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Bulk Entanglement Spectrum Reveals
Quantum Criticality
within a Topological State
Liang Fu
with Timothy Hsieh
Workshop on “Topological Phases of Matter”
Simons Center, Stony Brook, June 13, 2013
Outline
Introduction:
•
•
topological phase and topological phase transition
quantum entanglement
Bulk entanglement spectrum (BES):
•
•
extensive spatial partition
extract a type of topological phase transition from a wavefunction
Tensor network:
•
Hsieh and LF, arXiv:1305.1949
thanks to Xiao-liang Qi
analytical implementation of BES
with Timothy Hsieh and Xiao-liang Qi, in progress
thanks to Senthil
Conventional Phases and Phase Transitions
Spontaneous symmetry breaking:
free energy
paramagnet: <S>=0
ferromagnet: <S>≠0
•
symmetry distinguishes ordered and disordered phases
•
order parameter is continuous and small near critical point
order parameter
Conventional Phases and Phase Transitions
Spontaneous symmetry breaking:
free energy
paramagnet: <S>=0
ferromagnet: <S>≠0
•
symmetry distinguishes ordered and disordered phases
•
order parameter is continuous and small near critical point
Landau-Ginzburg paradigm:
•
•
order-disorder transition: how order parameter develops
critical phenomena from spatial/temporal fluctuations of order parameter
order parameter
Topological Phases
broadly defined
Examples:
quantum Hall state (integer & fractional)
spin liquid
string-net state
topological insulator
Haldane chain
SPT
...
Common features:
•
•
gapped state without symmetry breaking
topological quantum numbers
Topological Phases
broadly defined
Examples:
Theoretical descriptions:
quantum Hall state (integer & fractional)
Chern-Simons field theory
spin liquid
discrete gauge theory
string-net state
string-net theory
topological insulator
topological band theory
Haldane chain
matrix product state
SPT
cohomology
...
...
Common features:
•
•
gapped state without symmetry breaking
topological quantum numbers
Implications for Topological Phase Transitions
Examples:
Theoretical descriptions:
quantum Hall state (integer & fractional)
Chern-Simons field theory
spin liquid
discrete gauge theory
string-net state
string-net theory
topological insulator
topological band theory
Haldane chain
matrix product state
SPT
cohomology
...
...
Challenge #1: theories of topological phases usually do not presume a trivial state.
(additional information about trivial state must be included)
Implications for Topological Phase Transitions
Examples:
Theoretical descriptions:
quantum Hall state (integer & fractional)
Chern-Simons field theory
spin liquid
discrete gauge theory
string-net state
string-net theory
topological insulator
topological band theory
Haldane chain
matrix product state
SPT
cohomology
...
...
Challenge #1: theories of topological phases usually do not presume a trivial state.
(additional information about trivial state must be included)
Challenge #2: phase transition to different trivial states => different critical theories.
(e.g. a Z2 spin liquid can undergo topological phase transitions
to either a magnetic state or a valence bond solid.)
Read & Sachdev, 91; Moessner & Sondhi, 01
Implications for Topological Phase Transitions
Examples:
Theoretical descriptions:
quantum Hall state (integer & fractional)
Chern-Simons field theory
spin liquid
discrete gauge theory
string-net state
string-net theory
topological insulator
topological band theory
Haldane chain
matrix product state
SPT
cohomology
...
...
Challenge #1: theories of topological phases usually do not presume a trivial state.
(additional information about trivial state must be included)
Challenge #2: phase transition to different trivial states => different critical theories.
(e.g. a Z2 spin liquid can undergo topological phase transitions
to either a magnetic state or a valence bond solid.)
Is there anything generic about topological phase transitions?
