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Dec. 10, 2008 at KITP (3,1) (1,1) Bulk-Edge correspondence and Fractionalization As a topological (spin) insulator with strong interaction Y. Hatsugai i C ( A ) = A C Institute of Physics Univ. of Tsukuba JAPAN Plan With

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SLIDE 1

Bulk-Edge correspondence and Fractionalization

  • Y. Hatsugai

Institute of Physics

  • Univ. of Tsukuba

JAPAN

As a topological (spin) insulator with strong interaction

(1,1)

(3,1)

  • Dec. 10, 2008 at KITP

iγC(Aψ) =

  • C

Aψ

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SLIDE 2

Plan

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

Time Reversal operators with interaction

Global to Local : super-selection rule

Gapped spin liquid as a topological insulator with strong interaction Let us consider

Z2

With time reversal invariance

  • r

deconfined spinons in 2D & 3D ??

Θ2 = 1, −1

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SLIDE 3

Quantum Liquids without Symmetry Breaking

Quantum Liquids in Low Dimensional Quantum Systems Low Dimensionality, Quantum Fluctuations No Symmetry Breaking 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 ?

Topological Order

X.G.Wen

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SLIDE 4

How to understand gapped quantum liquids ?

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SLIDE 5

How to understand gapped quantum liquids ?

Bulk

classically featureless : need geometrical phase

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SLIDE 6

How to understand gapped quantum liquids ?

Bulk

classically featureless : need geometrical phase 1-st Chern number for QHE TKNN

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SLIDE 7

How to understand gapped quantum liquids ?

Bulk

classically featureless : need geometrical phase

Edge

low energy localized modes in the gap 1-st Chern number for QHE TKNN

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SLIDE 8

How to understand gapped quantum liquids ?

Bulk

classically featureless : need geometrical phase

Edge

low energy localized modes in the gap 1-st Chern number for QHE TKNN edge states for QHE Laughlin, Halperin, YH

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SLIDE 9

How to understand gapped quantum liquids ?

Bulk

classically featureless : need geometrical phase

Edge

low energy localized modes in the gap 1-st Chern number for QHE TKNN edge states for QHE Laughlin, Halperin, YH

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SLIDE 10

How to understand gapped quantum liquids ?

Bulk

classically featureless : need geometrical phase

Edge

low energy localized modes in the gap 1-st Chern number for QHE TKNN edge states for QHE Laughlin, Halperin, YH

Bulk-Edge correspondence

Common property of topological ordered states

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SLIDE 11

How to understand gapped quantum liquids ?

Bulk

classically featureless : need geometrical phase

Edge

low energy localized modes in the gap 1-st Chern number for QHE TKNN edge states for QHE Laughlin, Halperin, YH

Z2 Berry Phase as a Topological Order Parameter of bulk Entanglement Entropy to detect edge states (generic Kennedy triplets)

As for quantum spins Bulk-Edge correspondence

Common property of topological ordered states

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SLIDE 12

The RVB state by Anderson

Quantum Liquid (Example 1)

|Singlet Pair12 = 1 √ 2(| ↑1↓2 − | ↓1↑2)

|G =

  • J=Dimer Covering

cJ ⊗ij |Singlet Pairij

Local Singlet Pairs : (Basic Objects)

Purely Quantum Objects are basic Purely Quantum Objects are basic

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SLIDE 13

The RVB state by Anderson

Quantum Liquid (Example 1)

|Singlet Pair12 = 1 √ 2(| ↑1↓2 − | ↓1↑2)

|G =

  • J=Dimer Covering

cJ ⊗ij |Singlet Pairij

Local Singlet Pairs : (Basic Objects)

Purely Quantum Objects are basic Purely Quantum Objects are basic

Spins disappear as a Singlet pair

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SLIDE 14

The RVB state by Pauling

Quantum Liquid (Example 2)

Purely Quantum Objects are basic Purely Quantum Objects are basic |Bond12 = 1 √ 2(|1 + |2) = 1 √ 2(c†

1 + c† 2)|0

|G =

  • J=Dimer Covering

cJ ⊗ij |Bondij

Local Covalent Bonds : (Basic Objects)

Do Not use the Fermi Sea

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SLIDE 15

The RVB state by Pauling

Quantum Liquid (Example 2)

Purely Quantum Objects are basic Purely Quantum Objects are basic |Bond12 = 1 √ 2(|1 + |2) = 1 √ 2(c†

1 + c† 2)|0

|G =

  • J=Dimer Covering

cJ ⊗ij |Bondij

Local Covalent Bonds : (Basic Objects)

Do Not use the Fermi Sea

Delocalized charge as a covalent bond

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SLIDE 16

“Classical” Observables

Charge density, Spin density,...

