying the relaxation of the spin mod-
ulation, M1, we are specifically probing the spin transport time which may diverge
differently from the energy transport time close to the critical point. Such a possibility
has been discussed in the context of zero-temperature metal-insulator transitions [62]
and may play a role in deciding the universality class of the many-body localization
transition.
56
Naively, the Thouless energy is set by the relaxation rate of the longest-wavelength
spin density modulation, M1. If the scaling at the many-body localization transition
is such that the Thouless energy is of order the many-body level spacing, then at the
transition a nonzero fraction of the dynamic part of 〈M1〉 should be from its matrix
elements between adjacent energy levels, and this fraction should remain large as L is
increased. In each sample α, the contribution of a given eigenstate |n〉 to the dynamic
part of 〈M1〉 is given by
(∆M1)(n)α = 〈n|M †
1M1|n〉 − |〈n|M1|n〉|2 . (2.17)
In the ergodic phase, (∆M1)(n)α has significant contributions from matrix elements
with many other eigenstates, and the Thouless energy is a measure of the energy
range over which these contributions occur. To quantify this, we define Q(n)iα as the
contribution to the dynamic part of 〈M1〉 from the matrix elements between state n
and states n± i:
Q(n)iα = |〈n− i|M1|n〉|2 + |〈n|M1|n+ i〉|2 (2.18)
in sample α. Note that∑
i 6=0
Q(n)iα = (∆M1)
(n)α . (2.19)
We define P (n)α = Q
(n)1α /(∆M1)
(n)α as the fraction of the longest-wavelength “diffu-
sive” dynamics that is due to interference between adjacent (i = 1) many-body energy
levels. Fig. 2.14 shows this quantity averaged over disorder realizations and states.
If at the localization transition the Thouless energy ET is proportional to the
many-body level spacing δ, then [P ] should remain nonzero in the limit L → ∞.
We do indeed find a strong peak in this fraction near the many-body localization
transition, and that the fraction is large and not decreasing much as L is increased.
Note that the level spacing decreases by almost a factor of 4 for every increase of
57
L by two additional spins, so near the transition the Thouless energy is apparently
decreasing by almost the same factor as L is increased. This seems at least consistent
with ET ∼ δ scaling, and thus dynamic exponent z → ∞. In the localized phase, the
dynamics is due to spin-moves that are short-range in real space (probably of order
the localization length). These spin-hops involve pairs of many-body eigenstates that
become far apart (large i) for large L; this is why [P ] drops with increasing L in the
localized phase. Note that the peak in [P ] occurs a little below h = 3. If one ignores
L = 8, the location of this peak is apparently drifting to larger h with increasing L,
consistent with our other rough estimates of hc.
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
P(f
)
f
1016
Figure 2.15: Probability distribution of the dynamic fraction of 〈M1〉 for L = 10 and16. Close to the transition for h = 3.0, the distribution becomes broader and morebimodal with increasing L.
The dynamic fraction [f(n)α ] (Fig. 2.6) tends to 1 in the ergodic phase and decreases
to 0 in the localized phase. The probability distribution of f (n)α (P (f)) is strongly
58
peaked around 1 and 0 in these respective phases. At the phase transition, this
distribution could either be peaked at the critical fc, broadly distributed, or even
bimodal with peaks near both zero and one. In Fig. 2.15, we show P (f) for a
disorder strength h = 3.0 close to the estimated transition, for system sizes 10 and
16. This distribution P (f) becomes broader and more bimodal with increasing L.
This feature of the distribution is consistent with the indication from Fig. 6 that the
critical point may be governed by a strong disorder fixed point.
2.7 Entanglement
One of the intriguing outcomes of quantum mechanics is the notion of entanglement.
A collection of degrees of freedom (e.g., a system of many-particles or just two dis-
parate degrees of freedom of a single particle but with different quantum numbers
(spin and orbital angular momentum)) can exist in a state which has correlations not
describable by correlations between effective classical degrees of freedom. It is one of
the fundamental features which renders a system its quantum nature (and gives rise
to uncertainty). It is more easily described mathematically rather than through an
intuitive picture. Consider a pure state ψ of a system defined on a Hilbert space H.
On dividing the Hilbert space into two parts S and S such that, H = HS ⊗HS, the
state is expressible as a a linear combination of basis states of S and S
|ψ〉 =N∑
i=1
M∑
α=1
xiα|i〉S ⊗ |α〉S (2.20)
where |i〉S and |α〉S are the basis states of HS and HS while N and M are their
dimensions respectively. The simplest example is of two localized spin-12
degrees of
freedom in a singlet state.
|ψ〉 = | ↑〉S ⊗ | ↓〉S − | ↓〉S ⊗ | ↑〉S√2
59
Figure 2.16: S + S form an isolated system. S is the subsystem while S serves as theenvironment to the subsystem. l . L
A pure state is said to be quantum mechanically entangled in space H if it is not
expressible as a direct product of state vectors in HS and HS.
|ψ〉H 6= |φ〉S ⊗ |ξ〉S
If this is true, the degrees in these two parts of the space are entangled. This
idea of defining entanglement between exactly two parts is specifically called bipartite
entanglement. There is still no unambiguous way to define multipartite entanglement
and is a question of ongoing research.
A more precise mathematical definition of entanglement can be given in terms of
a density matrix. The density matrix ρ of a pure state is |ψ〉〈ψ|. If |ψ〉 is normalized
ρ satisfies the constraint
TrH [ρ] = 1 (2.21)
The reduced density matrix is evaluated from ρ by taking a partial trace over one
part of the Hilbert space
60
ρS = TrS [ρ]
ρS = TrS [ρ]
For the simple case of two spins, tracing out spin B for a product state (spin-triplet
with maximal spin) | ↑〉S ⊗ | ↑〉S the density matrix is
ρprodS =
1 0
0 0
while for the singlet state the reduced density matrix is
ρentangledS =
12
0
0 12
For these simple cases, the matrix only has diagonal entries. In general, there will
be off-diagonal matrix elements. One particular of way of probing entanglement is
by studying the spectrum of ρS - λi. For a general product state, the spectrum
is always λ = 1 while all the other eigenvalues are 0. In the the case of the singlet
state, the eigenvalues are λ1 = λ2 = 12. This is the maximally entangled state. The
contraint in Eq. 2.21 gives an additional meaning to the λs. The normalization and
positivity of λ allows us to treat the distribution of λ as a probability distribution. A
distribution dominated by one value of λ near 1 implies a less entangled state while
a more uniform distribution means closer to being maximal entanglement. What are
the physically measurable consequences of this distribution remains to be explored
(In analogy with energy-level statistics where they are tied to conductance fluctua-
tions in mesoscopic systems). Drawing from our knowledge of statistical physics, this
61
probability distribution can be assigned an entropy (a.k.a. von Neumann entropy),
called the entanglement entropy.