A Type of Topological Phase Transition
Since a topological state is distinct from a direct product state, there is
a canonical way to realize a topological phase transition:
Cutting the system into disconnected blocks
H=
��
i
λ=1
Hi,i+n
n
i: sites
n: range of interactions
H=
�
HR + λ
R
R: blocks
�
HR,R+1
λc
: topological phase transition
subject of this talk
λ=0
R
Example:
phase transition from v=1/3 fractional quantum Hall to band insulator
induced by external periodic potential
Wen & Wu, 93; Chen, Fisher & Wu, 93
Entanglement
Spectrum
Topological
Order and Quantum Entanglement
Ground state |Ψ�
All about ground state
A
B
ρA = TrB |Ψ��Ψ|
ρA = Tr B |Ψ��Ψ| ≡ e −HA
S(ρA ) = −TrρA log ρA
entanglement entropy:
Entanglement Hamiltonian HA with entanglement spectrum {ξi }
S(ρA ) = αL − γ
topological entanglement entropy
reduced density matrix:
(area law)
related to quantum dimension
Levin & Wen, Kitaev & Preskill, 06
In addition, entanglement entropy encodes information on quasi-particle statistics and spin.
Zhang, Grover, Turner, Oshikawa &Vishwanath, 12
Tu, Zhang & Qi, arXiv, 12
Tim Hsieh
Bulk Entanglement Spectrum (BES)
Entanglement
Spectrum
Topological
Order and Quantum Entanglement
Ground state |Ψ�
All about ground state
A
B
reduced density matrix:
ρA = TrB |Ψ��Ψ| ≡ e−HA
entanglement spectrum:
the full set of eigenvalues {ξi} of “entanglement Hamiltonian” HA
ρA = Tr B |Ψ��Ψ| ≡ e −HA
Entanglement Hamiltonian HA with entanglement spectrum {ξi }
• contains at least as much information as entanglement entropy
S(ρA ) =
�
ξi e−ξi
resembles thermal average of “energy” at T=1
i
• generally contains more information (universal and non-universal)
Li & Haldane, 08
Tim Hsieh
Bulk Entanglement Spectrum (BES)
Entanglement Spectrum and Edge States
Spectrum
Ψ�
left-right spatial partition
A
B
ρA = Tr B |Ψ��Ψ| ≡ e −HA
For topological phases with gapless edge states, entanglement spectrum resembles
Li & Haldane; Bernevig, Regnault et al; Dubail & Read; Yao & Qi
edge-state
Hamiltonian
Hspectrum.
with
entanglement
spectrum {ξi }
A
• low-lying entanglement spectra correspond to edge excitations
• boundary-local nature of quantum entanglement
Qi, Katsura & Ludwig
Swingle & Senthil
Entanglement Spectrum and Edge States
Spectrum
Ψ�
left-right spatial partition
A
B
ρA = Tr B |Ψ��Ψ| ≡ e −HA
good for studying the edge,
not for the bulk
For topological phases with gapless edge states, entanglement spectrum resembles
Li & Haldane; Bernevig, Regnault et al; Dubail & Read; Yao & Qi
edge-state
Hamiltonian
Hspectrum.
with
entanglement
spectrum {ξi }
A
• low-lying entanglement spectra correspond to edge excitations
• boundary-local nature of quantum entanglement
Qi, Katsura & Ludwig
Swingle & Senthil
Extensive Partition and Bulk Entanglement
ρA = TrB |Ψ��Ψ| ≡ e−HA
A: grey region
B: white region
Partition is extensive in ALL directions
Working hypothesis: HA is a bulk, local Hamiltonian, with bulk entanglement spectrum
(BES) and bulk entanglement states defined on A
Extensive Partition and Bulk Entanglement
ρA = TrB |Ψ��Ψ| ≡ e−HA
A: grey region
B: white region
Partition is extensive in ALL directions
what do we learn about the bulk?
Working hypothesis: HA is a bulk, local Hamiltonian, with bulk entanglement spectrum
(BES) and bulk entanglement states defined on A
Extensive Partition and Bulk Entanglement
ρA = TrB |Ψ��Ψ| ≡ e−HA
A: grey region
B: white region
Partition is extensive in ALL directions
what do we learn about the bulk?
Working hypothesis: H is a bulk, local Hamiltonian, with bulk entanglement spectrum
topological
phase
(BES) and bulk entanglement
states defined
on Atransition
A
Begin with groundTuning
statethe
|Ψ�Partition
with C = 1, denoting top
nontrivial order. Below, CA = 1 indicates that the gro
Entanglement Hamiltonian H undergoes a topological phase transition as we tune
HAgeometry
has ofnontrivial
order.
partition (with fixed periodicity L) between two extremely asymmetric limit.
