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Quantum phase transitions
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• cond-mat/0109419
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
E
g
True level crossing:
Usually a first-order transition
g
Avoided level crossing which
becomes sharp in the infinite
volume limit:
second-order transition
Why study quantum phase transitions ?
T
Quantum-critical
gc
• Theory for a quantum system with strong correlations:
describe phases on either side of gc by expanding in
deviation from the quantum critical point.
g
• 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.
Important property of ground state at g=gc :
temporal and spatial scale invariance;
characteristic energy scale at other values of g:  ~ g  g c
z
Outline
I.
Quantum Ising Chain
II.
Landau-Ginzburg-Wilson theory
Mean field theory and the evolution of the
excitation spectrum.
III. Superfluid-insulator transition
Boson Hubbard model at integer filling.
IV. Tilting the Mott insulator
Density wave order at an Ising transition.
V.
Bosons at fractional filling
Beyond the Landau-Ginzburg-Wilson paradigm.
I. Quantum Ising Chain
I. Quantum Ising Chain
Degrees of freedom: j  1
N qubits, N "large"
 , 
or
 j

1
  
j
2
j

j
j
,  j

1
 
j
2

Hamiltonian of decoupled qubits:
H 0   Jg   xj
j
j

j
2Jg

j
Coupling between qubits:
H1   J   zj  zj 1
j

j

j

 

j 1
j 1
Prefers neighboring qubits
are either 
Full Hamiltonian
j

j 1
or 
j

j 1
(not entangled)

H  H 0  H1   J  g xj   zj  zj 1

j
leads to entangled states at g of order unity


Experimental realization
LiHoF4
Weakly-coupled qubits
g
1
Ground state:
G 

1

2g
   

Lowest excited states:
j

  j  

Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p
p  e
  p
ipx j
j
j
 pa 
1
Excitation energy   p     4 J sin 2 
O g
2 
 
 
Excitation gap   2 gJ  2 J  O g 1



a
p
Entire spectrum can be constructed out of multi-quasiparticle states

a
Dynamic Structure Factor S ( p, ) :
Weakly-coupled qubits  g
Cross-section to flip a  to a  (or vice versa)
while transferring energy  and momentum p
S  p,  
Z      p  
Quasiparticle pole
Three quasiparticle
continuum
~3
Structure holds to all orders in 1/g

At T  0, collisions between quasiparticles broaden pole to
a Lorentzian of width 1   where the phase coherence time  
is given by
1


2k B T

e kBT
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
1
Ground states:
G 
Strongly-coupled qubits  g
1

g

2
  

Ferromagnetic moment
N0  G  z G  0
Second state G  obtained by   
G  and G  mix only at order g N
Lowest excited states: domain walls
dj 
   j    

Coupling between qubits creates new “domainwall” quasiparticle states at momentum p
p  e
ipx j
  p

dj
j
 pa 
2
Excitation energy   p     4 Jg sin 2 
O g
2 
 
 
Excitation gap   2 J  2 gJ  O g 2


a
p

a
Dynamic Structure Factor S ( p, ) :
Strongly-coupled qubits  g
1
Cross-section to flip a  to a  (or vice versa)
while transferring energy  and momentum p
S  p,  
N 02  2       p 
2
Two domain-wall
continuum

~2
Structure holds to all orders in g
At T  0, motion of domain walls leads to a finite phase coherence time   ,
and broadens coherent peak to a width 1   where
1


S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
2k B T

e kBT
Entangled states at g of order unity
Z ~  g  gc 
1/ 4
“Flipped-spin”
Quasiparticle
weight Z
A.V. Chubukov, S. Sachdev, and J.Ye,
Phys. Rev. B 49, 11919 (1994)
gc
g
N0 ~  gc  g 
1/ 8
Ferromagnetic
moment N0
P. Pfeuty Annals of Physics, 57, 79 (1970)
gc
g
Excitation
energy gap 
 ~ g  gc
gc
g
Dynamic Structure Factor S ( p, ) :
Critical coupling
 g  gc 
Cross-section to flip a  to a  (or vice versa)
while transferring energy  and momentum p
S  p,  