Sentanglement = −N∑
i=1
λi lnλi (2.22)
Thus, a product state has zero entanglement entropy while a maximally entangle-
ment state maximizes entropy. (Smaxentanglement = lnN ; Note this has the right scaling
for entropy to be extensive. We should expect the thermal state to be similar to a
maximally entangled state) There are other measures which have also been studied
like the Rényi entropy. This is analogous to the moments of the distribution of λ
SqRényi =1
1− qln TrS (ρ
qS) =
1
1− qln
(
N∑
i=1
λqi
)
(2.23)
Entanglement is now beginning to be understood as the fundamental source of
local decoherence in a system. Consider a spin which is coupled to a collection of
other spins just like our spin chain. Even though this entire system is isolated from
external noise, under the dynamics of its own hamiltonian any particular spin or
cluster of spins generically loses the coherence of the intial state by entangling with
the rest of the spins with time. Eventually, if the system is ergodic any subsystem
must reach the equilibrium state described by the Gibb’s distribution
ρS = ρGibbs =e−
HS
kBT
Z(2.24)
Assuming the entire system starts from a normalized pure state |ψ〉 =∑NM
n=1 cn|n〉
with a well-defined mean energy E with small fluctuations around it. The corre-
sponding density matrix is given by
ρ0 =∑
n,m
c∗ncm|n〉〈m|
62
|n〉 and |m〉 are the energy eigenstates of the entire system. At a later time, the
reduced density matrix for S is given by
ρS(t) =∑
n,m
c∗ncm exp(−i(En −Em)t) TrS (|n〉〈m|) (2.25)
For n 6= m, the relative phases are randomized with time and hence, average to
zero. For the purposes of thermalization, the behaviour of the diagonal term (n = m)
becomes important. One of the consequences of Eigenstate thermalization hypothesis
is that TrS (|n〉〈n|) is typically independent of n.
ρS(∞) ≈ TrS (|n〉〈n|)
What should be the entanglement properties of the eigenstate for the final equilib-
rium state to be thermal? For an arbitrary state given by Eq. 2.20 the density matrix
is given by 1
ρ =∑
i,j
∑
α,β
xiαx∗jβ|i〉〈j| ⊗ |α〉〈β| (2.26)
⇒ ρS = TrS ρ =
M∑
β=1
〈β|ρ|β〉 (2.27)
=N∑
i,j=1
M∑
α=1
xiαx∗jα|i〉〈j| (2.28)
=
N∑
i,j=1
Uij |i〉〈j| (2.29)
where U = X¯X¯† where U is a N ×N matrix while X is a N ×M matrix. Inspired
by the Berry conjecture where the amplitude of a quantum chaotic wavefunction is a
gaussian random function and the results from random matrix theory, let us consider
1I will drop the subscript S and S from now on. Use of Roman letters would mean a state in S
while of Greek letters would imply a state in S.
63
the case where the elements of the matrix X are drawn from a gaussian distribution
and X is a complex matrix.
P (Xij) ∝ exp(
−Tr(
X¯X¯†))
(2.30)
The properties of such a random matrix has been studied extensively in statistics
and goes by the name Wishart matrix. The spectrum of the Wishart random matrix
with the normalization constraint∑N
i=1 λi = 1 had been studied from the point of
view of quantum entanglement in a random pure state as far back as 1978 [63]. The
probability distribution of the eigenvalues is
PWishart(λi) ∝ δ
(
N∑
i=1
λi − 1
)
N∏
i=1
λM−Ni
∏
j<k
(λj − λk)2 (2.31)
The average entropy of a subsystem was calculated by several authors [63–65]
in the limit N ≪ M and they found the state to be very close to being maximally
entangled.
Srandom = lnN − N
2M(2.32)
Thus, a random pure state has an extensive amount of entropy in a subsystem.
The treatment in terms of a random pure state assumes the state being sampled
uniformly from the space of states without any reference to the Hamiltonian. This
misses out that in a microcanonical ensemble the states are limited to an energy shell.
There has been some work done mostly from a mathematical physics point of view
where the microcanonical condition was imposed [66, 67] along with studying the
dynamics [68, 69]. But these also fall short of studying realistic quantum many-body
hamiltonians and in the presence of disorder. They are successful at proving the
approach to thermal equilibrium under some very general assumptions.
64
In Fig. 2.17 the entanglement entropy of the energy eigenstates averaged over
the mid one-third states and disorder realizations are shown. The eigenstates were
evaluated with open boundary conditions and the entanglement spectrum is for one
half of the system traced out.
Figure 2.17: Entanglement entropy as a function of system size: L = 8 to 14. Thelegend indicates the disorder strengths. The dashed line has slope ln(2).
Sentanglement = −∑
i
λi lnλi
At low disorder when the system is thermal, entanglement entropy tends to its
thermal value ∼ L2ln(2) which is extensive in subsystem size. The entropy per site
is ln 2 consistent with the infinite temperature result. While deep in the localized
phase, the entanglement entropy is independent of system size for this one-dimensional
system.
65
Besides the entanglement entropy, even the distribution of entanglement spectrum
has distintive features in the 2 phases. In the localized phase, it is dominated by a
few values close to 1 showing that the eigenstates have very low entanglement. While
the delocalized phase the spectrum is distributed much more evenly.
Figure 2.18: Average of the logarithm of the entanglement spectrum plotted versus i(It is defined such λi > λi+1) for L = 14. The legend indicates the disorder strengths.For stronger disorder, λi ≈ 0 within machine precision for larger i and hence are leftout of the plot.
2.8 Summary
This study of the exact many-body eigenstates of our model 2.1 has demonstrated
some of the properties of the ergodic and localized phases. We also find a rough
estimate of the localization transition using various different diagnostics. Based on
earlier work by Oganesyan and Huse [39], the many-body energies go from having
66
GOE to Poisson level statistics with increasing disorder. The scaling of the probability
distributions of the long-distance spin correlations suggests that the transition might
be governed by an infinite-randomness fixed point with dynamic critical exponent
z → ∞. We also study the relaxation of spin modulation under the dynamics of the
Hamiltonian. In this case our results are consistent with ET ∼ δ scaling at criticality,
in apparent agreement with our earlier conclusion of z → ∞ at the transition. These
results suggest that efforts to develop a theory of this interesting phase transition
should consider the possibility of a strong disorder renormalization group approach
[70]. Of course, the model we have studied is only one-dimensional, and the behavior
of this transition in higher dimensions might be different in important ways.
