(1,1)

Dec. 10, 2008 at KITP

(3,1)

(4,0)
(2,2)
(0,4)

(1,3)

Bulk-Edge correspondence
and Fractionalization
As a topological (spin) insulator
with strong interaction

Y. Hatsugai
Institute of Physics
Univ. of TsukubaJAPAN

i C (A ) =

A
C

Plan

With time reversal invarianceZ2 Berry phase for a topological order parameter
Fractionalization for the Bulk in 1D & 2D

Entanglement Entropy to detect edge states(effective) Description by the Edges :Fractionalization at the Edges in 1D

deconfined spinons in 2D & 3D ??

Time Reversal operators with interaction
Global to Local : super-selection rule2

= 1, orLet us consider
Gapped spin liquid as a topological insulator
with strong interaction1Quantum Liquids without Symmetry Breaking
Quantum Liquids in Low Dimensional Quantum Systems
Low Dimensionality, Quantum Fluctuations
Topological Order
No Symmetry Breaking
X.G.Wen
No Local Order Parameter
Various Phases & Quantum Phase Transitions
Gapped Quantum Liquids in Condensed Matter
Integer & Fractional Quantum Hall States
Dimer Models of Fermions and Spins
Integer spin chains
Valence bond solid (VBS) states
Half filled Kondo LatticeQuantum Liquids without Symmetry Breaking
Quantum Liquids in Low Dimensional Quantum Systems
Low Dimensionality, Quantum Fluctuations
Topological Order
No Symmetry Breaking
X.G.Wen
No Local Order Parameter
Various Phases & Quantum Phase Transitions
Gapped Quantum Liquids in Condensed Matter
Integer & Fractional Quantum Hall States
Dimer Models of Fermions and Spins
Integer spin chains
Valence bond solid (VBS) states
Half filled Kondo Lattice

How to understand gapped quantum liquids ?

How to understand gapped quantum liquids ?

How to understand gapped quantum liquids ?

Bulkclassically featureless : need geometrical phase

How to understand gapped quantum liquids ?

Bulkclassically featureless : need geometrical phase1-st Chern number for QHE TKNN

How to understand gapped quantum liquids ?

Bulkclassically featureless : need geometrical phase1-st Chern number for QHE TKNN

Edge

low energy localized modes in the gap

How to understand gapped quantum liquids ?

Bulkclassically featureless : need geometrical phase1-st Chern number for QHE TKNN

Edge

low energy localized modes in the gap
edge states for QHE

Laughlin, Halperin, YHHow to understand gapped quantum liquids ?

Bulkclassically featureless : need geometrical phase1-st Chern number for QHE TKNN

Edge

low energy localized modes in the gap
edge states for QHE

Laughlin, Halperin, YHHow to understand gapped quantum liquids ?

Bulk-Edge correspondence
Common property of topological ordered states

Bulk

classically featureless : need geometrical phase1-st Chern number for QHE TKNN

Edge

low energy localized modes in the gap
edge states for QHE

Laughlin, Halperin, YHHow to understand gapped quantum liquids ?

Bulk-Edge correspondence
Common property of topological ordered states

Bulk

classically featureless : need geometrical phase1-st Chern number for QHE TKNN

Edge

low energy localized modes in the gap
edge states for QHE

Laughlin, Halperin, YH

As for quantum spins
Z2 Berry Phase as a Topological Order Parameter of bulk
Entanglement Entropy to detect edge states (generic
Kennedy triplet)Quantum Liquid (Example 1)

The RVB state by Anderson|Singlet Pair12
|G =

1
=
(|
2
cJ

J=Dimer Covering

1 2

|

1 2

ij |Singlet Pairij

Local Singlet Pairs :(Basic Objects)
PurelyPurelyQuantum
QuantumObjects
Objectsare
arebasic
fundamental

Quantum Liquid (Example 1)

The RVB state by Anderson|Singlet Pair12|G =Spins disappear
as a Singlet pair

1
=
(|
2
cJ

J=Dimer Covering

1 2

|

1 2 )

ij |Singlet Pairij

Local Singlet Pairs :(Basic Objects)

PurelyPurelyQuantum
QuantumObjects
Objectsare
arebasic
fundamental

Quantum Liquid (Example 2)

The RVB state by Pauling|Bond121
1 ††=(|1 + |2 ) =(c1 + c2 )|0
2
2

|G =

cJ
J=Dimer Covering

ij |Bondij

Do Not use the Fermi Sea

Local Covalent Bonds :(Basic Objects)
PurelyPurelyQuantum
QuantumObjects
Objectsare
arebasic
fundamental

Quantum Liquid (Example 2)

The RVB state by Pauling|Bond121
1 ††=(|1 + |2 ) =(c1 + c2 )|0
2
2

|G =

Delocalized charge
as a covalent bondcJ
J=Dimer Covering

ij |Bondij

Do Not use the Fermi Sea

Local Covalent Bonds :(Basic Objects)

PurelyPurelyQuantum
QuantumObjects
Objectsare
arebasic
fundamental

Quantum Interference for the Classification“Classical” Observables
Charge density, Spin density,...