~  c p
c p
2
2
2

7 / 8

No quasiparticles --- dissipative critical continuum

H I   J  g ix   iz iz1

i
Quasiclassical
dynamics
Quasiclassical
dynamics
 ( ) 
i

  dt
k

 zj  t  ,  kz  0   ei t
0
A
T
7/4
1  i /  R  ...
 k T

 R   2 tan  B
16 

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).
 
z
j
z
k
~
1
jk
P. Pfeuty Annals of
Physics, 57, 79 (1970)
1/ 4
Outline
I.
Quantum Ising Chain
II.
Landau-Ginzburg-Wilson theory
Mean field theory and the evolution of the
excitation spectrum.
III. Superfluid-insulator transition
Boson Hubbard model at integer filling.
IV. Tilting the Mott insulator
Density wave order at an Ising transition.
V.
Bosons at fractional filling
Beyond the Landau-Ginzburg-Wilson paradigm.
II. Landau-Ginzburg-Wilson theory
Mean field theory and the evolution of the
excitation spectrum
Outline
I.
Quantum Ising Chain
II.
Landau-Ginzburg-Wilson theory
Mean field theory and the evolution of the
excitation spectrum.
III.
Superfluid-insulator transition
Boson Hubbard model at integer filling.
IV.
Tilting the Mott insulator
Density wave order at an Ising transition.
V.
Bosons at fractional filling
Beyond the Landau-Ginzburg-Wilson paradigm.
III. Superfluid-insulator transition
Boson Hubbard model at integer filling
Bosons at density f  1
Weak interactions:
superfluidity
Strong interactions:
Mott insulator which
preserves all lattice
symmetries
LGW theory: continuous quantum transitions between these states
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
I. The Superfluid-Insulator transition
Boson Hubbard model
Degrees of freedom: Bosons, b†j , hopping between the
sites, j , of a lattice, with short-range repulsive interactions.
U
†
H  t  bi b j -  n j   n j ( n j  1) 
2 j
ij
j
nj  b b
†
j j
M.PA. Fisher, P.B. Weichmann, G.
Grinstein, and D.S. Fisher Phys.
Rev. B 40, 546 (1989).
For small U/t, ground state is a superfluid BEC with
superfluid density

density of bosons
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
nj  3
nj  7 / 2
t

U
Ground state has “density wave” order, which
spontaneously breaks lattice symmetries
Excitations of the insulator: infinitely long-lived, finite energy
quasiparticles and quasiholes
p2
Energy of quasi-particles/holes:  p ,h  p    p ,h 
2m*p ,h
Excitations of the insulator: infinitely long-lived, finite energy
quasiparticles and quasiholes
p2
Energy of quasi-particles/holes:  p ,h  p    p ,h 
2m*p ,h
Excitations of the insulator: infinitely long-lived, finite energy
quasiparticles and quasiholes
p2
Energy of quasi-particles/holes:  p ,h  p    p ,h 
2m*p ,h
Boson Green's function G ( p, ) :
Insulating ground state
Cross-section to add a boson
while transferring energy  and momentum p
G  p, 
Z      p  
Quasiparticle pole
~3
Continuum of
two quasiparticles +
one quasihole

Similar result for quasi-hole excitations obtained by removing a boson
Entangled states at g  t / U of order unity
Z ~  gc  g 

Quasiparticle
weight Z
A.V. Chubukov, S. Sachdev, and J.Ye,
Phys. Rev. B 49, 11919 (1994)
gc
g
 p ,h ~  gc  g  for g  gc

Excitation
energy gap 
 p ,h =0 for g  gc
g
rs ~  g  gc 
Superfluid
density rs
ggcc
g
( d  z  2)
Crossovers at nonzero temperature
Quasiclassical
dynamics
Quasiclassical
dynamics
Relaxational dynamics ("Bose molasses") with
phase coherence/relaxation time   given by
1