A
L
A
!"#$%
!"#$%
!"#$&
Begin with groundTuning
statethe
|Ψ�Partition
with C = 1, denoting top
nontrivial order. Below, CA = 1 indicates that the gro
Entanglement Hamiltonian H undergoes a topological phase transition as we tune
HAgeometry
has ofnontrivial
order.
partition (with fixed periodicity L) between two extremely asymmetric limit.
A
L
A
A
!"#$%
!"#$%
Argument:
•
!"#$%
!"#$%
!"#$&
!"#$&
require L>correlation length
spectrum of HA is gapped in the two extreme limit:
(islands are far apart & finite # of degrees of freedom per island)
•
•
ground state of HA is topologically ordered in one limit; trivial in the other
discreteness of topological order implies a gap-closing phase transition in between.
Begin with groundTuning
statethe
|Ψ�Partition
with C = 1, denoting top
nontrivial order. Below, CA = 1 indicates that the gro
Entanglement Hamiltonian H undergoes a topological phase transition as we tune
HAgeometry
has ofnontrivial
order.
partition (with fixed periodicity L) between two extremely asymmetric limit.
A
L
A
A
!"#$%
!"#$%
Argument:
•
•
•
!"#$%
!"#$%
!"#$&
!"#$&
require L>correlation length
Accessing
topological
phase
transition
from
(islands are far apart & finite # of degrees of freedom per island)
a
single
ground
state
wavefunction
ground state of H is topologically ordered in one limit; trivial in the other
spectrum of HA is gapped in the two extreme limit:
A
discreteness of topological order implies a gap-closing phase transition in between.
in with ground state |Ψ� with C = 1, denoting topologically
trivial order. Below, CA = 1 indicates that the ground state o
Phase Transition Point
has nontrivial order.
A
B
We argue that for “typical”
!"#$&
!"#$%
!"#$% topological states, at a symmetric partition
(A and B related by translation or reflection etc), HA must be gapless.
!"#$&
ning geometry realizes topological phase transition!
mmetric partition must yield gapless BES for irreducible C . (n
to satisfy CA = CB and irreducibility)
Spectrum
in withEntanglement
ground state
|Ψ� with C = 1, denoting topologically
GroundBelow,
state |Ψ� CA = 1 indicates that the ground state o
trivial order.
Phase Transition Point
has nontrivial order.
A
B
Step 1: A Conjecture
A
B
ρA = Tr B |Ψ��Ψ| ≡ e −HA
Step 1: A Conjecture
Entanglement Hamiltonian HBES
A with entanglement spectrum {ξi }
!"#$&
We argue that for “typical”
!"#$&
!"#$%
!"#$% topological states, at a symmetric partition
decomposition
(A and B Schmidt
related by
translation or reflection etc), HA must be gapless.
H
�
A
ξ
− 2i
HB
|Ψ� =
e |ψi �A ⊗ |ψ̃
Schmidt decomposition:
i �B
ning geometry
realizes
topological
phase
transition!
i
If BES
is gapped, the largest-weight state |ψ � ⊗ |ψ̃ �
in the
Schmidt
decomposition
possesses
same topological
When bulk entanglement spectrum
is gapped,
we expect
that thethe
largest-weight
state order as
BES
possesses
the same
topological
order
as |Ψ�.
the Tim
original
state
Hsieh ground
Bulk
Entanglement
Spectrum (BES)
ht state |ψ0 �A ⊗ |ψ̃0 �B in
the
he same topological order as
0 A
0 B
mmetric partition must yield gapless BES for irreducible C . (n
Typical topological orders cannot be equally divided into two.
to satisfy
CA = CB and irreducibility)
As such, HA is gapless at symmetric partition.
Tim Hsieh
HA
HB
Bulk Entanglement Spectrum (BES)
Spectrum
in withEntanglement
ground state
|Ψ� with C = 1, denoting topologically
GroundBelow,
state |Ψ� CA = 1 indicates that the ground state o
trivial order.
Phase Transition Point
has nontrivial order.