O = n ± n⇥ , · · ·⇥O⇤G = ⇥G|O|G⇤ = ⇥G |O|G ⇤ = ⇥O⇤G|G ⇤ = |G⇤ei“Quantum” Observables !
Quantum Interferences:

G1 |G2 ⇥ = G⇥1 |G⇥2 ⇥ei( 1Probability Ampliture (overlap)|Gi = |Gi e

Aharonov-Bohm Effects
Phase (Gauge) dependent
G|G + dG⇥ = 1 + G|dG⇥

2)

i i

A = G|dG⇥ :Berry Connectioni =A:Berry Phase

Use Quantum Interferences To Classify Quantum Liquids

Examples: RVB state by Anderson|Singlet Pair12
|G =

1
=
(|
2

cJ
J=Dimer Covering

Spins disappear
as a Singlet pair

1 2

|

1 2 )

ij |Singlet Pairij

No Long Range Orderin Spin-Spin Correlation

Local Singlet Pair is a Basic Object

How to Characterize the Local Singlet Pair ?
1|G⌅ = ⌃ (|⇥i ⇤j ⌅2|⇤i ⇥j ⌅)Use Berry Phase to characterize the Singlet!
Singlet does not carries spin but does Berry phasesinglet pair = ⇥ mod 2

Examples: RVB state by Anderson|Singlet Pair12
|G =

1
=
(|
2

cJ
J=Dimer Covering

Spins disappear
as a Singlet pair

1 2

|

1 2 )

ij |Singlet Pairij

No Long Range Orderin Spin-Spin Correlation

Local Singlet Pair is a Basic Object

How to Characterize the Local Singlet Pair ?
1|G⌅ = ⌃ (|⇥i ⇤j ⌅2|⇤i ⇥j ⌅)Use Berry Phase to characterize the Singlet!
Singlet does not carries spin but does Berry phasesinglet pair = ⇥ mod 2

Z2 Berry phases for gapped quantum spins
generic Heisenberg Models (with frustration)

H=
ij

Jij S i · S j

Time Reversal Invariant

N Si

[H,

1
=
N

Si

N] = 0N = (i2
N =(Mostly N: even12Ny ) ⇥ (i y ) · · · (i y )K

)N

2
N =1

(probability 1/2 in HgTe)Z2 Berry phases for gapped quantum spins
Define a many body hamiltonian by local twist as a parameter

H(x = ei )C = {x = e | : 0i

Si · Sj

2 }

U(1)

1
(e i Si+ Sj + e+i Si Sj+ ) + Siz Sjz Only link <ij>2

Calculate the Berry Phases using the Entire Many Spin Wavefunction numericallyZ2 quantization
C =Require excitation Gap!

⇥

C

A =

⇥

C

⇤|d⇤⇥ =

0:⇥

mod 2⇥Z2

Time Reversal ( Anti-Unitary ) InvarianceBerry Connection and Gauge Transformation
Parameter Dependent Hamiltonian
H(x)| (x) = E(x)| (x) ,H(x) Y(x) =% (x) Y(x)(x)| (x) = 1.

dBerry Connections A = |d = | dxdx.
i C (A ) =A
Berry Phases
C(Abelian)
Phase Ambiguity of the eigen state| (x)= | (x) ei (x)d= A + id = A + i dx
dx

AGauge Transformation

Berry phases are not well-defined without
C (A

)=

C (A

)+
C

Well Defined up to mod 2
C (A

)

C (A

specifying the gauged2) mod 2(integer) if ei is single valued

Anti-Unitary Operator and Berry Phases
Anti-Unitary Operator
K:= KU ,
U :

|

=

|

=

J

(Time Reversal, Particle-Hole)Complex conjugate
Unitary(parameter independent)
CJ CJ =