  Universal number 
k BT
Q2   
Conductivity (in d=2) =


h  kBT 
1 K  20.9kHz 
S. Sachdev and J. Ye,
Phys. Rev. Lett. 69, 2411 (1992).
K. Damle and S. Sachdev
Phys. Rev. B 56, 8714 (1997).
  universal function
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).
Outline
I.
Quantum Ising Chain
II.
Landau-Ginzburg-Wilson theory
Mean field theory and the evolution of the
excitation spectrum.
III. Superfluid-insulator transition
Boson Hubbard model at integer filling.
IV. Tilting the Mott insulator
Density wave order at an Ising transition.
V.
Bosons at fractional filling
Beyond the Landau-Ginzburg-Wilson paradigm.
IV. Tilting the Mott insulator
Density wave order at an Ising transition
Applying an “electric” field to the Mott insulator
V0=10 Erecoil perturb = 2 ms
V0= 16 Erecoil perturb = 9 ms
V0= 13 Erecoil perturb = 4 ms
V0= 20 Erecoil perturb = 20 ms
What is the
quantum state
here ?
H  t 
ij


U
b b  b b   ni  ni  1   E  ri ni
2 i
i
†
i j
†
j i
ni  bi†bi
U  E ,t
E,U
Describe spectrum in subspace of states resonantly
coupled to the Mott insulator
S. Sachdev, K. Sengupta, and S.M. Girvin, Physical Review B 66, 075128 (2002)
Important neutral excitations (in one dimension)
Important neutral excitations (in one dimension)
Nearest neighbor dipole
Important neutral excitations (in one dimension)
Creating dipoles on nearest neighbor links creates a
state with relative energy U-2E ; such states are not
part of the resonant manifold
Important neutral excitations (in one dimension)
Nearest neighbor dipole
Important neutral excitations (in one dimension)
Nearest-neighbor dipoles
Dipoles can appear resonantly on non-nearest-neighbor links.
Within resonant manifold, dipoles have infinite on-link
and nearest-link repulsion
Charged excitations (in one dimension)
Effective Hamiltonian for a quasiparticle in one dimension (similar for a quasihole):


H eff   3t b†j b j 1  b†j 1b j  Ejb†j b j 
j
Exact eigenvalues  m  Em ; m  

Exact eigenvectors  m  j   J j  m  6t / E 
All charged excitations are strongly localized in the plane perpendicular electric field.
Wavefunction is periodic in time, with period h/E (Bloch oscillations)
Quasiparticles and quasiholes are not accelerated out to infinity
A non-dipole state
State has energy 3(U-E) but is connected to resonant
state by a matrix element smaller than t2/U
State is not part of resonant manifold
Hamiltonian for resonant dipole states (in one dimension)
d †  Creates dipole on link


H d   6t  d †  d  (U  E ) d †d
Constraints: d †d  1 ; d †1d 1d †d  0
Determine phase diagram of Hd as a function of (U-E)/t
Note: there is no explicit dipole hopping term.
However, dipole hopping is generated by the
interplay of terms in Hd and the constraints.
Weak electric fields: (U-E)
t
Ground state is dipole vacuum (Mott insulator)
First excited levels: single dipole states
0
d† 0
d †dm† 0
t
Effective hopping between dipole states
d† 0
t
t
t
dm† 0
0
If both processes are permitted, they exactly cancel each other.
The top processes is blocked when , m are nearest neighbors
t2
 A nearest-neighbor dipole hopping term ~
is generated
U E
Strong electric fields: (E-U)
t
Ground state has maximal dipole number.
Two-fold degeneracy associated with Ising density wave order:
d1† d3† d5† d7† d9† d11†
0
or
Eigenvalues
(U-E)/t
d2† d4† d6† d8† d10† d12†
0
Ising quantum critical point at E-U=1.08 t
0.220
Equal-time structure
factor for Ising order
parameter
N=8
N=10
N=12
N=14
N=16
0.216
0.212
3/4
S /N
0.208
0.204
0.200
-1.90
-1.88
-1.86

