量子液体とトポロジカル秩序 Quantum liquids & topological order Use of the Bulk-edge correspondence

(1)

筑波大学大学院数理物質科学研究科物理学専攻

初貝安弘

量子液体とトポロジカル秩序

Quantum liquids & topological order Use of the Bulk-edge correspondence

物理科学特別講義（バルクエッジ対応の物理）

Physics of bulk-edge correspondence, Hiroshima Univ. June 27-29, 2012

バルクエッジ対応の物理：背景２

background 2

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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 S=1 Spin chain ( Haldane Phase )

Integer spin chains (S=1,2) with dimerization Generic valence bond solid (VBS) states

Dimer models (Rokhsar-Kivelson) Half filled Kondo Lattice

How to Classify the Quantum Liquid Phases New way to think about !

Topological/quantum orders

(3)

Order & Symmetry

Magnetic Order

Disordered Ordered

(4)

Order & Symmetry

Magnetic Order

special direction! ( symmetry broken)

(5)

Order & Symmetry

Magnetic Order

No special direction (symmetric)

(6)

Order & Symmetry

Magnetic Order

m( ⇤ ⇤ r )

⇥ r

Local magnetic structure around Local order parameter :

Ferromagnetic / Antiferromagnetic, ...

(7)

Order & Symmetry

Magnetic Order

m( ⇤ ⇤ r )

⇥ r

Local magnetic structure around Local order parameter :

Ferromagnetic / Antiferromagnetic, ...

m(r ) ⇥ =0 ?

⌅ m(r )m(r ) ⇧ ⇥ 0 ? ( | r r | ⇥⇤ )

Spontaneous symmetry breaking Long Range Order

(8)

Order & Symmetry

Disordered

Quantum Disordered Gapped

(9)

Order & Symmetry

Disordered

Quantum Disordered

No Observables to Characterize

Gapped

(10)

Order & Symmetry

Disordered

Quantum Disordered

Quantum Liquids & Spin Liquids No Observables to Characterize

Gapped

(11)

Order & Symmetry

Disordered

Quantum Disordered Gapped

Quantum Liquids & Spin Liquids

(12)

Order & Symmetry

Disordered

Quantum Disordered Gapped

Gapped: Nothing in the gap

(13)

Order & Symmetry

Disordered

Quantum Disordered Gapped

Gapped: Nothing in the gap

No low lying excitations

(14)

Order & Symmetry

Disordered

Quantum Disordered

No Response against small perturbation

Gapped

Gapped: Nothing in the gap

No low lying excitations

(15)

Order & Symmetry

Disordered

Quantum Disordered

No Response against small perturbation

Gapped

Gapped: Nothing in the gap No low lying excitations

?? ^{打てども響かず}

(16)

Order & Symmetry

Disordered

Quantum Disordered

No Response against small perturbation

Gapped

Gapped: Nothing in the gap No low lying excitations

?? ^{打てども響かず}

c.f. gapless modes:

acoustic phonons zero sounds

spin waves

(17)

Order & Symmetry

Disordered

Quantum Disordered

Need Novel Kind of Orders :

No Response against small perturbation

Gapped

Gapped: Nothing in the gap No low lying excitations

?? ^{打てども響かず}

spin waves

(18)

Order & Symmetry

Disordered

Quantum Disordered

Need Novel Kind of Orders :

No Response against small perturbation

Gapped

Gapped: Nothing in the gap No low lying excitations

?? ^{打てども響かず}

spin waves

Topological Order

(19)

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)

Local disturbance induces a finite cost

(20)

Excitations and Local Order Parameter

(21)

Excitations and Local Order Parameter

(22)

Excitations and Local Order Parameter

E O(1)

Excitation Gap

(23)

Excitations and Local Order Parameter

Gapless excitation:

E O(1)

Excitation Gap

Need some mechanism

(24)

Excitations and Local Order Parameter

Nambu-Goldstone mechanism Gapless excitation:

E O(1)

Excitation Gap

(25)

Excitations and Local Order Parameter

breaking of continuous symmetry based on the local order parameter

E O(1)

Excitation Gap

(26)

Excitations and Local Order Parameter

Slowly varying its local order parameter : spin waves Lieb-Schultz-Mattis

E O(1)

Excitation Gap

(27)

Excitations and Local Order Parameter

E O(1)

Excitation Gap

Any kinds of Fermi surfaces

(28)

Excitations and Local Order Parameter

Appearance of Edge states

E O(1)

Excitation Gap

(29)

Excitations and Local Order Parameter

Topological Origin

Quantum Hall Effects & Haldane spin chains

quantum/topological order

E O(1)

Excitation Gap

(30)

Excitations and Local Order Parameter

Topological Origin

Quantum Hall Effects & Haldane spin chains

quantum/topological order

E O(1)

Excitation Gap

fermions as boundary states in high dimensions

( domain wall fermions)

(31)

How to characterize the phase

Without Symmetry Breaking Quantum Liquids are Featureless !!