CJ |J

J

=

|

J

CJ |J

,

=

|J

|

=1

|J

Berry Phases and Anti-Unitary Operation
A =
A

=

|d
|d

C (A

=

CJ dCJJ

dCJ CJ +J=J

CJ dCJ =

A

J

)=

C (A

CJ dCJ =0)

Anti-Unitary Invariant State and2Z2 Berry Phase
N =1
Anti-Unitary Symmetry
Invariant State[H(x), ] = 0|,

ex. Unique Eigen State=||

To be compatible with the ambiguity,

=|

i

e

Gauge
Equivalent(Different
Gauge)

the Berry Phases have to be quantized as
C (A
C (A

)=
)=

0
C (A

Z2 Berry phase

mod 2

)

C (A

), mod2

Numerical Evaluation of the Berry Phases (incl. non-Abelian)(1) Discretize the periodic parameter space
x0 , x1 , · · · , xN = x0i nn+1 = n +xn = e0 = 0,nn

N = 2⇥

0(2) Obtain eigen vectors H(x )| i = E i (x )| innnn(3) Define Berry connection in a discretized formjin | n+1 ⌅non-Abelian An = Im log det Dn , {Dn }ij = ⇤(4) Evaluate the Berry phase
N

1

=
n=0

non-Abelian

An = Im log⇥ 0 | 1 ⇤⇥ 1 | ⇤ · · · (= Im log det D1 D2 · · · Dn)

Independent of the choice of the phase
Gauge invariant
after the discretization
Convenient for Numerics

| n⇥

i

| n⇥ enLuscher ’82 (Lattice Gauge Theory)
King-Smith & Vanderbilt ’93 (polarization in solids)
T. Fukui, H. Suzuki & YH ’05 (Chern numbers)

Introduce interaction between singlets

Adiabatic Continuation & the QuantizationZ2-quantization of the Berry phases protects fromcontinuous change
Adiabatic Continuation in a gapped system
Renormalization Group in a gapless system

Local Order Parameters of Singlet Pairs
1D AF-AF,AF-F Dimers

Y.H., J. Phys. Soc. Jpn. 75 123601 (2006)

Strong Coupling Limit of the AF Dimer link is a gapped
unique ground state.

AF-AF case
AF-AF

Strong bonds
:

bonds

F-AF case
F-AF
Hida

AF bonds:

bonds

Local Order Parameters of the Haldane Phase

Heisenberg Spin Chains with integer S
No Symmetry Breaking by the Local Order Parameter“String Order”: Non-Local Order Parameter!

S=1 (S i ) = S(S + 1), S = 1
2

H=J
ij⇥

Si · Sj + D(Siz )2
Y.H., J. Phys. Soc. Jpn. 75 123601 (2006)

i

P

P

P

P

P

P

P

P

















D < DCD > DC

Describe the Quantum Phase Transition locallyc.f. S=1/2, 1D dimers, 2D with Frustrations, Ladders
t-J with Spin gap

Topological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 02 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

















set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 02 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

P

P

P

P

P

P

P

P

set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 02 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

















set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 02 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

















set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 02 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

P

P

P

P

P

P

P

P

set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 02 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

















set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 02 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

P

P

P

P

P

P

P

P

set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 02 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

















set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 2
0 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
a local spin twist on a link as a generic param: dimerization: dimerization strength
he definition of the Berry phase[1]. Understrength
this(c)(b) the following term Si+ Sj− + Si− Sj+ in
n twist,(3,1)(2,2)+ − (1,3)+
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
j ,±= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
the local spin twists extracts a local structure oftum system. By this quantized Berry phase, one(c)e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases on the local link of (a) the S = 1
0-bond”, “π-bond”, or “undefined”. It has a reperiodic N = 14 and (b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phasess against the small perturbations unless the enis π on the bold line while that is 0 on the other line. We labetween the ground state
andtheexcitedstatesTopological Quantum Phase
Transitions
withtranslation
invariancebeltheregionofthedimerized
Heisenberg
chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

P

P

P

P

P

P

P

P

















set of two numbers as (n, m). The phase boundaries in theTopological Classification of Gapped Spin Chains