-1.84
-1.82
(U-E)/t
S. Sachdev, K. Sengupta, and S.M. Girvin, Physical Review B 66, 075128 (2002)
-1.80
Non-equilibrium dynamics in one dimension
Start with the ground state at E=32 on a chain with open boundaries.
Suddenly change the value of E and follow the evolution of the wavefunction
Critical point at E=41.85
Non-equilibrium dynamics in one dimension
Non-equilibrium response is maximal near the Ising critical point
K. Sengupta, S. Powell, and S. Sachdev, Physical Review A 69, 053616 (2004)
Outline
I.
Quantum Ising Chain
II.
Landau-Ginzburg-Wilson theory
Mean field theory and the evolution of the
excitation spectrum.
III. Superfluid-insulator transition
Boson Hubbard model at integer filling.
IV. Tilting the Mott insulator
Density wave order at an Ising transition.
V.
Bosons at fractional filling
Beyond the Landau-Ginzburg-Wilson paradigm.
V. Bosons at fractional filling
Beyond the Landau-Ginzburg-Wilson
paradigm
L. Balents, L. Bartosch, A. Burkov, S. Sachdev, K. Sengupta, to appear.
Bosons at density f  1
Weak interactions:
superfluidity
Strong interactions:
Mott insulator which
preserves all lattice
symmetries
LGW theory: continuous quantum transitions between these states
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Bosons at density f  1/2
Weak interactions: superfluidity
 sc  0
Strong interactions: Candidate insulating states

1
2
(
+
)
All insulating phases have density-wave order r  r    rQ eiQ.r with rQ  0
Q
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001)
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
Ginzburg-Landau-Wilson approach to
multiple order parameters:
F  Fsc   sc   Fcharge  rQ   Fint
Fsc   sc   r1  sc  u1  sc 
2
4
2
4
Fcharge  rQ   r2 rQ  u2 rQ 
Fint  v  sc
2
2
rQ 
Distinct symmetries of order parameters permit
couplings only between their energy densities
S. Sachdev and E. Demler, Phys. Rev. B 69, 144504 (2004).
Predictions of LGW theory
Superconductor
 sc  0, rQ  0
First Charge-ordered insulator
order
 sc  0, rQ  0
transition
r1  r2
Superconductor
 sc  0, rQ  0
Coexistence
(Supersolid)
 sc  0, rQ  0
Charge-ordered insulator
 sc  0, rQ  0
r1  r2
Superconductor
" Disordered "
Charge-ordered insulator
 sc  0, rQ  0
 sc  0, rQ  0
 sc  0, rQ  0
r1  r2
Superfluid insulator transition of hard core bosons at f=1/2
A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)
Large scale (> 8000 sites) numerical study of the destruction of superfluid order at
half filling with full square lattice symmetry
H  J   Si S j  Si S j   K
ij
 S
ijkl 

i
g=
S j Sk Sl  Si S j Sk Sl 
Boson-vortex duality
Quantum
mechanics of twodimensional
bosons: world
lines of bosons in
spacetime