67
Chapter 3
Energy transport in disordered
classical spin chains
3.1 Classical many-body localization?
Setting aside the question of the existence of a many-body localization transition
(i.e., assuming it does exist), one might wonder about its nature, e. g., the univer-
sality class. On the one hand, the theoretical analysis of Basko, et al. [25] relies
entirely on quantum many-body perturbation theory. Rather generally, however, one
expects macroscopic equilibrium and low-frequency dynamic properties of interact-
ing quantum systems at nonzero temperature to be describable in terms of effective
classical models. This expectation is certainly borne out in a variety of symmetry-
breaking phase transitions with a diverging correlation length, such as, e. g., a finite
temperature Néel ordering of spin-1/2 moments. One can begin to understand the
microscopic mechanism behind such a many-body “correspondence principle" as a
consequence of an effective coarse-graining, whereby the relevant degrees of freedom
68
are correlated spins moving together in patches that grow in size as the phase tran-
sition is approached and therefore become “heavy” and progressively more classical.
Further extension of these ideas to general, non-critical, dynamical response is more
involved: roughly speaking, it requires that the typical many-body level spacing in
each patch be much smaller than the typical matrix element of interactions with other
patches. If this is true (as it is in most models at finite temperature, though not nec-
essarily in the insulating phase analysed by Basko and collaborators) one replaces
microscopic quantum degrees of freedom with macroscopic classical ones, which typ-
ically obey “hydrodynamic” equations of motion at low frequencies [71]. Since it is
expected that the many-body localization transition is accompanied by a diverging
correlation length (akin to the Anderson transition) one might expect some sort of
classical description to emerge en-route from the localized phase to the diffusive phase.
It was this thinking that initially motivated us to consider the possibility of classical
many-body localization.
The process by which collective classical (hydro-) dynamics emerges from a mi-
croscopic quantum description is subtle and may or may not be relevant to the many-
body localization discussed above. A somewhat less subtle, but apparently largely
unexplored related question, is whether nonlinear, interacting and disordered classical
many-body systems are capable of localization at nonzero temperature. To be precise,
a many-body classical dynamical system with a local Hamiltonian (including static
randomness) should show hydrodynamic behavior, e.g. energy diffusion, provided the
local degrees of freedom are nonlinear and interacting, and the disorder is not too
strong. In this regime, the isolated system can function as its own heat bath and
relax to thermal equilibrium. Diffusive energy transport must stop if the interactions
between the local degrees of freedom are turned off. How is this limit approached?
Can there be a classical many-body localization transition where the energy diffusiv-
69
ity vanishes while the interactions remain nonzero? These are the basic questions we
set out to investigate in this work.
Our preliminary conclusion is that classical many-body systems with quenched
randomness and nonzero nonlinear interactions do generically equilibrate, so there is
no generic classical many-body localized phase. Our picture of why this is true is
that generically a nonzero fraction of the nonlinearly interacting classical degrees of
freedom are chaotic and thus generate a broad-band continuous spectrum of noise.
This allows them to couple to and exchange energy with any other nearby degrees of
freedom, thus functioning as a local heat bath. Random classical many-body systems
generically have a nonzero density of such locally-chaotic “clusters”, and thus the
transport of energy between them is over a finite distance and can not be strictly zero,
resulting in a nonzero (although perhaps exponentially small) thermal conductivity.
Quantum systems, on the other hand, can not have a finite cluster with a truly
continuous density of states: the spectrum of a finite cluster is always discrete. Thus
the mechanism that we propose forbids a generic classical many-body localized phase,
yet it does not appear to apply to the quantum case. The proposed existence of the
many-body insulator in quantum problems is then a remarkable manifestation of
quantum physics in the macroscopic dynamics of highly-excited matter. In this work
we shall primarily focus on macroscopic low frequency behavior, postponing detailed
analysis of local structure of noise and its relation to transport. Our conclussions are
broadly consistent with findings of Dhar and Lebowitz [72] although given the rather
major differences in models, methods and, most importantly, the extent to which the
strongly localized regime is probed we refrain from making direct comparisons.
We study energy transport in a simple model of local many-body Hamiltonian dy-
namics that has both strong static disorder and interactions: a classical Heisenberg
spin chains with quenched random fields. For simplicity, we consider the limit of infi-
nite temperature, defined by averaging over all initial conditions with equal weights.
70
Our systems conserve the total energy and should exhibit energy diffusion; they have
no other conservation laws. The energy diffusion coefficient, D, can be deduced from
the autocorrelations of the energy current (as explained below) and is shown in Fig.
3.1 as a function of the strength of the spin-spin interactions, J . The mean-square
random field is ∆2, and as we vary J we keep 2J2 + ∆2 = 1, as explained below.
The limit J → 0 is where the interactions vanish, so there is (trivially) no energy
transport.
-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2log10J
-6
-5
-4
-3
-2
-1
log10D
Figure 3.1: Disorder-averaged energy diffusion constant D as a function of the spin-spin interaction J . The line has slope 8 on this log-log plot.
As the interaction J is decreased, the thermal diffusivityD decreases very strongly;
we have been able to follow this decrease in D for about 5 orders of magnitude before
the systems’ dynamics become too slow for our numerical studies. For most of this
range, we can roughly fit D(J) with a power-law, D ∼ Jγ , with a rather large
exponent, γ ∼= 8, as illustrated in Fig. 3.1. This large exponent suggests that the
asymptotic behavior at small interaction J may be some sort of exponential, rather
than power-law, behavior, consistent with the possibility that the transport is actually
71
essentially nonperturbative in J . In principle, it is also possible to fit these data to a
form with a nonzero critical Jc, so that D(J < Jc) = 0 — such fits prove inconclusive
as they produce estimates of Jc considerably smaller than the values of J where we
can measure a nonzero D. Since we are not aware of any solid theory for the behavior
of D(J), these attempts at fitting the data are at best suggestive. The large range
of variation of the macroscopic diffusion constant D across a rather modest range of
J is the most clearly remarkable and robust finding that we wish to present in this
work.
Our model and general methods employed will be presented and discussed in the
next Section. Much of what we present is based on the analysis of energy current
fluctuations in isolated rings. For various reasons we have found it beneficial to focus
on these rather than fluctuations of the energy density or on current carrying states
in open systems (we have spot-checked for quantitative agreement among these three
methods). In section 3.3 we present our results for macroscopic transport starting
from short time behavior that is relatively easy to understand and working up to long
time, DC behavior that is both difficult to compute and as of yet poorly understood.