“Quantum” Observables !

Quantum Interferences: Probability Ampliture (overlap) Aharonov-Bohm Effects Phase (Gauge) dependent

Quantum Interference for the Classification

Use Quantum Interferences To Classify Quantum Liquids

:Berry Connection :Berry Phase

OG = G|O|G = G′|O|G′ = OG′ |G′ = |Geiφ

O = n↑ ± n↓, · · ·

G|G + dG = 1 + G|dG

A = G|dG iγ =

  • A

|Gi = |G′

ieiφi

G1|G2 = G′

1|G′ 2ei(φ1−φ2)

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SLIDE 17

Examples: RVB state by Anderson

No Long Range Order in Spin-Spin Correlation |G =

  • J=Dimer Covering

cJ ⊗ij |Singlet Pairij

|Singlet Pair12 = 1 √ 2(| ↑1↓2 − | ↓1↑2)

Local Singlet Pair is a Basic Object

Spins disappear as a Singlet pair

How to Characterize the Local Singlet Pair ?

|G = 1 √ 2(|↑i↓j − |↓i↑j)

Use Berry Phase to characterize the Singlet! Singlet does not carries spin but does Berry phase

γsinglet pair = π mod 2π

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SLIDE 18

Examples: RVB state by Anderson

No Long Range Order in Spin-Spin Correlation |G =

  • J=Dimer Covering

cJ ⊗ij |Singlet Pairij

|Singlet Pair12 = 1 √ 2(| ↑1↓2 − | ↓1↑2)

Local Singlet Pair is a Basic Object

Spins disappear as a Singlet pair

How to Characterize the Local Singlet Pair ?

|G = 1 √ 2(|↑i↓j − |↓i↑j)

Use Berry Phase to characterize the Singlet! Singlet does not carries spin but does Berry phase

γsinglet pair = π mod 2π

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SLIDE 19

generic Heisenberg Models (with frustration)

Z2 Berry phases for gapped quantum spins H =

  • ij

JijSi · Sj

Time Reversal Invariant

ΘN = (iσ1

y) ⊗ (iσ2 y) · · · (iσN y )K

Θ2

N = (−)N

[H, ΘN] = 0

Mostly N: even Θ2

N = 1

(probability 1/2 in HgTe)

ΘNSiΘ−1

N = −Si

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SLIDE 20

Z2 Berry phases for gapped quantum spins

H(x = eiθ) C = {x = eiθ|θ : 0 → 2π}

Si · Sj → 1 2(e−iθSi+Sj− + e+iθSi−Sj+) + SizSjz

numerically

Calculate the Berry Phases using the Entire Many Spin Wavefunction

γC =

  • C

Aψ =

  • C

ψ|dψ =

  • π

: mod 2π

Z2 quantization Time Reversal ( Anti-Unitary ) Invariance Require excitation Gap! Only link <ij>

Define a many body hamiltonian by local twist as a parameter U(1)

Z2

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SLIDE 21

Parameter Dependent Hamiltonian Berry Connections Berry Phases Phase Ambiguity of the eigen state Berry phases are not well-defined without specifying the gauge Well Defined up to mod

Berry Connection and Gauge Transformation

Aψ = ψ|dψ = ψ| d

dxψdx.

|ψ(x) = |ψ′(x)eiΩ(x) Aψ = A′

ψ + idΩ = A′ ψ + idΩ

dx dx

(Abelian)

iγC(Aψ) =

  • C

Aψ

γC(Aψ) = γC(Aψ′) +

  • C

dΩ

Gauge Transformation

2π × (integer) if eiΩ is single valued

2π

γC(Aψ) ≡ γC(Aψ′) mod 2π

H(x)|ψ(x) = E(x)|ψ(x), ψ(x)|ψ(x) = 1.

H(x)

= (x)

  • (x)
  • (x)
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SLIDE 22

Anti-Unitary Operator Berry Phases and Anti-Unitary Operation

AΨ = Ψ|dΨ =

  • J

C∗

JdCJ

AΘΨ = ΨΘ|dΨΘ =

  • J

CJdC∗

J = −AΨ

Anti-Unitary Operator and Berry Phases

  • J

C∗

JCJ = Ψ|Ψ = 1

|Ψ =

  • J

CJ |J |ΨΘ = Θ|Ψ =

  • J

C∗

J |JΘ,

|JΘ = Θ|J

γC(AΘΨ) = −γC(AΨ)

Θ = KUΘ, K : Complex conjugate UΘ : Unitary

(parameter independent) (Time Reversal, Particle-Hole)

