Subir Sachdev
Science 286, 2479 (1999).
Quantum phase transitions in atomic gases and condensed matter
Transparencies online at http://pantheon.yale.edu/~subir
Quantum Phase Transitions Cambridge University Press
What is a quantum phase transition ?
Non-analyticity in ground state properties as a function of some control parameter g
E
g
True level crossing:
Usually a first-order transition
E
g
Avoided level crossing which becomes sharp in the infinite
volume limit:
second-order transition
T Quantum-critical
Why study quantum phase transitions ?
• Theory for a quantum system with strong correlations: describe phases on either side of gc by expanding in deviation from the quantum critical point.
• Critical point is a novel state of matter without quasiparticle excitations
• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.
ggc
~ zcg g
Important property of ground state at g=gc : temporal and spatial scale invariance;
characteristic energy scale at other values of g:
OutlineI. The quantum Ising chain.
II. The superfluid-insulator transition
III. Quantum transitions without local order parameters: fractionalization.
IV. Conclusions
I. Quantum Ising Chain
Degrees of freedom: 1 qubits, "large"
,
1 1 or , 2 2
j j
j jj j j j
j N N
0
Hamiltonian of decoupled qubits:
xj
j
H J 2J
j
j
1 1
Coupling between qubits:
z zj j
j
H Jg
1 1
Prefers neighboring qubits
are
(not entangle
d)j j j j
either or
1 1j j j j
0 1H H H Full Hamiltonian
leads to entangled states at g of order unity
Weakly-coupled qubits Ground state:
2
G
g
Lowest excited states:
jj
Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p
Entire spectrum can be constructed out of multi-quasiparticle states
jipxj
j
p e
2 2
2
Excitation energy 4 sin2
Excitation gap 2 2
pap gJ O g
J gJ O g
p
a
a
p
1g
Dynamic Structure Factor :
Cross-section to flip a to a (or vice versa)
while transferring energy
and momentum
( , )
p
S p
,S p Z p
Three quasiparticlecontinuum
Quasiparticle pole
Structure holds to all orders in g
At 0, collisions between quasiparticles broaden pole to a Lorentzian of width 1 where the
21is given by
Bk TB
T
k Te
phase coherence time
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
~3
Weakly-coupled qubits 1g
Ground states:
1 2
G
g
Lowest excited states: domain walls
jjd Coupling between qubits creates new “domain-
wall” quasiparticle states at momentum pjipx
jj
p e d
2 1
1
Excitation energy 4 sin2
Excitation gap 2 2
pap J O g
gJ J O g
p
a
a
p
Second state obtained by
and mix only at order N
G
G G g
0
Ferromagnetic moment
0zN G G
Strongly-coupled qubits 1g
Dynamic Structure Factor :
Cross-section to flip a to a (or vice versa)
while transferring energy
and momentum
( , )
p
S p
,S p 22
0 2N p
Two domain-wall continuum
Structure holds to all orders in 1/g
At 0, motion of domain walls leads to a finite ,
21and broadens coherent peak to a width 1 where Bk TB
T
k Te
phase coherence time
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
~2
Strongly-coupled qubits 1g
Entangled states at g of order unity
ggc
“Flipped-spin” Quasiparticle
weight Z
1/ 4~ cZ g g
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)
ggc
Ferromagnetic moment N0
1/80 ~ cN g g
P. Pfeuty Annals of Physics, 57, 79 (1970)
ggc
Excitation energy gap ~ cg g
Dynamic Structure Factor :
Cross-section to flip a to a (or vice versa)
while transferring energy
and momentum
( , )
p
S p
,S p
Critical coupling cg g
c p
7 /8~ c p
No quasiparticles --- dissipative critical continuum
Quasiclassical dynamics
Quasiclassical dynamics
0
7 / 4
( ) , 0
1Phase coherence time given by 2 ta
1 ...
n16
z z i tj
B
kk
k
i dt t e
AT
T
i
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).
Crossovers at nonzero temperature
II. The Superfluid-Insulator transition
†
†
Degrees of freedom: Bosons, , hopping between the
sites, , of a lattice, with short-range repulsive interactions.