One particularly interesting observation we make here is that of a subdiffusive behav-
ior over a substantial time range at weaker interactions, apparently distinct from the
much discussed mode-coupling behavior well representative of linear diffusion in the
presence of disorder. We discuss some afterthoughts and open problems in the Sum-
mary, with some important additional details in Sections 3.6, 3.5 (such as quantitative
explorations of finite-size effects, roundoff, many-body chaos and self-averaging).
3.2 Model, trajectories and transport
The classical motion of N interacting particles is usually defined by a system of
coupled differential equations of motion. The “particles" we study here are classical
72
Heisenberg spins – three-component unit-length vectors, Si, placed at each site i of a
one-dimensional lattice. With a standard angular momentum Poisson bracket and a
Hamiltonian, H , the equations of motion are
∂Sj∂t
= Hj × Sj , (3.1)
where Hj = ∂H/∂Sj is the total instantaneous field acting on spin Sj . The Hamilto-
nians we consider are all of the form
H =∑
j
(hj · Sj + JSj · Sj+1), (3.2)
with uniform pairwise interaction J between nearest-neighbor spins, and quenched
random magnetic fields, hj . For almost all of the results in this work, we choose the
random fields to be hj = hjnj , where the hj are independent Gaussian random
numbers with mean zero and variance ∆2, while the nj are independent randomly-
oriented unit vectors, uniformly distributed in orientation. Because of the random
fields, total spin is not conserved and we can focus on energy diffusion as the only
measure of transport in this system. For J = 0 and ∆ > 0 any initial distribution of
energy is localized, as the spins simply precess indefinitely about their local random
fields, so the diffusivity is D = 0. In the opposite limit, where ∆ = 0 and J > 0,
there is diffusive transport withD ∼ J (with nonlinear corrections due to the coupling
between energy and spin diffusion [71, 73]). We are interested in the behavior of D as
one moves between these two limits, especially as one approaches J = 0 with ∆ > 0.
Given initial spin orientations, it is in principle straightforward to integrate the
equations of motion numerically, thus producing an approximate many-body trajec-
tory. Correlation functions can then be computed and averaged over a such trajec-
tories and over realizations of the quenched random fields. The transport coefficients
can thereby be estimated via the fluctuation-dissipation relations.
73
3.2.1 The Model
Before we embark on this program, however, we start by making a change to the
model’s dynamics (3.1), but not to its Hamiltonian (3.2), in order to facilitate the
numerical investigation of the long-time regime of interest to us, where the diffusion
is very slow. In order to get to long times with as little computer time as possible,
we want our basic time step to be as long as is possible. What we are interested in
is not necessarily the precise behavior of any specific model, but the behavior of the
energy transport in a convenient model of the type (3.2). Since we are studying energy
transport, it is absolutely essential that the numerical procedure we use does conserve
total energy (to numerical precision) and that the interactions and constraints remain
local. Thus we modify the model’s dynamics to allow a large time step while still
strictly conserving total energy.
We change the equations of motion (3.1) of our model so that the even- and
odd-numbered spins take turns precessing, instead of precessing simultaneously. We
will usually have periodic boundary conditions, so we thus restrict ourselves to even
length (thus bipartite) chains. We use our basic numerical time step as the unit of
time (and the lattice spacing as the unit of length). During one time step, first the
odd-numbered spins are held stationary, while the even-numbered spins precess about
their instantaneous local fields,
Hr(t) = hr + JSr−1(t) + JSr+1(t) , (3.3)
by the amount they should in one unit of time according to (3.1). Note that since
the odd spins are stationary, these local fields on the even sites are not changing while
the even spins precess, so that this precession can be simply and exactly calculated,
and the total energy is not changed by this precession. Then the even spins are
stopped and held stationary in their new orientations while the odd spins “take their
74
turn” precessing, to complete a full time step. Although this change in the model’s
dynamics from a continuous-time evolution to a discrete-time map is substantial,
we do not expect it to affect the qualitative long-time, low-frequency behavior of the
model that is our focus in this work. In particular, we clearly observe correct diffusive
decay of local correlations for weak disorder and essentially indefinite precession of
spins at very strong disorder.
We have decided to use parameters so that the mean-square angle of precession
of a spin during one time step is one radian (at infinite temperature), which seems
about as large as one can make the time step and still be roughly approximating
continuous spin precession. This choice dictates that the parameters satisfy
2J2 +∆2 = 1 . (3.4)
We will generally describe a degree of interaction by quoting the J ; the strength
∆ of the random field varies with J as dictated by (3.4).
3.2.2 Observables
The basic observable of interest, the instantaneous energy ei(t) at site i is
ei(t) = hi · Si(t) +J
2(Si−1(t) · Si(t) + Si+1(t) · Si(t)) . (3.5)
Note that with this definition, the interaction energy corresponding to a given
bond is split equally between the two adjacent sites. When updating the spin at
site i, only the energies of the three adjacent sites, ei and ei±1, change, due to the
change in the interaction energies involving spin i. This rather simple pattern of
rearrangement of energy allows for an unambiguous definition of the energy current
at site i during the time step from time t to t + 1. If site i is even, so it precesses
first, then the current is
75
ji(t) = J [Si(t+ 1)− Si(t)] · [Si+1(t)− Si−1(t)] , (3.6)
while for i odd,
ji(t) = J [Si(t+ 1)− Si(t)] · [Si+1(t + 1)− Si−1(t+ 1)] . (3.7)
We are working at infinite temperature, or alternatively at β = (kBT )−1 = 0. The
conventionally-defined thermal conductivity vanishes for β → 0 [71]. Instead, here
we define the DC thermal conductivity κ so that the average energy current obeys
j = κ∇β (3.8)
in linear response to a spatially- and temporally-uniform small gradient in β =
1/(kBT ). The Kubo relation then relates this thermal conductivity at β = 0 to the
correlation function of the energy current via
κ =∑
t
C(t) , (3.9)
where
C(t) =∑
i
[〈j0(0)ji(t)〉] (3.10)
is the autocorrelation function of the total current, where the square brackets,
[. . .], denote a full average over instances of the quenched randomness (“samples”)
and the angular brackets, 〈. . .〉, denote an average over initial conditions in a given
sample and time average within a given run. For our model (3.2) the average energy
per site obeys
d[〈e〉]dβ
=J2 +∆2
3(3.11)
76
at β = 0, and the energy diffusivity D is then obtained from the relation
κ
D=d[〈e〉]dβ
. (3.12)
In a numerical study, if a quantity (such as κ) is non-negative definite, then it is
helpful to measure it if possible as the square of a real measurable quantity. We use
this approach here, noting that
κ = limL,t→∞
1
Lt[〈
t∑
τ=1
L∑
i=1
ji(τ)2〉] . (3.13)
For a particular instance of the random fields in a chain of even length L with
periodic boundary conditions and a particular initial condition I run for time t, we
thus define the resulting estimate of κ as
κI(t) =1
Lt
t∑
τ=1
L∑
i=1
ji(τ)2 . (3.14)
If these estimates are then averaged over samples and over initial conditions for
a given L and t, this results in the estimate κL(t) = [〈κI(t)〉] . These estimates κL(t)
must then converge to the correct DC thermal conductivity κ in the limits L, t→ ∞.