  • J

dC∗

JCJ +

  • J

C∗

JdCJ =0

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SLIDE 23

Anti-Unitary Symmetry Invariant State

  • ex. Unique Eigen State

To be compatible with the ambiguity, the Berry Phases have to be quantized as

Anti-Unitary Invariant State and Z2 Berry Phase

[H(x), Θ] = 0

∃ϕ,

|ΨΘ = Θ|Ψ = |Ψeiϕ ≃ |Ψ

Gauge Equivalent(Different Gauge)

γC(AΨ) = −γC(AΘΨ) ≡ −γC(AΨ), mod2π

γC(AΨ) = π

mod 2π

Z2 Berry phase

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SLIDE 24

Numerical Evaluation of the Berry Phases (incl. non-Abelian)

(1) Discretize the periodic parameter space

x0, x1, · · · , xN = x0 xn = eiθn

θn+1 = θn + ∆θn

θ0 = 0, θN = 2π

∀∆θn → 0

(2) Obtain eigen vectors (3) Define Berry connection in a discretized form

King-Smith & Vanderbilt ’93 (polarization in solids) Luscher ’82 (Lattice Gauge Theory)

  • T. Fukui, H. Suzuki & YH ’05 (Chern numbers)

An = Im logψn|ψn+1 H(xn)|ψi

n = Ei(xn)|ψi n

(4) Evaluate the Berry phase

Gauge invariant after the discretization

An = Im log det Dn, {Dn}ij = ψi

n|ψj n+1

Independent of the choice of the phase

|ψn → |ψn′eiΩn

= Im logψ0|ψ1ψ1|ψ · · · = Im log det D1D2 · · · Dn

γ =

N−1

  • n=0

An

non-Abelian non-Abelian ( ) Convenient for Numerics

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SLIDE 25

Z2-quantization of the Berry phases protects from

continuous change

Adiabatic Continuation & the Quantization

Adiabatic Continuation in a gapped system Renormalization Group in a gapless system

Introduce interaction between singlets

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SLIDE 26

Strong bonds : bonds AF bonds : bonds

Local Order Parameters of Singlet Pairs

1D AF-AF,AF-F Dimers

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

AF-AF F-AF AF-AF case

π

F-AF case

π

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

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SLIDE 27

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

H = J

  • ij

Si · Sj + D

  • i

(Sz

i )2

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

  • D < DC

D > DC

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

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

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SLIDE 28

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

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SLIDE 29

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

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SLIDE 30

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

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SLIDE 31

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

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SLIDE 32

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

slide-33
SLIDE 33

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

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SLIDE 34

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

slide-35
SLIDE 35

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

slide-36
SLIDE 36

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

Z2Berry phase

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

  • : dimerization strength

: dimerization strength

  • Topological Quantum Phase Transitions with translation invariance
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SLIDE 37

Topological Classification of Gapped Spin Chains

✦ S=1,2 dimerized Heisenberg model

J1 = cos θ, J2 = sin θ

(2,0) (1,1) (0,2) (4,0) (3,1) (2,2) (1,3) (0,4)

S = 2 N = 10 S = 1 N = 14

(4,0) (3,1) (2,2) (1,1)

(2,0) (0,2)

: S=1/2 singlet state : Symmetrization

Berry phase

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

Reconstruction of valence bonds!

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SLIDE 38

Topological Classification of Gapped Spin Chains (cont.)

✦ S=2 Heisenberg model with D-term

0.5 1 1.5 2

S = 2 N = 10

Reconstruction of valence bonds!

:0 magnetization

Berry phase

Red line denotes the non trivial Berry phase

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

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SLIDE 39

Topological Classification of Generic AKLT (VBS) models

|{φi,j} =

  • ij
  • eiφij/2a†

ib† j − e−iφij/2b† ia† j

Bij |vac(4)

H({φi,i+1}) =

N

  • i=1

2Bi,i+1

  • J=Bi,i+1+1

AJP J

i,i+1[φi,i+1],

Twist the link of the generic AKLT model

Berry phase on a link (ij)

γij = Bijπ mod 2π

The Berry phase counts the number of the valence bonds! S=1/2 objects are fundamental in S=1&2 spin chains FRACTIONALIZATION S=1/2

Contribute to the Entanglement Entropy as of Edge states

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

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SLIDE 40

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

(a) (b)

J=1: J=2: = 0:

  • =

:

  • ฀

฀ ฀ ฀

  • (c)

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

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SLIDE 41

Entanglement Entropy to detect edge states

A B

direct calculation of spectrum with bondaries

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SLIDE 42

Mixed State From Entanglement

Direct Product State Entangled State

Partial Trace How much the State is Entangled between A & B?