- ( 1)2
j
i j jj j
ji j
j
b
b b n n
jU nH t
† j j jn b b
Boson Hubbard model
For small U/t, ground state is a superfluid BEC with
superfluid density density of bosons
M.PA. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher
Phys. Rev. B 40, 546 (1989).
What is the ground state for large U/t ?
Typically, the ground state remains a superfluid, but with
superfluid density density of bosons
The superfluid density evolves smoothly from large values at small U/t, to small values at large U/t, and there is no quantum phase transition at any intermediate value of U/t.(In systems with Galilean invariance and at zero temperature, superfluid density=density of bosons always, independent of the strength of the interactions)
What is the ground state for large U/t ?
Incompressible, insulating ground states, with zero superfluid density, appear at special commensurate densities
3jn tU
7 / 2jn Ground state has “density wave” order, which spontaneously breaks lattice symmetries
Excitations of the insulator: infinitely long-lived, finite energy quasiparticles and quasiholes
2
, , *,
Energy of quasi-particles/holes: 2p h p h
p h
ppm
Boson Green's function : Cross-section to add a boson while transferring energy and momentum
( , )
p
G p
Continuum of two quasiparticles +
one quasihole
,G p Z p
Quasiparticle pole
~3
Insulating ground state
Similar result for quasi-hole excitations obtained by removing a boson
Entangled states at of order unity
ggc
Quasiparticle weight Z
~ cZ g g
ggc
Superfluid density s
( 2)~ d zs cg g
ggc
Excitation energy gap
,
,
~ for
=0 for p h c c
p h c
g g g g
g g
/g t U
Quasiclassical dynamics
Quasiclassical dynamics
Relaxational dynamics ("Bose molasses") with phase coherence/relaxation time given by
1 Universal number 1 K 20.9kHzBk T
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).K. Damle and S. SachdevPhys. Rev. B 56, 8714 (1997).
Crossovers at nonzero temperature
2
Conductivity (in d=2) = universal functionB
Qh k T
M.P.A. Fisher, G. Girvin, and G. Grinstein, Phys. Rev. Lett. 64, 587 (1990).K. Damle and S. Sachdev Phys. Rev. B 56, 8714 (1997).
II. Quantum transitions without local order parameters: fractionalization
S=1/2 spins on coupled 2-leg ladders
jiij
ij SSJH
'JJ
e.g. SrCu2O3
''J'''J
Ground state for J large
2
1
S=0 quantum paramagnet
Elementary excitations of paramagnet
For large J, there is a stable S=1, neutral, quasiparticle excitation: its two S=1/2
constituent spins are confined by a linear attractive potential
,S p
Elementary excitations of paramagnet
The gap to all excitations with non-zero S remains finite across this transition, but the gap to spin singlet excitations
vanishes. There is no local order parameter and the transition is described by a Z2 gauge theory
For smaller J, there can be a confinement-deconfinement transition at which the S=1/2 spinons are liberated: these are neutral, S=1/2 quasiparticles
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991). X.G. Wen, Phys. Rev. B 44, 2664 (1991).T. Senthil and M. P. A. Fisher Phys. Rev. B 62, 7850 (2000).
,S p
P.W. Anderson, Science 235, 1196 (1987).
Fractionalization in atomic gases
= Two F=1 atoms in a spin singlet pair
“Ordinary” spin-singlet insulator
Quasiparticle excitation
Quasiparticle carries both spin and “charge”
E. Demler and F. Zhou, cond-mat/0104409
Quasiparticle excitation in a fractionalized spin-singlet insulator
Quasiparticle carries “charge” but no spin
Spin-charge separation
Conclusions
I. Study of quantum phase transitions offers a controlled and systematic method of understanding many-body systems in a region of strong entanglement.
II. Atomic gases offer many exciting opportunities to study quantum phase transitions because of ease by which system parameters can be continuously tuned.
III. Promising outlook for studying quantum systems with “fractionalized” excitations (only observed so far in quantum Hall systems in condensed matter).