3.2.3 Finite-size and finite-time effects
In a sample of length L, we expect finite-size effects to become substantial on time
scales
t > tL = CDL2/Deff , (3.15)
where Deff is the effective diffusion constant at those time and length scales, and
we find CD ∼= 10 (remarkably Eq. 3.15 remains valid more or less with the same
value of CD across the entire range of parameters – see Section 3.5). With periodic
boundary conditions (which is the case in our simulations) this means that κL(t)
77
saturates for t > tL to a value different from (and usually above) its true DC value in
the infinite L limit, while with open boundary conditions (no energy transport past
the ends of the chain) the infinite-time limit of κL(t) is instead identically zero for
any finite L. We simply avoid this purely hydrodynamic finite-size effect by using
chains of large enough length L, which is relatively easy, especially in the strongly-
disordered regime of interest, where Deff is quite small. Thus, when the subscript L
is dropped, this means the results being discussed are at large enough L so that they
are representative of the L→ ∞ limit.
For the smallest values of J that we have studied, the system is essentially a
thermal insulator, and the Deff is so small that finite-size effects are just not visible
at accessible times even for small values of L, such as L = 10. Instead, given the way
we are estimating κ, a finite-time effect, due to the sharp “cutoffs” in time at time
zero and t in (3.14), dominates the estimates κL(t) ∼ J2/t in this small-J regime. To
explain this better we can rewrite the definition of κ as
κL(t) =1
Lt
t∑
τ=1
t∑
τ ′=1
C(τ − τ ′) =2
Lt
t/2∑
tav=1
κ∗L(tav), (3.16)
where we have assumed an even t (there is an additional term otherwise) and
κ∗L(τ) ≡∑τ
−τ C(τ′). Localization, i.e. zero DC conductivity, implies a rapidly van-
ishing κ∗ as well as κ at long times. The latter however acquires a tail, κL(t) ∼ 1/t
whose amplitude is set by the short-time values of κ∗L.
For the intermediate values of J that are of the most interest to us in this work,
there is also another, stronger finite-time effect due to an apparently power-law “long-
time tail” in the current autocorrelation function, C(τ), as we discuss in detail below.
Importantly, at long times this intrinsic finite-time effect dominates the extrinsic,
cutoff-induced, 1/t effect discussed above, so κL(t) remains a useful quantity to study
in this regime.
78
3.3 Results: Macroscopic diffusion
3.3.1 Current autocorrelations
Since the total current is not dynamically stationary, its autocorrelation function,
C(t), should decay in time. In a strongly disordered dynamical system we expect the
DC conductivity, which is the sum over all times of this autocorrelation function, to
be very small due to strong cancelations between different time domains (i.e., C(t)
changes sign with varying t). The basic challenge of computing the DC thermal
conductivity κ boils down to computing (and understanding) this cancelation.
The autocorrelation function C(t) has three notable regimes as we vary J and t.
First, C(t) is positive and of order J2 at times less than or of order one, as illustrated
in Fig. 3.2. It quickly becomes negative at larger times. For small J it is negative
and of order J3 in magnitude for times of order 1/J (see Fig. 3.3). For very small
J , this negative portion of C(t) almost completely cancels the short-time positive
portion, resulting in an extremely small κ∗ (see inset in Fig. 3.3). This cancelation is
a hallmark of strong localization and can be observed, e.g. in an Anderson insulator
where it is nearly complete. While the very short time behavior at small J is easily
reproduced analytically by ignoring dynamical spin-spin correlations, the behavior out
to times of order 1/J is representative of correlated motion of few spins (likely pairs).
Although likely non-integrable, this motion is nevertheless mostly quasiperiodic — we
recorded indications of this in local spin-spin correlation functions (not shown here).
Finally, there is apparently a power-law long-time tail with a negative amplitude:
C(t) ∼ −t−1−x, with an exponent that we find is approximately x ∼= 0.25 over an
intermediate range of 0.2 . J . 0.4 (and more generally, perhaps).
To observe this with the least amount of effort it is best to average C(t) at long
times over a neighborhood of t (see Figs. 3.4, 3.5) or to measure κL(t) and compute its
“exponential derivative”, η(t) ≡ κL(t)−κL(2t), at a sequence of points tn = 2n, n = 1,
79
1 2 3 4 5t0.00
0.01
0.02
0.03
0.04
0.05
CJ2
Figure 3.2: Short time behavior of C(t) for J = 0.32 (red, noticeably different trace)and J = 0.08, 0.12, 0.16 (these are almost identical data in this plot). Note rescalingof the vertical axis by J2.
2, 3 . . . (see Fig. 3.6). The apparent value x ∼= 0.25 of this exponent is something
that we do not understand yet theoretically. However, we find that it does provide a
good fit to the data over a wide dynamic range, providing some support for our use
of it to extrapolate to infinite time and thus estimate the DC thermal conductivity,
as discussed below.
3.3.2 DC conductivity: extrapolations and fits
Our extrapolations of the DC conductivity will be based entirely on the long time
behavior of κL(t) evaluated at a set of times tn = 2n with integer n and for large
enough L to eliminate finite-size effects (so we drop the subscript L). We start
by describing the procedure used to arrive at the numerical estimates of the DC
conductivity, then turn to the subject of uncertainties.
80
2 3 4 5 6 7 8 t J
-0.015
-0.010
-0.005
0.000
0.005CJ3
2 4 6 8t J0.00
0.02
0.04
0.06
0.08Κ*J2
Figure 3.3: Current autocorrelations on medium time scales ∼ 1/J for J = 0.32, 0.16,0.12, 0.08, from top (red) to bottom (green) trace at tJ = 2). Note the rescaling ofboth the vertical and time axes. The inset shows near cancellation between short andmedium times.