Entanglement Entropy

A B

System = A ⊕ B State =

  • ΨA ⊗ ΨB

|ΨAB = 1 √ D

D

  • j

|Ψj

A ⊗ |Ψj B

|ΨAB = |ΨA ⊗ |ΨB

ρAB = 1 D

D

  • jk

|Ψj

AΨk A| ⊗| Ψj BΨk B|

Pure State Mixed State

SA = −log ρA = log D ρAB = |ΨABΨAB| ρA = TrB ρAB = 1 D

D

  • j

|Ψj

AΨj A|

Entanglement Entropy :

D = 1

Vidal, Latorre, Rico, Kitaev ‘02

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SLIDE 43

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

E.E. & Edge states (Gapped)

(of spins, fermions...)

A B

Entanglement Entropy > (# edge states) Log DE

  • S. Ryu & YH, Phys. Rev. B73, 245115 (2006)

(Fermions)

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SLIDE 44

EE of the Generic VBS States (S=1,2,3,...)

Fractionalization : Emergent as edge states (Quantum Resources for qbits)

  • H. Katsura, T.Hirano & YH, Phys. Rev. B76, 012401 (2007)

T.Hirano & YH, J. Phys. Soc. Jpn. 76, 1 13601 (2007)

HV BS =

N

  • i=1
  • Si ·

Si+1 + αHS

extra,

  • S2

i = S(S + 1)

HS=1

extra =

  • i

1 3( Si · Si+1)2 HS=2

extra =

  • i

2 9( Si · Si+1)2 + 1 63( Si · Si+1)3 +

|VBS =

L

  • j=0

(a†

jb† j+1 − b† ja† j+1)S|vac

S EE

Effective Boundary spins

Degrees of Freedom

1 2 S 2 Log 2 Seff=1/2 22=4 2 Log 3 Seff=1 32=9 2 Log (S+1) Seff=S/2 (S+1)2

SL = −log ρρ → 2 log(S + 1), (L → ∞)

Boundary Spins: S/2

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SLIDE 45

Another Models

(Pi + P −1

i

) = S1,i · S2,i + S1,i+1 · S2,i+1 + S1,i · S1,i+1 + S2,i · S2,i+1 + S1,i · S2,i+1 + S2,i · S1,i+1 + 4(S1,i · S2,i)(S1,i+1 · S2,i+1) + 4(S1,i · S1,i+1)(S2,i · S2,i+1) − 4(S1,i · S2,i+1)(S2,i · S1,i+1).

Spin ladder model with four-spin cyclic exchange W e set parameters as

Ferromagnetic Rung Singlet Dominant Collinear Spin K J Dimer LRO Dominant Vector Chirality θ Scalar Chiral LRO

J = 2K

Self dual at the point of

  • A. Lauchli, G. Schmid and M. Troyer (2003)
  • T. Hikihara, T. Momoi and X. Hu (2003)

H =

  • i

{JrS1,i · S2,i + Jl(S1,i · S1,i+1 + S2,i · S2,i+1) + K(Pi + P −1

i

)}

  • J = Jr = Jl = cos θ

K = sin θ

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SLIDE 46

Rung singlet phase

Topologically equivalence

θ = 6

Hr =

  • i=1

S1,i · S2,i

Rung singlets

Berry phase remains the same

θ = 2.6

Hps =

  • i∈odd

(S1,i × S2,i) · (S1,i+1 × S2,i+1)

Plaquette singlet (PS)

Vector chirality phase

Adiabatic deformation

  • I. Maruyama, T. Hirano, YH, arXiv:0806.4416
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SLIDE 47

Bulk-Edge correspondence for spins

Bulk: Z2 Berry phases Edge: Entanglement Entropy & low energy states in the gap

S = 1/2 is always fundamental ( electron spin )

K : complex conjugate

ΘN = (iσ1

y) ⊗ (iσ2 y) · · · (iσN y )K

ΘL

2 = −1

ΘR

2 = −1

Global TR Local (edge) TR

Θ2

N = 1

ΘN

Θedges = ΘL ⊗ ΘR ΘL, ΘR

slide-48
SLIDE 48

Bulk-Edge correspondence for spins

Bulk: Z2 Berry phases Edge: Entanglement Entropy & low energy states in the gap

S = 1/2 is always fundamental ( electron spin )

K : complex conjugate

ΘN = (iσ1

y) ⊗ (iσ2 y) · · · (iσN y )K

ΘL

2 = −1

ΘR

2 = −1

Global TR Local (edge) TR

Θ2

N = 1

ΘN

Θedges = ΘL ⊗ ΘR ΘL, ΘR

3D ?

slide-49
SLIDE 49

Thank you