T.Hirano, H.Katsura &YH, Phys.Rev.B77 094431’08✦

S=1,2 dimerized Heisenberg model

2

J1 = cos , J2 = sin(a)ian Berry connection obtained by the single(2,0)
(1,1)
(0,2)
ormalized ground state |GS(φ)! of H(φ) as"GS(φ)|∂φ |GS(φ)!. This Berry phase is real
2
ntized to 2
0 or π (mod 2π) if the Hamiltonian
invariant under the anti-unitary operation Θ,
S =that
1 Nthe=Berry
14 phase isS = 2 N = 10(b)φ), Θ](a)= 0 [3]. Note
(4,0)
(3,1)
(2,2)
(1,3)
(0,4)
(1,1)
(0,2)
d if the (2,0)
gap between the groundstate and thetates vanishes while varying the parameter φ.
(3,1)
a local spin twist on a link as a generic paramhe definition of the Berry phase [1]. Under this
(4,0)
+ −
− +
(c)
(b) the following(3,1)
n twist,
term
S
S
+
S
i j
i Sj in(1,1)(3,1)(2,2)+ − (1,3)+(4,0)
iltonian(4,0)
is replacedwith eiφ
Si Sj + e−iφ Si− S(0,4)
(2,2)
j ,
±
= Six ±iSiy . The Berry phase defined by the re0
0.5
1
1.5
2
(0,4) (3,1) (2,2)
the(2,0)local spin twists extracts a (0,2)
local structure of(1,3)
tum system. By this quantized Berry phase, one
(4,0)
(c)
(0,4)
e a link-variable.
Then each link has one of three
FIG. 1: The Berry phases (1,3)
on the local link of (a) the S = 1: S=1/2
singlet: Symmetrization
0-bond”, “π-bond”, or “undefined”.
It hasa re- state
periodic N = 14and(2,2)(b) the S = 2 periodic N = 10 dimerproperty
that 0.5
the Berry phasehas topological
0
1
1.5
2 ized Heisenberg chains, and (c) the S = 2 periodic N = 10
ss against the small perturbations unless the en(0,4) Heisenberg chain with single-ion anisotropy. The Berry phaseis π on the bold line
while that is 0 on the other line. We la(1,3)
between the ground state and the excited states
bel the region of the dimerized Heisenberg chains using the
nFIG.
order
to
calculate
the
Berry
phase
numerically,
1: The Berry phases on the local link of (a) the S = 1

Z Berry phase

Reconstruction of valence bonds!

set of two numbers as (n, m). The phase boundaries in the2

Topological Classification of Gapped Spin Chains (cont.)btained by the single|GS(φ)! of H(φ) ass Berry phase is realπ) if the ✦Hamiltonian-unitary operation Θ,at the Berry phase isground state and theying the parameter φ.
2
k as a generic paramphase [1]. Under thisrm Si+ Sj− + Si− Sj+ in
eiφ Si+ Sj− + e−iφ Si− Sj+ ,hase defined by the reacts a local structure of
tized Berry phase, one
ch link has one of threendefined”. It has a rephase has topologicalrbations unless the enand the excited states
rry phase numerically,
erry phase[1, 33]. It iseter space of φ into N(a)T.Hirano,
H.Katsura&YH, Phys.Rev.B77 094431’08
(1,1)
(0,2)

(2,0)

S=2 Heisenberg model with D-term
(b)
(4,0)

Z Berry phase
(c)
0

(3,1)

(2,2)

(1,3)

(0,4)

Red line denotes the non trivial Berry phase

S = 2 N = 10
0.5

1

1.5

2

FIG. 1: The Berry phases on the local link of (a) the S = 1
periodic N = 14 and (b) the S = 2 periodic N = 10 dimerized Heisenberg chains, and (c) the S = 2 periodic N = 10
Heisenberg chain with single-ion anisotropy. The Berry phaseis π on the bold line while that is 0 on the other line. We la:0 magnetization
bel the region of the dimerized Heisenberg chains using theset of two numbers as (n, m). The phase boundaries in thefinite size system are θc1 = 0.531237, θc2 = 0.287453 andθc3 = 0.609305, respectively. The Berry phase in (a) and (b)has an inversion symmetry with respect to θ = π/4.

Reconstruction of valence bonds!