A typical instantaneous value of the energy current is set by the strength of the
exchange, J . As a consequence κ(t) ∼ J2 for small J at short and intermediate times
(t of order 1/J or less). Given the time-dependence at intermediate and long times,
as discussed above, we adopt the variable s = (1 + Jt)−0.25 as a convenient “scaling”
of time for displaying our results. These rescalings “collapse” the observed values of
κ(t) for short to intermediate times across the entire range of J studied, as shown in
Fig. 3.7.
The extrapolated values of the DC conductivity decrease strongly as J is reduced.
Extrapolation of κ(t) to s = 0 and thus DC is fairly unambiguous for J ≥ 0.32, as
can be seen in Fig. 3.7. To display the long-time results at smaller J , in Figs. 3.8,3.9
we instead show κ/J10. Here one can see that as we go to smaller J the extrapolation
to the DC limit (s = 0) becomes more and more of “a reach” as J is reduced.
81
2 3 4 5log10 t
-9
-8
-7
-6
-5
-4log10 H-CL
Figure 3.4: Long-time tail in the current autocorrelation function for J = 0.20, 0.24,0.28, 0.32, 0.40 shown bottom to top in yellow, light green, dark green, light blue anddark blue, respectively.
The outcomes of these extrapolations and rough estimates of the uncertainties are
summarized in Table 3.1.
J κ δ κ L log2 T samples0.64 0.18J2 0.01J2 5000 20 10000.56 0.09J2 0.01J2 2000 20 10000.48 0.045J2 0.005J2 1000 20 40000.40 0.020J2 0.003J2 1000 20 10000.36 0.014J2 0.003J2 1000 21 22000.32 50J10 8J10 1000 21 160000.28 70J10 10J10 1000 24 9120.24 95J10 20J10 1000 25 5580.20 130J10 30J10 1000 26 11790.18 175J10 50J10 500 27 20000.16 250J10 100J10 500 27 15500.14 400J10 200J10 500 27 5200.12 600J10 400J10 500 27 1116
Table 3.1: Extrapolated estimates of the DC conductivity κ, estimated uncertainties,length L of samples, and the number of time steps T of the runs.
82
1 2 3 4 5log10 t
-3.6
-3.4
-3.2
-3.0
-2.8
-2.6
log10 H-C t54L
Figure 3.5: To estimate the exponent we multiply the data by t5/4 (and also displaylines with slope ±0.05). Although these data do not exclude an exponent that varieswith J , we interpret these results as supportive of a single exponent x ≈ 0.25 atasymptotically long times but with a more pronounced short-time transient at smallerJ .
There are several sources of uncertainty in the estimates of the DC thermal con-
ductivity κ reported in Table 3.1. These can be separated into those originating with
the measured values of κL(t) and those due to the extrapolation to DC.
The statistical uncertainties in the measured values of κL(t) were estimated (and
shown in the figures) from sample-to-sample fluctuations which we find follow gaussian
statistics to a good approximation for these long (large L) samples. We did look for
a possible systematic source of error originating with roundoff and its amplification
by chaos (see Section 3.6) and found it not to be relevant for the values of J and t
studied.
The uncertainties in our estimates of the DC κ from the extrapolation procedure
begin with the assumed value of the long-time powerlaw, x ∼= 0.25. Clearly, using
a different exponent will change the extrapolated DC values of κ somewhat. This
83
3 4 5 6 7log10 t
-5.5
-5.0
-4.0
-3.5
-3.0
log10 Η
Figure 3.6: Long time tails as seen from η(t) for J = 0.20, 0.24, 0.28, 0.32, 0.36, 0.40,0.48 (bottom to top). Black line is a guide to the eye with slope −1/4. Note that theshort-time transients are stronger here, as compared to the auto-correlation data inFig. 3.4.
uncertainty increases with decreasing J as the ratio of the κL(t) at the last time point
to the extrapolated value increases. At our smallest J values, the curvature in our κ vs.
s plots due to the crossover to the earlier-time insulating-like ∼ 1/t ∼ s4 dependence
becomes more apparent and further complicates the extrapolation. Although we have
experimented some with different schemes for extrapolating to DC, including different
choices of exponent x, in the end the following procedure appeared to capture the
overall scale of the diffusion constant, and with a generous estimate of the uncertainty:
i) we start by removing early data with s & 0.5 to focus strictly on the long-time
behavior; ii) this long time dependence is further truncated by removing 5 latest points
and then fitted to a polynomial∑4
0 ansn to better capture the curvature apparent in
the data – these fits are shown in Fig. 3.9 and a0 are the DC values reported in Table
3.1; iii) the uncertainty is estimated as the greater of statistical error in the last point
84
0.0 0.2 0.4 0.6 0.8 1.0s0.00
0.05
0.10
0.15
ΚJ2
Figure 3.7: Variation of κ(t) for J = 0.64, 0.56, 0.48, 0.40, 0.36, 0.32, 0.28, 0.24, 0.20,0.18, 0.16, 0.14, 0.12 plotted vs. s = (1+Jt)−0.25. Lines are merely guides to the eye,and statistical errors are too small to be seen on most of these points. This figure isused for obtaining J ≥ 0.36 entries in Table 3.1.
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14s0
1000
2000
3000
4000
5000ΚJ10
Figure 3.8: Refer to the caption of Fig. 3.9
85
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14s0
200
400
600
800
1000ΚJ10
Figure 3.9: Same data as in Fig. 3.7, but now scaled and displayed in a way thatallows one to see the extrapolations to s = 0 (t→ ∞) for small J . Note the differentrescaling schemes used in preceeding and current plots to focus on “collapse” of shorttime data (previous plot) vs. long time extrapolations (present plot). As before blacklines are drawn through the data for guiding the eyes. Colored lines are results ofpolynomial fits, as explained in the text. Figs. 3.8 and 3.9 are used for obtainingentries in Table 3.1 for J ≤ 0.32.
(which is negligible for most of our data) and the difference between a0 and a simple
linear extrapolation performed on latest five data points not included in (ii).
Overall, we deem the values presented in Table 3.1 as “safe” since all extrapolated
κ’s differ by at most a factor 2 from κ’s actually measured, in other words our extrap-
olations are reasonably conservative (with the exception of two smallest J ’s where
the extrapolation yields stronger reductions).
