Physics of bulk-edge correspondence, Hiroshima Univ. June 27-29, 2012
物理科学特別講義（バルクエッジ対応の物理）

バルクエッジ対応の物理：背景２background 2

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

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Order & Symmetry

Magnetic Order
Ordered
DisorderedOrder & Symmetry

Magnetic Order
Ordered
Disordered

special direction! ( symmetry broken)

Order & Symmetry

Magnetic Orderspecial direction! ( symmetry broken)

Ordered
Disordered
No special direction (symmetric)Order & Symmetry

Magnetic Orderspecial direction! ( symmetry broken)

Ordered
Disordered
Local order parameter : m(⇤⇤ r)

No special direction (symmetric)

Local magnetic structure around ⇥r
Ferromagnetic / Antiferromagnetic, ...

Order & Symmetry

Magnetic Orderspecial direction! ( symmetry broken)

Ordered
Disordered
Local order parameter : m(⇤⇤ r)

No special direction (symmetric)

Local magnetic structure around ⇥r
Ferromagnetic / Antiferromagnetic, ...
Spontaneous symmetry breakingm(r)⇥ =0 ?

Long Range Order⌅m(r)m(r )⇧ ⇥0?

(|rr | ⇥⇤ )

Order & Symmetry
Quantum Disordered
Disordered

GappedOrder & Symmetry
Quantum Disordered

GappedDisordered

No Observables to CharacterizeOrder & Symmetry
Quantum Disordered

GappedDisordered

No Observables to Characterize
Quantum Liquids & Spin Liquids

Order & Symmetry

Quantum Liquids & Spin Liquids

Quantum Disordered

Disordered

GappedOrder & Symmetry

Quantum Liquids & Spin Liquids

Quantum Disordered

Disordered

Gapped: Nothing in the gap

GappedOrder & Symmetry

Quantum Liquids & Spin Liquids

Quantum Disordered

Disordered

Gapped: Nothing in the gap
No low lying excitations

GappedOrder & Symmetry

Quantum Liquids & Spin Liquids

Quantum Disordered

GappedDisordered

Gapped: Nothing in the gap
No low lying excitations
No Response against small perturbationOrder & Symmetry

Quantum Liquids & Spin Liquids

Quantum Disordered

GappedDisordered

Gapped: Nothing in the gap
No low lying excitations
No Response against small perturbation??

打てども響かずOrder & Symmetry

Quantum Liquids & Spin Liquids

Quantum Disordered

GappedDisordered

Gapped: Nothing in the gap
No low lying excitations
No Response against small perturbation??

打てども響かずc.f. gapless modes:
acoustic phonons
zero sounds
spin waves

Order & Symmetry

Quantum Liquids & Spin Liquids

Quantum Disordered

GappedDisordered

Gapped: Nothing in the gap
No low lying excitations
No Response against small perturbation??

打てども響かず

Need Novel Kind of Orders :c.f. gapless modes:
acoustic phonons
zero sounds
spin waves

Order & Symmetry

Quantum Liquids & Spin Liquids

Quantum Disordered

GappedDisordered

Gapped: Nothing in the gap
No low lying excitations
No Response against small perturbation??

打てども響かず

Need Novel Kind of Orders :c.f. gapless modes:
acoustic phonons
zero sounds
spin waves

Topological OrderExcitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite costExcitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite costExcitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite costExcitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Gapless excitation: Need some mechanism

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Gapless excitation: Need some mechanism
Nambu-Goldstone mechanism

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Gapless excitation: Need some mechanism
Nambu-Goldstone mechanism
breaking of continuous symmetry based on the local order parameter

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Gapless excitation: Need some mechanism
Nambu-Goldstone mechanism
breaking of continuous symmetry based on the local order parameter
Slowly varying its local order parameter : spin waves

Lieb-Schultz-Mattis

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Gapless excitation: Need some mechanism
Nambu-Goldstone mechanism
breaking of continuous symmetry based on the local order parameter
Slowly varying its local order parameter : spin waves