A
B
Step 1: A Conjecture
A
B
ρA = Tr B |Ψ��Ψ| ≡ e −HA
Step 1: A Conjecture
Entanglement Hamiltonian HBES
A with entanglement spectrum {ξi }
!"#$&
We argue that for “typical”
!"#$&
!"#$%
!"#$% topological states, at a symmetric partition
decomposition
(A and B Schmidt
related by
translation or reflection etc), HA must be gapless.
H
�
A
ξ
− 2i
HB
|Ψ� =
e |ψi �A ⊗ |ψ̃
Schmidt decomposition:
i �B
ning geometry
realizes
topological
phase
transition!
i
If BES
is gapped, the largest-weight state |ψ � ⊗ |ψ̃ �
in the
Schmidt
decomposition
possesses
same topological
When bulk entanglement spectrum
is gapped,
we expect
that thethe
largest-weight
state order as
BES
possesses
the same
topological
order
as |Ψ�.
the Tim
original
state
Hsieh ground
Bulk
Entanglement
Spectrum (BES)
ht state |ψ0 �A ⊗ |ψ̃0 �B in
the
he same topological order as
0 A
0 B
mmetric Phase
partition
must from
yield agapless
BES for
irreducible
C . (n
transition
single ground
state
at a
Typical topological orders cannot be equally divided into two.
to satisfy
CA =partition,
CB and no
irreducibility)
symmetric
need to tune to criticality
As such, HA is gapless at symmetric partition.
Tim Hsieh
HA
HB
Bulk Entanglement Spectrum (BES)
Application to Chern Insulator
Application to Chern Insulator
�
of Free
Chern
Insulator
H = Hamiltonian
H c c
fermion
H is also free fermion. Peschel
BES
of system:
Chernentanglement
Insulator
†
ij i j
A
�ij�
SquareBES
latticeof
model:
= (cos kx + cos ky − µ)σz + sin kx σx + sin ky σy ,
Chern H(k)
Insulator
Chern number =1 for 0<µ<2
When 0 < µ < 2, ground state has Chern number C = 1.
BES
Free fermion system → entanglement Hamiltonian
also free
BES
BES
fermion
A (Peschel)
A
A
A
A
A
A
A
A
0
0
0
Tim Hsieh
Г
Г
Bulk Entanglement Spectrum (BES)
X
X
Г
M
M
Г
X
Г
M
Г
BES is linearly dispersing near Γ : 2D Dirac fermion
•
BES is
linearly
at symmetric
partition,partition,
bulk entanglement
spectrum
= network
of dispersing
edge
At symmetric
entanglement
edge modes
percolate
a la states:
edge-state
percolation => “entanglement
percolation”.
Chalker-Coddington
network model.
•
•
asymmetric
partition:partition,
gapped
At symmetric
near Γ
entanglement edge modes percolate a la
Chern
number =2: can be gapped
at symmetric
Chalker-Coddington
network
model. partition.
Plateau transition indeed
Tim Hsieh
Bulk Entanglement Spectrum (BES)
described by Dirac Hamiltonian.