boson representation of the spin operaors as Si = ai bi ,&as † |φ"
†
†
†
†
†
−
z
t
(ai bi+1
[36].
This
state
has
nonzeroaverageofvectorspinchiralS
=
a
b
,
and
S
=
(a
a
−
b
b
)/2.
a
and
b
satisfy
i
i
i
i
i
i
i
i
i
i
Topological Classification of Generic
AKLT(VBS) models
wheretha|k††obviousitythe#Scommutation
the[atwist= 0 allor π.
relation= [bi , bj ] = δφijijwithi × S j · ẑ" unless
i , aj ] parameter
triplet
b
since
the
st
T.Hirano,
H.Katsura&YH,
Phys.Rev.B77
094431’08
†
This
state
is a zero-energy
groundstateof the following'
othercommutators
vanishing[37]. The
constraint
ai ai +
orthogonalt(1/2)
Twist the
link
of
the
generic
AKLT
model
†
Hamiltonian:bi bi = 2S is imposed to reproduce the dimension of thethe (B
Berry
pk
−
12
spin S Hilbert spaceN at eachsite. In general, the ground
2Bi,i+1
Next
we
sha
$
$
of
the
c
state of the chiral AKLT model having B
valence
bonds
J ij
B
AJ Pi,i+1[φi,i+1 ],(5) arbitrary
H({φi,i+1 }) =Im#k|l"on the link (ij) is written as
with a loca
i=1 J=Bi,i+1 +1† or odd)"#s and t† a!
Bij
iφij /2 † †−iφij /2 † †parities.
|{φ
}"
=
e
a
b
−
e
b
a
|vac",
J(4)
i,j
where
(· · · )ijijwhere AJ is the arbitrary positive coefficient. Pi,i+1 [0]
state
rep
"ij#
using
the
b
is the polynomial in S i · S i+1 and act as a projection
we
can
o
as
|φ"
=
Berry
phase
on
a
link
(ij)
operatorprojecting
the bond
spinofJvector
S ichiral+ S i+1[36]. This
state has nonzero
averagespin
i,i+1 =
using
th
where
|k"
=
S=1/2
onto
the
subspaceofspinmagnitudeJ.
The
replacement
ity #S i × S j · ẑ" unless the twist parameter φij = 0 or π.
ij
ij + −
phaseoftripletbonds+This− state −
+
−
+
iφ
−iφ
i,i+1
i,i+1
zero-energy
groundSi Si+1 + SiisSai+1→e
Si Si+1state+ e of the following
Si Si+1 in'B12 (
the k=0edg(1/2)
JHamiltonian:produces
Pi,i+1the[φi,i+1] in Eq.of(5).
i · S i+1phase
TheSBerrycountsnumberthe valence bonds!
(B12is−trunck) ta
2Bshow
i,i+1
N
Now we shall explicitlythat the Berry phase of of the conne$$
propertyJ], valence
(5)
H({φ
=
the VBS
state
extracts
the localAnumberof theJ Pi,i+1 [φi,i+1
i,i+1 })
Im#k|l"= 0
stressed
S=1/2
objectsarei=1fundamental
in now
S=1&2
spinthechainsJ=Bi,i+1 +1
bonds B2π).
Let usconsideror odd)andij as Bij π(modrestricte
local twist of the parameters φij = φδij,12 andJ rewrite parities. Thi
where AJ is the arbitrary positive coefficient.
Pi,i+1 [0]to the generali
Contributestate represe
theis ground
state
|{φ
}"
as
|φ".
To
calculate
the
Berry
i,j
the
polynomial in S i · S i+1 and actEntanglement
as a projection Entropy
FRACTIONALIZATION
longobtai
as t
we
can
phase
of theVBS state,
relation
operatorprojecting
the thebondfollowing
spin J i,i+1= S i is+ useful:
S i+1
Now,
using therelas of Edge states= B ⇥ mod 2⇥

2D, Ladders (S=1/2), t-J (spin gapped)

Y.H., J. Phys. Soc. Jpn. 75 123601 (2006), J. Phys. Cond. Matt.19, 145209 (2007)
J=1:

(a)

(b)

(c)

J=2:

G = 0:

G = P:

A

B

Entanglement Entropy to detect edge states
direct calculation of spectrum with bondaries

Entanglement Entropy

Mixed State From Entanglement

Vidal, Latorre, Rico, Kitaev ‘02

Direct Product State

System = A ⊕ B!
State =ΨA ⊗ ΨB|ΨAB ! = |ΨA ! ⊗ |ΨB !

Entangled State

D
!
1
j
j
|ΨAB ! = √|ΨA ! ⊗ |ΨB !
D j
Partial Trace

ρAB = |ΨAB !"ΨAB |ρA = TrB ρAB
=

D
!
1

D

j

j
j
|ΨA !"ΨA |A

B

Pure State
D=1

Mixed State

1 ! j
ρAB =
|ΨA !"ΨkA | ⊗ |ΨjB !"ΨkB |DDjk

How much the State is Entangled between A & B?