Use Quantum Interference!

(32)

The RVB state by Anderson

Quantum Liquid

|Singlet Pair₁₂ = 1

2 (| ^{1 2} | ^{1 2} )

|G =

J=Dimer Covering

c_J _ij |Singlet Pair_ij

Dimer covering

with non orthogonal

& local states

(33)

The RVB state by Anderson

Quantum Liquid

|Singlet Pair₁₂ = 1

2 (| ^{1 2} | ^{1 2} )

|G =

J=Dimer Covering

c_J _ij |Singlet Pair_ij

Local Singlet Pairs :

(Fundamental Objects)

Spins disappear as a Singlet pair

Local Quantum Objects for characterization

(34)

The RVB state by Pauling

One more Quantum Liquid

|Bond₁₂ = 1

2 (|1 + |2 ) = 1

2 (c^†₁ + c^†₂)|0

|G =

J=Dimer Covering

c_J _ij |Bond_ij

Local Covalent Bonds : (Fundamental Objects)

Do Not use the Fermi Sea

Delocalized charge as a covalent bond

Local Quantum Objects for characterization

(35)

Tools to play with the Quantum Liquids

Local Characterization

Quantized Berry Phases as

a Topological Local Order Parameter Global Characterization

Entanglement Entropy as a Novel Tool

for the Condensed Matter Physics

Quantum Liquids Featureless !!

Closely related to the edge states

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「対称性の破れ」で物質相の記述に本当に十分？

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「対称性の破れ」で物質相の記述に本当に十分？

真に量子的な相

いかなる対称性の破れもない

古典的な秩序変数が存在しない

なおかつ多種多様な物質相が存在する！！

(38)

「対称性の破れ」で物質相の記述に本当に十分？

真に量子的な相

いかなる対称性の破れもない

古典的な秩序変数が存在しない

なおかつ多種多様な物質相が存在する！！

量子液体相、スピン液体

秩序変数対称性の破れ

(39)

「対称性の破れ」で物質相の記述に本当に十分？

真に量子的な相

いかなる対称性の破れもない

古典的な秩序変数が存在しない

なおかつ多種多様な物質相が存在する！！

量子液体相、スピン液体

トポロジカル秩序

秩序変数対称性の破れ

量子秩序

(40)

「対称性の破れ」で物質相の記述に本当に十分？

真に量子的な相

いかなる対称性の破れもない

古典的な秩序変数が存在しない

なおかつ多種多様な物質相が存在する！！

トポロジカルな量子相転移

量子液体相、スピン液体

トポロジカル秩序

秩序変数対称性の破れ

量子秩序

(41)

量子液体相、スピン液体相とは？？

(42)

量子液体相、スピン液体相とは？？

整数量子ホール相、分数量子ホール相、

量子スピンホール相：トポロジカル絶縁体

フラストレートしたスピン系、ダイマー系

RVB状態、Haldane スピン系、VBS状態

近藤格子、Kitaev 模型、Levin-Wen模型

(43)

量子液体相、スピン液体相とは？？

整数量子ホール相、分数量子ホール相、

量子スピンホール相：トポロジカル絶縁体フラストレートしたスピン系、ダイマー系 RVB状態、Haldane スピン系、VBS状態近藤格子、Kitaev 模型、Levin-Wen模型

トポロジカルな量子的秩序変数

(44)

量子液体相、スピン液体相とは？？

整数量子ホール相、分数量子ホール相、

量子スピンホール相：トポロジカル絶縁体フラストレートしたスピン系、ダイマー系 RVB状態、Haldane スピン系、VBS状態近藤格子、Kitaev 模型、Levin-Wen模型

トポロジカルな量子的秩序変数

ベリー位相チャーン数

エンタングルメント

エッジ状態

(45)

A way to the Quantum liquids

Example: frustration

Relax local degrees of freedom

(46)

Frustration : residual entropy spin

charge

H = X

H _loc

local degeneracy

# deg.

e

^cN

c : ^const.

N : system size

⇡

# low energy modes

Quantum effects

(47)

Frustration : residual entropy spin

charge

H = X

H _loc

local degeneracy

# deg.

e

^cN

c : ^const.