Application to Tensor Network States
•
Tensor network: economical representation of ground state
States
Matrix (MPS)
Product
(MPS)
describes aStates
large class
of non-chiral topological phases
duct States
(MPS)
recently
developed for quantum Hall states
β
σ
sor Tαβ
S
σ
σ
Tensor Tαβ
α
MPS
σ�σ
1 1 ...σ
σ σNN |ψ�
σ
α
β
≡ Tr (T σ1 ...T σN )
� ≡N |ψ�
Tr (T
) N)
..σ
≡ Tr...T
(T 1 ...T
wavefunction:
ψ=
ψ =ψ =
Tim Hsieh
σ
σ: physical
degrees of freedom
σ
β
Bulk Entanglement Spectrum (BES)
Tim Hsieh
Xie, Gu & Wen
Verstraete
Levin & Wen
Zalatel & Mong
Regnault & Bernevig
ɑ,β: αauxiliary
β degrees of freedom
ψ(σ1 , σ2 , ...σN ) = Tr(T σ1 ...T σN )
Bulk Entanglement Spectrum (BES)
Application to Tensor Network States
•
Tensor network state: economical representation of ground state
States
Matrix (MPS)
Product
(MPS)
describes aStates
large class
of non-chiral topological phases
duct States
(MPS)
recently
developed for quantum Hall states
Xie, Gu & Wen
Verstraete
Levin & Wen
Zalatel & Mong
Regnault & Bernevig
uct States (MPS)
σ
Tensor Tαβ
β
σ
sor Tαβ
or
S
σ
α
MPS
σ
Tαβ
σ�σ
1 1 ...σ
σ σNN |ψ�
σ
σ1
σ
α
α
ɑ,β: αauxiliary
β degrees of freedom
βσ
β
≡ Tr (T ...T
� ≡N |ψ�
Tr (T
) N)
..σ
≡ Tr...T
(T 1 ...T
wavefunction:
σ
σ:
physical
degrees
of freedom
Matrix Product
States
(MPS)
σN
ψ =ψ =
β
)
σ
Tensor Tαβ
ψ=
ψ(σ1 , σ2 , ...σN ) = Tr(T σ1 ...T σN ) α
σN |ψ� ≡ Tr (T σ1 ...T σN )
MPS
ψ=
�
σB
•
σ
�σ1 ...σN |ψ� ≡ Tr (T σ1 ...T σN )
Tensor network approach to bulk entanglement
ψ=
�σB |ψ��ψ|σB �
B
B
ρA =
�
σB
Tim Hsieh
Bulk Entanglement Spectrum (BES)
Tim Hsieh
Bulk Entanglement Spectrum (BES)
�σB |ψ��ψ|σB �
β
From Entanglement
Hamiltonian
to Partition Function
Partition
Function of Entanglement
Hamiltonian
Partition Function of Entanglement Hamiltonian
Partition function of HA = tensor network in one higher dimension:
Z = tr (e
−βHA
)=
β
tr (ρA )
−βHA
tr ∞
(e
β = 1/N,ZN=→
)=
B
A
β
tr (ρA )
A
B
A
periodic boundary condition in β direction
• mapping from d-dimensional quantum criticality at topological phase
transition to (d+1)-dimensional classical statistical mechanics
• tensor indices involve both physical and auxiliary degrees of freedom
Tim Hsieh
B
Bulk Entanglement Spectrum (BES)
B
A
B
B
From Entanglement
Hamiltonian
to Partition Function
Partition
Function of Entanglement
Hamiltonian
Partition Function of Entanglement Hamiltonian
β
) = tr (ρA )
Partition function of HA = tensor network in one higher dimension:
−βHA
Z = tr (e
−βHA
)=
β
tr (ρA )
−βHA
tr ∞
(e
β = 1/N,ZN=→
)=
B
A
β
tr (ρA )
A
B
A
ange perspective
from
physical
to
virtual
degrees
of
B
periodic boundary condition in β direction
• Change perspective from physical to auxiliary degrees of freedom:
aluatewillbuilding
block
want to evaluate
building block, to eliminate physical degrees of freedom
γ
δ
α
β
=
Tim Hsieh
Tim Hsieh
Bulk Entanglement Spectrum (BES)
Bulk Entanglement Spectrum (BES)
B
B
A
B
Topological
Phase Transition:
Auxiliary Becomes Real
Quantum
to Classical
Mapping
•
auxiliary degrees of freedom in tensor network representation of ground state
become actors at topological phase transition
•
tensor in ground state determines criticality at the transition
Tim Hsieh
Bulk Entanglement Spectrum (BES)
Example: Spin-1 Chain
Spin-1 Chain
with Spin-1
AKLTChain
MPS as representative
Spin-1
Haldane phase, with AKLT
MPS asChain
representative
Haldane chain: a symmetry-protected
topological phase
�
�
�
�
(Gu
&
Wen;
Pollmann,
Berg,
Turner
&
Oshikawa)
Haldane phase, with AKLT MPS as +representative
2 + 0 Haldane
1 z phase,
with2 AKLT
MPS as represen
−
−
2 Spin-1
1
2
√
T
=
σ
,
T
=
−
σ
,
T
=
−
σ
Chain
+
0
z
−
−�
�
� 3
3
3
√
σ •, T
=
−
σ
,
T
=
−
σ
2
1
2
+
0
zof AKLT
−
−
matrix product
stateσ +
representation
wavefunction
2 + 0
1 z −
+
√
T
=
,
T
=
−
σ
,
T
=
−
σ
3 Haldane
3
3 3AKLT MPS
√
phase, with
as
representative
σ ,T = −
σ ,T =
Physical:3 spin-1. Virtual:3 spin-1/2. T =
3
�
�
2 Virtual:
1 z −
2 −
Physical:
spin-1.
spin-1/2.