Entanglement Entropy : SA = −"log ρA # = log D

E.E. & Edge states (Gapped)(of spins, fermions...)A

B

Partial Trace induces effective edge states
Requirement: Finite Energy Gap for the Bulk

The effective edge states contribute to the E.E.
Let us assume that the edge states has degrees of freedom DE

Entanglement Entropy > (# edge states) Log DE
S. Ryu & YH, Phys. Rev. B73, 245115 (2006)(Fermions)EE of the Generic VBS States (S=1,2,3,...)
H. Katsura, T.Hirano & YH, Phys. Rev. B76, 012401 (2007)
N
T.Hirano & YH, J. Phys. Soc. Jpn. 76, 113601 (2007)
S⇧i · S⇧i+1 + Hextra⇧i2 = S(S + 1)
HV BS =
S
, S
⇤1
S=1
⇤i · S⇤i+1 )2i=1
Hextra =
(S
3
i
⇤ 2
1 ⇤ ⇤
S=2
2
3
⇤
⇤
H
=
(
S
·
S
)
+
(
S
·
S
)
i
i+1
i
i+1
L
extra
9
63
i
† †
† †
|VBS⇥ =
(aj bj+1 bj aj+1 )S |vac⇥
j=0
SL = ⇧log ⌃ ⇥ 2 log(S + 1), (L ⇥ ⇤)Boundary Spins: S/2

Degrees ofFreedom

S

EE

Effective
Boundary spins

1

2 Log 2

Seff=1/2

22=4

2

2 Log 3

Seff=132=9S

2 Log (S+1)

Seff=S/2(S+1)2Fractionalization : Emergent as edge states(Quantum Resources for qbits)

Another Models
Spin ladder model with four-spin cyclic exchange
H=i{Jr S 1,i · S 2,i + Jl (S 1,i · S 1,i+1 + S 2,i · S 2,i+1 ) + K(Pi + Pi 1 )}(Pi + Pi−1 ) = S 1,i · S 2,i + S 1,i+1 · S 2,i+1 + S 1,i · S 1,i+1+ S 2,i · S 2,i+1 + S 1,i · S 2,i+1 + S 2,i · S 1,i+1+ 4(S 1,i · S 2,i )(S 1,i+1 · S 2,i+1 )+ 4(S 1,i · S 1,i+1 )(S 2,i · S 2,i+1 )− 4(S 1,i · S 2,i+1 )(S 2,i · S 1,i+1 ).

K

Scalar Chiral LRO
Dominant
Collinear Spin

Dominant
Vector Chirality

Dimer LROθJ

Ferromagnetic

Rung
SingletWe set parameters as
J = Jr = Jl = cos
K = sin

Self dual at the point of J = 2K
T. Hikihara, T. Momoi and X. Hu (2003)
A. Lauchli, G. Schmid and M. Troyer (2003)

Adiabatic deformation

Rung singlet phase

I. Maruyama, T. Hirano, YH, arXiv:0806.4416

Vector chirality phase

=6

Hr =
i=1

= 2.6

S 1,i · S 2,i

Rung singlets

Hps =(S 1,i ⇥ S 2,i ) · (S 1,i+1 ⇥ S 2,i+1 )i odd

Plaquette singlet (PS)

Berry phase remains the same
Topologically equivalence

Energy spectrum with boundaries (diagonal)
M. Arikawa, S. Tanaya, I. Maruyama, YH, unpublishedopen BC
periodic BC

gap
for periodic system

3 fold degenerate
S=1 excitation(Kennedy⇥ 1.08 exp(
Triplet)

N

)

3.22

Ex. for Haldane spin chain
Kennedy ’90
M. Hagiwara, K. Katsumata, I. Affleck, and B. Halperin, ‘90
Interaction between effective boundary spinsHef f =SR · SLBulk-Edge correspondence for spins
2
N = 1

N = (i12Ny ) ⇥ (i y ) · · · (i y )KK : complex conjugate

Bulk: Z2 Berry phases

Edge: Entanglement Entropy& low energy states in the gap
S = 1/2 is always fundamental ( electron spin )edges =

Global TR

=
2
L =

R

2

1
1

L

R

N

Local (edge) TR

L,R

Bulk-Edge correspondence for spins
2
N = 1

N = (i12Ny ) ⇥ (i y ) · · · (i y )KK : complex conjugate

Bulk: Z2 Berry phases

Edge: Entanglement Entropy& low energy states in the gap
S = 1/2 is always fundamental ( electron spin )edges =

Global TR

=
2
L =

R

2

3D ?

1
1

L

R

N

Local (edge) TR

L,R

Thank you