N : system size

⇡

Quantum effects

(48)

Residual Entropy to form gapped quantum liquid

H = X

H _loc

local degeneracy

Gap opening

Stabilization to relax local entropy

Quantum effects

(49)

H = X

H _loc

local degeneracy

Gap opening

Quantum effects

(50)

Formation of local quantum object Multimer as a generic dimer

H = X

H _loc

local degeneracy

Gap opening

Covalent molecular orbital

c.f. Tamura ’06, Matsuda-Motome ’06, Katsufuji

trimer tetramer

Quantum effects

(51)

Energy band & gap : physicist & chemist ? physicist

Sorry if I’m wrong

itinerant electrons

(52)

Energy band & gap : physicist & chemist ?

make energy band metal

physicist

itinerant electrons hopping

(53)

make energy band metal

physicist

(54)

Covalent molecular orbital physicist

(55)

Peierls instability

physicist

Opening gap stabilize

hopping

(56)

physicist

chemist

form molecules first hopping

(57)

physicist

chemist

(58)

physicist

chemist

(59)

physicist

chemist

(60)

physicist

make bands of molecules

stabilize

chemist

form molecules first

Dimer & Multimer non orthogonality

(61)

physicist

stabilize

chemist

Dimer & Multimer Adiabatic process Insulator

non orthogonality

EF

(62)

physicist

stabilize

chemist

quantum objects to be respected non orthogonality

EF

(63)

physicist

stabilize

chemist

quantum objects to be respected non orthogonality

short range entanglement

EF

(64)

H = X

H _loc

local degeneracy

Local anti-ferromagnetism

Local charge ordering

(65)

Multimer as a generic dimer

H = X

H _loc

local degeneracy

Gap opening

by quantum effect Stabilization to relax

local entropy Covalent molecular orbital

Singet formation

Dimer formation

singlet pair

covalent bond

(66)

H = X

H _loc

local degeneracy

Gap opening

Singet formation

Dimer formation

singlet pair

covalent bond

RVB: Anderson

(67)

H = X

H _loc

local degeneracy

Gap opening

Singet formation

Dimer formation

singlet pair

covalent bond

RVB: Anderson

RVB: Pauling

(68)

Quantum effects to relax classical frustration

Quantum & local Gapped

# deg.^⇡

e

^cN

Classical frustration

Quantum effects

finite low energy states (topological)

Gapped quantum liquid

(boundary condition dependent) gapped

“Topological insulators” (c.f. QHE, QSHE, top.super, Haldane chain, ladders... )

(69)

Gapped quantum liquids

Still Lots of variety

Absence of fundamental symmetry breaking

No Response against small perturbation

No low lying excitations

??

gapless modes:

spin waves

Zoo

Nambu-Goldstone

Fermi surface

massless Dirac fermions with/without doubling

(70)

Gapped quantum liquids

Something for classification Still Lots of variety

??

gapless modes:

spin waves

Zoo

Nambu-Goldstone

Fermi surface

(71)

Gapped quantum liquids

Something for classification

Topological order Berry connections

Edge states & entanglement

Still Lots of variety

??

gapless modes:

spin waves

Zoo

Nambu-Goldstone

Fermi surface

(72)

How to understand gapped quantum liquids ?

Bulk state

Edge state

Bulk-Edge correspondence

Bulk without boundary is boring

Characteristic localized states with edges

QHE, Spin chains, Graphene, QSHE, Andreev bound st., photonic crystals, cold atoms ...

YH’93

(73)

Quantum Liquids & Entanglement

Gapped quantum liquids featureless

bulk

(74)

Quantum Liquids & Entanglement

Divide into A & B

A B

bulk

(75)

Quantum Liquids & Entanglement

Divide into A & B

A B

bulk

How much the states are entangled

between A & B?

(76)

Quantum Liquids & Entanglement

Divide into A & B

A B

bulk

How much the states are entangled between A & B?

Entanglement entropy

⇢_AB = | ^ABih ^AB|

⇢_A = Tr_B ⇢_AB S_A = hlog ⇢_Ai

Characterize the gapped quantum liquids

(77)

Quantum Liquids & Entanglement

Divide into A & B

A B

bulk

How much the states are entangled between A & B?

Edge states near the boundary between A & B

(78)

Quantum Liquids & Entanglement

Divide into A & B

A B

bulk

How much the states are entangled between A & B?

Effective edge states reflect quantum entanglement Edge states near the boundary between A & B

(79)

local deg. freedom, entanglement and edge states

In-gap edge states

# local degree of freedom How to relax local entropy

Quantum effects with boundaries

Entangled & gapped

# deg.^⇡

e

^cN ^{# deg.}^⇡

^e

^V^e^/V⁰

V_e : volume (length) of edges

(80)

Quantum entanglement EATS local degrees of freedom and makes the system gapped

# deg.^⇡

e

^cN ^{# deg.}^⇡

^e

^V^e^/V⁰

(81)

# deg.^⇡

e

^cN ^{# deg.}^⇡

^e

^V^e^/V⁰

Superconductivity :Anderson mechanism

NG bosons are eaten by gauge field and gapped.

(82)

# deg.^⇡

e

^cN ^{# deg.}^⇡

^e

^V^e^/V⁰

Superconductivity :Anderson mechanism

NG bosons are eaten by gauge field and gapped.

They revive again as edge states near the boundaries

(83)

Physics of bulk-edge correspondence, Hiroshima Univ. June 27-29, 2012

量子液体とトポロジカル秩序 Quantum liquids & topological order Use of the Bulk-edge correspondence