T+ =
σ + , T 0 = Trace
− √ σout
, Tevery
=−
σ
N spins.
3
3
3
Trace out
everyVirtual:
N spins.
Physical:
spin-1.
spin-1/2.
Virtual: spin-1/2.
Physical: spin-1. Virtual: spin-1/2.
{
N columns
γ
β
γ
δ
α
β
N columns
�
�
1
δ
N
Bγδ,αβ = 1 − −
�
�
3
� �N
� �N
α N
β
1
1
BNγδ,αβ
= 1− −
I+ −
P,
columns
P is 3projection onto3singlet.
{
{
• symmetric partition: trace outγ every Nδ spins
N spins.
α
3
�N �
�
1
I+ −
3
�N
P,
Odd N: FM interaction; Even N: AFM interaction
P is projection onto singlet.
Tim Hsieh
Bulk Entanglement Spectrum (BES)
Odd N: FM interaction; Even N: AFM interaction
Tim Hsieh
Bulk Entanglement Spectrum (BES)
Tim Hsieh
Tim Hsieh
Bulk Entanglement Spectrum (BES)
Bulk Entanglem
Large-N Limit in 1D Topological States
Large-N (Limit )
(
N
Bγδ,αβ
B
)
(
�N �
�
=
�
∞
1 N
(− 13 )N P
≈ I + (− ) P ≈ e
3
�
1
1− −
3
1
I+ −
3
)
�N
P
Using e A e B ≈ e A+B for small A, B,
≈ tr (e −β H̃ )
1 N�
H̃ ≡ −(− )
Pi,i+1 .
3
Z
i
•
•
trace out large blocks: weak interactions
among
auxiliary degrees of freedom
Tim Hsieh
Bulk Entanglement Spectrum (BES)
directly extract entanglement Hamiltonian from tensor network
Quantum-Classical
Quantum to Classical
Mapping
Mapping
B N for even N maps to ‘six-vertex’ model with vertex
Vklij = δki δlj + λδli δkj
→ 4-state Potts model → continuum theory level-1 SU(2) WZW
(Affleck 1985) (thanks to Senthil)
alternating bond strength undergoes a
• Indeed, Haldane chain with
Tim Hsieh
Bulk Entanglement Spectrum (BES)
phase transition described by SU(2)1 WZW
Summary/Outlook
•
bulk entanglement spectrum from extensive spatial partition probes a type
of topological phase transition, using a single wavefunction without tuning.
•
tensor network approach: lattice implementation of quantum criticality
Summary/Outlook
ase transition by either tuning geometry of
bulk entanglement spectrum from extensive spatial partition probes a type
ment • temperature
%
of topological phase transition, using a single wavefunction without tuning.
•
tensor network approach: lattice implementation of quantum criticality
'()$%
Why this works?
B
topological phase transition as
“entanglement percolation”
generic topological state encodes
fluctuations towards a trivial state
'()$*
'()$*
Summary/Outlook
•
bulk entanglement spectrum from extensive spatial partition probes a type
of topological phase transition, using a single wavefunction without tuning.
•
tensor network approach: lattice implementation of quantum criticality
Ongoing and future work:
•
tensor renormalization group
•
application to 2D topological and SPT phases
•
bulk entanglement spectrum with translation symmetry
levin & Nave, Gu & Wen ....
Phase Transition via Entanglement Temperature
Phase Transition via Entanglement Temperature
'()$%
!"#$%
A
'()$%
B
'()$*
'()$%
!"&$%
'()$*
Fix the Hilbert space: As ≡ A at symmetric partition.
�
−H
�
−1
−H
/T
A
A
At asymmetric partition, ρ = e
→ρ =Z e
'()$*