y
x
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60,
1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);
Boson-vortex duality
Classical statistical
mechanics of a
“dual” threedimensional
superconductor:
vortices in a
“magnetic” field
z
y
x
Strength of “magnetic” field = density of bosons
= f flux quanta per plaquette
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60,
1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);
Boson-vortex duality
Statistical mechanics of dual superconductor is invariant
under the square lattice space group:
Tx , Ty : Translations by a lattice spacing in the x, y directions
R : Rotation by 90 degrees.
Magnetic space group:
TxTy  e 2 if TyTx ;
1
1
1
y
R Ty R  Tx ; R Tx R  T
; R 1
4
Strength of “magnetic” field = density of bosons
= f flux quanta per plaquette
Boson-vortex duality
Hofstädter spectrum of dual “superconducting” order
At density f =p / q (p, q relatively
prime integers) there are q species
of vortices,  (with =1 q),
associated with q gauge-equivalent
regions of the Brillouin zone
Magnetic space group:
TxTy  e 2 if TyTx ;
R 1Ty R  Tx ; R 1Tx R  Ty1 ; R 4  1
Boson-vortex duality
Hofstäder spectrum of dual “superconducting” order
At density f =p / q (p, q relatively
prime integers) there are q species
of vortices,  (with =1 q),
associated with q gauge-equivalent
regions of the Brillouin zone
The q vortices form a projective representation of the space group
Tx :   
1
; Ty :   e2 i f 
1 q
2 i mf
R : 

e

m
q m1
See also X.-G. Wen, Phys. Rev. B 65, 165113 (2002)
Boson-vortex duality
The  fields characterize both superconducting and charge order
Superconductor insulator :   0   0
Charge order:
Status of space group symmetry determined by
2 p
density operators rQ at wavevectors Qmn 
 m, n 
q
q
rQ  ei mnf   *  n e2 i mf
mn
1
Tx : rQ  rQ e
iQ xˆ
;
Ty : rQ  rQ e
R : r  Q   r  RQ 
iQ yˆ
Boson-vortex duality
The  fields characterize both superconducting and charge order
Competition between superconducting and charge orders:
"Extended LGW" theory of the  fields with the action
invariant under the projective transformations:
Tx :   
1
; Ty :   e
2 i f

1 q
2 i mf
e

R : 

m
q m1
Immediate benefit: There is no intermediate “disordered”
phase with neither order (or without “topological” order).
L. Balents, L. Bartosch, A. Burkov, S. Sachdev, K. Sengupta, to appear.
Analysis of “extended LGW” theory of projective representation
Superconductor
 sc  0, rQ  0
First Charge-ordered insulator
order
 sc  0, rQ  0
transition
r1  r2
Superconductor
 sc  0, rQ  0
Coexistence
(Supersolid)
 sc  0, rQ  0
Charge-ordered insulator
 sc  0, rQ  0
r1  r2
Superconductor
" Disordered "
Charge-ordered insulator
 sc  0, rQ  0
 sc  0, rQ  0
 sc  0, rQ  0
r1  r2
Analysis of “extended LGW” theory of projective representation
Superconductor
 sc  0, rQ  0
First Charge-ordered insulator
order
 sc  0, rQ  0
transition
r1  r2
Superconductor
 sc  0, rQ  0
Coexistence
(Supersolid)
 sc  0, rQ  0
Superconductor
 sc  0, rQ  0
Charge-ordered insulator
 sc  0, rQ  0
r1  r2
Second Charge-ordered insulator
order
 sc  0, rQ  0
transition
r1  r2
Analysis of “extended LGW” theory of projective representation
Spatial structure of insulators for q=2 (f=1/2)

1
2
(
+
)
q
q
a  b unit cells;
,
, ab , all integers
a
b
q
Analysis of “extended LGW” theory of projective representation
Spatial structure of insulators for q=4 (f=1/4 or 3/4)
q
q
a  b unit cells;
,
, ab , all integers
a
b
q
Summary
I.
Ferromagnet/paramagnet and superfluid/insulator quantum
phase transitions.
II.
Phases flanking the critical point have distinct stable
quasiparticle excitations; these quasiparticles disappear at the
quantum critical point.
III. Landau-Ginzburg-Wilson theory provides a theoretical
framework for simple quantum phase transitions with a single
order parameter.
IV. Multiple order parameters:
* LGW theory permits a “disordered” phase which is often
unphysical.
* Berry phase effects are crucial.
* Quantum fields transform under projective representations of symmetry
groups and via exchange of emergent of gauge bosons.