3.4 Further explorations and outlook
In summary, we considered a rather generic model of classical Hamiltonian many-
body dynamics with quenched disorder, and explored the systematic variation in
86
the thermal diffusivity between conducting and insulating states. We found a rapid
variation of the diffusion constant and presented quantitative estimates of the latter
across more than 5 orders of magnitude of change. The origin of this behavior may be
traced to spatial localization of classical few-body chaos. Qualitatively, such a scenario
is rather plausible at very low J , where most spins are spectrally decoupled due to
disorder and essentially just undergo independent Larmor precessions. As long as J
is nonzero, however, there will always be a fraction of spins in resonance with some
of their immediate neighbors. These clusters are then deterministically chaotic and
thus generate broad-band noise, which allows them to exchange energy with all other
nearby spins. Importantly, in the entire parameter range studied this heterogeneous
regime eventually gives way at long time to a more homogeneous conducting state in
the DC limit. Thus, we suspect that internally generated but localized noise always
causes nonzero DC thermal transport even in the strongly disordered regime, as long
as the spin-spin interaction J is nonzero.
Additionally, we also discovered and characterized an apparent, novel finite-time
(frequency) correction to diffusion, with the diffusivity varying asD(ω) ≈ D(0)+a|ω|x
with x ∼= 0.25. Previous theoretical work on corrections to diffusion due to quenched
disorder [74] have instead found a correction with exponent x = 1/2, which is quite
inconsistent with our numerical results. This powerlaw behavior is apparently not
due to the localization of chaos discussed above, as it persists well into the strongly
conducting regime (larger J) and also exists in models without a strong disorder limit
at all (e.g. with random fields of equal magnitude but random direction; data not
shown). So far we have not found a theoretical understanding of these interesting
corrections to simple thermal diffusion.
87
3.5 Finite size effects
0.0 0.2 0.4 0.6 0.8 1.0s0.00
0.02
0.04
0.06
0.08ΚJ2
0.0 0.1 0.2 0.3 0.4s0
102030405060
ΚJ10
Figure 3.10: Finite size effects for J = 0.16, 0.32, 0.40: 100 vs. 20 spins for J = 0.40,10 and 20 vs. 100 spins for J = 0.32, 20 vs. 100 spins for J = 0.16. Red color isused to indicate the data influenced by finite size effects according to Eq. 3.15 withCD = 10. Inset: J = 0.16 data replotted.
We have checked that all of the extrapolations above are free from finite size
effects (by comparing against simulations on smaller, and in some cases, larger sam-
ples). Nevertheless, it is interesting to consider the expected hydrodynamic size effects
somewhat quantitatively, via Eq. 3.15. To illustrate this we display in Fig. 3.10 some
results on shorter systems for J = 0.16, 0.32, 0.40 that do show size effects: due to the
periodic boundary conditions, the conductivity in smaller rings saturates in the DC
limit at a value corresponding to the AC value at a “frequency” 2π/t∗ corresponding
to ∼= CDD(2π/L)2, with D ∼= 3κ. Our results are qualitatively consistent this with
CD ∼= 10 or slightly larger. It perhaps remarkable that despite orders of magnitude
88
of variation in the diffusion constant in going from J = 0.4 to J = 0.16 the crossover
from bulk to finite system behavior is characterized by roughly the same constant
CD ∼= 10.
3.6 Chaos amplification of round-off errors
No numerical study of a nonlinear classical dynamical system is complete without
some understanding of the interplay of discretization and round-off errors and chaos.
We are studying a Hamiltonian system that conserves total energy, so the chaos is
only within manifolds of constant total energy in configuration space. Thus although
round-off errors introduce tiny violations of energy conservation, these changes in
the total energy are not subsequently amplified by the system’s chaos; we have nu-
merically checked that this is indeed the case. As a result of this precise energy
conservation the energy transport computation remains well-defined. The simulation
is far less stable within an equal-energy manifold, where nearby trajectories diverge
exponentially due to chaos. In particular, this means that the component of any
round-off error that is parallel to the equal-energy manifold is exponentially ampli-
fied by the chaos. At large J this happens rather quickly, while for small J the chaos
is weaker and longer individual trajectories can be retraced back to their respective
initial conditions. However, at small J very long runs are necessary to extrapolate
to the DC thermal conductivity: in the end all of our extrapolations are done in
the regime where all individual many-body trajectories are strongly perturbed by
chaos-amplified round-off errors.
Ultimately, however, we are only concerned with the stability of the current auto-
correlations that enter in the Kubo formula for κ. Although the precise trajectories
may diverge due to chaos-amplified round-off errors, this need not have a strong effect
on C(t). To study this issue quantitatively we simulated roundoff noise of different
89
strength in our computations. Specifically, we add extra random noise to the compu-
tation without altering the total energy by multiplying the angle each spin precesses
in each time step by a factor of 1 + ηi(t), where the ηi(t) are independent random
numbers uniformly distributed between P and −P (P = noise strength).
In 400 rings of 500 spins coupled with J = 0.14 we simulated the same ini-
tial condition with different ηi(t), and with different values of simulated noise P =
100, 10−1, 10−2, 0 – these results are presented in Fig. 3.11 below.
0.0 0.2 0.4 0.6 0.8 1.0s0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07ΚJ2
0.00 0.02 0.04 0.06 0.08 0.10s0
100020003000400050006000ΚJ10
Figure 3.11: Roundoff effects at J = 0.14: low frequency, long-time conductivity islarger for larger values of P but is essentially indistinguishable between P = 0 andP = 0.01. Inset: data with P = 0.1, 0.01, 0.
As expected, the long-time insulating behavior is weakened by the presence of
noise. Quantitatively, however, we observe little or no difference between results
obtained in the presence of simulated noise with P = 10−2 vs. ones obtained for
intrinsic noise (which at double precision corresponds to Pintr . 10−15). Clearly, this
statement heavily depends on the duration of the simulation, value of J , etc. Judging
from Fig. 3.11 roundoff errors are not a serious source of uncertainty in our results in
90
main text. Interestingly, it is also possible for strong noise to suppress κ, as indeed
happens at shorter times, which can be traced here to a sort of “dephasing” of sharp
response of quasiperiodic localized states.
3.7 Summary
Our work on the dynamics of interacting, disordered systems has given some insight
into the physics of the many-body localization transition. But, it has also raised
some challenging theoretical questions. This preliminary work seems to suggest that
though the many-body localization transition is a finite-temperature transition, there
is no effective classsical description of it. Hence, quantum dynamics even at finite
temperature can be distinct from classical dynamics when the system is isolated. It is
imperative to establish universality for this class of phase transitions. Unfortunately,
there doesn’t appear to be any obvious theoretical tool to address this question.
We believe that the transition is a strong-disorder fixed point where the disorder
is relevant at long length scales with a dynamic critical exponent z → ∞. Thus,
one possible direction to proceed would be to perform a real-space strong-disorder
renormalization group but at finite temperature. A controlled understanding of this
method only exists for quantum ground states. Other numerical methods may also
prove to be useful in understanding the physics of this phenomena. Specifically, in the
localized phase where the entanglement of a subsystem with the rest of the system
is low, methods like time-dependent density-matrix renormalization group can study
larger system sizes.
Given the ubiquitous nature of disorder in real systems, many-body localization
may be a phenomena not very unaccesible to certain experimental systems. Specifi-
cally, experiments in cold atoms where the system is isolated to a very good approx-
imation. Dynamic measurements of transport and relaxation in such systems look
91
the most promising to explore many-body localization. Even in solid state settings,
where the coupling to a heat-bath is weak, it is conceivable that many-body local-
ization can strongly alter the conduction properties in low dimensions. For instance,
many-body localization may be underlying some highly nonlinear low-temperature
current-voltage characteristics measured in certain thin films [37].
92
Chapter 4
Conclusion and Future outlook
There are many unsettled aspects of localization involving interacting degrees of free-
dom in the presence of disorder which remain to be explored. Our work presented in
this thesis has probably only scratched the surface of this edifice. The salient features
of our work can be highlighted in the following points:
1. Some of the interesting aspects of the many-body localization transition are ac-
cessible to computational techniques currently prevalent. Contrary to “conven-
tional ” equilibrium phase transitions which do not exist in d = 1, many-body
localization transition is expected to exist in all dimensions. This makes it quite
amenable for further numerical work, for example using DMRG-like techniques.
Leaving aside the aspects of the transition, the many-body localized phase in
itself can host interesting phenomena arising due to the interplay of disorder
and interactions. Since the localized eigenstates are relatively less entangled,
the localized phase is more numerically tractable.
93
2. Quantum dynamics of an isolated system at high (extensive) energies can be
different from effective classical Langevin dynamics. The conventional under-
standing suggests that the quantum nature of a system becomes irrelevant for
highly excited states. It is indeed true that on coupling to an external heat
bath a classical description is sufficient to capture the low-frequency dynamics
at such an energy scale. But, dynamics of an isolated system which is many-
body localized doesn’t fit into this paradigm. An isolated system can maintain
its quantum coherence for relatively long times (large T2 time) under its own
dynamics. The lack of many-body localization in a classical system with dis-
order strongly suggests that although the quantum transition is at nonzeor
temperature there is no classical description of it.
3. The work by Basko, Aleiner and Altshuler [25] had put forward a picture of
many-body localization which is an analogue of single-particle Anderson local-
ization but in the many-particle Fock space. We find based on our numerics that
there is a sense in which the highly excited states are localized in real space as
well i.e., a local operator creating an excitation in an eigenstate, with an exten-
sive amount of energy, has exponentially decaying support in real space. This
has implications for the growth with time of the entanglement of a subsystem in
real space (in which the Hamiltonian is local) with the remainder of the system
in the many-body localized insulator. In a single-particle Anderson insulator
the entanglement remains finite at long times after starting in a product state.
While in the many-body localized insulator due to the finite interactions the
entanglement grows without limit as the logarithm of time [56].
4. We have shown that the critical point between the localized and the ergodic
phases may belong to the infinite randomness class. On coarse-graining in a
renormalization group sense, this would imply the system appears increasingly
94
disordered at larger length scales. Also, there is evidence that the dynamical
critical exponent z of this transition is ∞. Recent analytical studies suggest that
this is indeed the case at the transition [70]. Thus the relaxation time diverges
exponentially on approaching the critical point (assuming that the correlation
length critical exponent ν is bounded from below: ν > 2/d > 0 [75]).
τ ∼ (h− hc)−zν (4.1)
4.1 Question of Universality
The notion of universality is extremely crucial for a full theory of a phase transition.
In our work and others’ the existence of the two phases which have distinct dynamical
behaviour has been established. Strictly speaking, a renormalization group treatment
to describe the critical behaviour seems essential for establishing the universal prop-
erties of the transition. But such a method is relatively challenging, though there
have been some efforts in this direction [70]. One needs to coarse grain to “flow ” to a
state which is an excited state with an extensive amount of energy. Real-space renor-
malization group technique appears to be amenable for such a treatment but it may
only be a controlled calculation in the localized phase. Also, performing an RG for
the long-time hamiltonian dynamics is fraught with pitfalls (such as vanishing energy
denominators in the perturbative treatment) and is a challenging open question.
4.2 Symmetries
An intriguing question pertaining to the many-body localization transition is the
effect of various kinds of global symmetries. Symmetries play an important role in
distinguishing the different universality classes of finite temperature equilibrium tran-
sitions. Do they play an analogous role in the many-body localization transition? The
95
symmetries correspond to different globally conserved quantities, and their transport
can tend to zero as one approaches the critical point from the ergodic phase in distinct
ways. Also, if the system is susceptible to a symmetry-breaking transition at equilib-
rium could it affect this dynamical transition? It is not possible to have many-body
localization in the presence of a spontaneously broken continuous symmetry due to
non-localizability of the resulting Goldstone mode, except maybe for cases where the
Goldstone mode is gapped because of the Anderson-Higgs mechanism. When the
symmetry is discrete, it may very well be that the many-body localization transition
is not affected by such a discrete spontaneous symmetry breaking.
4.3 Topological order
There is an interesting class of ground state quantum phase transitions on tuning a
parameter of the hamiltonian where the transition is not accompanied by the breaking
of any symmetry. The transition manifests itself through a change in the topological
properties of the system [76]. In the presence of disorder the topological order in the
ground state is expected to be robust at least for weak disorder, but the existence of
a many-body localization transition could bear interesting dynamical effects [77, 78].
Hamiltonians with topological order are also predicted to be particularly useful for
performing fault-tolerant quantum computation [79]. Hence, many-body localization
combined with topological order in Kitaev-like models could putatively change the
bounds for quantum error-correction by a significant amount.
4.4 Decoherence
Decoherence is a major issue affecting almost all experimental realizations under study
for the purposes of quantum computation. Interactions result in entangling the qubit
with the environment and other qubits, giving rise to decoherence. Thus, having
96
strong enough interactions allowing for sufficient control of the qubits combined with
long coherence times for performing many quantum operations with high fidelity
becomes a challenging problem. In this regard spin-echo techniques have been able
to undo some of the decoherence due to interactions. A many-body localized state
can possibly be used as an effective quantum memory because of the slow growth of
entanglement, perhaps rendering spin-echo methods more effective.
97
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