Any kinds of Fermi surfaces

Lieb-Schultz-Mattis

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Gapless excitation: Need some mechanism
Nambu-Goldstone mechanismbreaking of continuous symmetry based on the local order parameter
Slowly varying its local order parameter : spin waves

Any kinds of Fermi surfaces
Appearance of Edge states

Lieb-Schultz-Mattis

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Gapless excitation: Need some mechanism
Nambu-Goldstone mechanism
breaking of continuous symmetry based on the local order parameter
Slowly varying its local order parameter : spin waves

Lieb-Schultz-Mattis

Any kinds of Fermi surfaces
Appearance of Edge states
Topological Origin

quantum/topological order

Quantum Hall Effects & Haldane spin chains

Excitations and Local Order Parameter

Discrete Symmetry Breaking: (ex. Ising order)
Local disturbance induces a finite cost
E O(1)

Excitation Gap

Gapless excitation: Need some mechanism
Nambu-Goldstone mechanism
breaking of continuous symmetry based on the local order parameter
Slowly varying its local order parameter : spin waves

Any kinds of Fermi surfaces
Appearance of Edge states
Topological Origin

quantum/topological order

Lieb-Schultz-Mattisfermions as boundary statesin high dimensions( domain wall fermions)

Quantum Hall Effects & Haldane spin chains

Quantum Liquids are Featureless !!
How to characterize the phase
Without Symmetry Breaking
Use Quantum Interference!

Quantum Liquid

The RVB state by Anderson|Singlet Pair12
|G =

1=
(|2cJ

J=Dimer Covering

1 2|1 2 )ij |Singlet Pairij

Dimer coveringwith non orthogonal& local states

Quantum Liquid

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

1=
(|2cJ

J=Dimer Covering

1 2|1 2 )ij |Singlet Pairij

Local Singlet Pairs :(Fundamental Objects)

Local Quantum Objects for characterization

One more Quantum Liquid

The RVB state by Pauling|Bond1211 ††=(|1 + |2 ) =(c1 + c2 )|022|G =Delocalized charge
as a covalent bondcJ
J=Dimer Covering

ij |Bondij

Do Not use the Fermi Sea

Local Covalent Bonds :(Fundamental Objects)

Local Quantum Objects for characterization

Tools to play with the Quantum Liquids

Quantum Liquids
Featureless !!
Local Characterization
Quantized Berry Phases asa Topological Local Order Parameter
Global Characterization
Entanglement Entropy as a Novel Toolfor the Condensed Matter Physics
Closely related to the edge states

「対称性の破れ」で物質相の記述に本当に十分？

「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！！

「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「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

対称性の破れトポロジカル秩序

量子秩序

「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

対称性の破れトポロジカル秩序

量子秩序

トポロジカルな量子相転移

量子液体相、スピン液体相とは？？

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

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

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

エッジ状態
チャーン数
エンタングルメント

A way to the Quantum liquids
Example: frustration
Relax local degrees of freedom

Frustration : residual entropy X
spin

H=Quantum effects

Hloc

local degeneracy# deg.

N : system sizec : const.
cN⇡e# low energy modes

chargeFrustration : residual entropy X
spin

H=Quantum effects

Hloc

local degeneracy# deg.

N : system sizec : const.
cN⇡e# low energy modes

chargeResidual Entropy to form gapped
quantumliquidX

H=

Hloc

local degeneracy

Quantum effects
Gap opening
Stabilization to relax
local entropy

Residual Entropy to form gapped
quantumliquidX

H=

Hloc

local degeneracy

Quantum effects
Gap opening
Stabilization to relax
local entropy

Residual Entropy to form gapped
quantum
liquidX

tetramerH=

Hloclocal degeneracy

Quantum effects

trimer

Gap opening
Stabilization to relax
local entropy
Formation of local quantum object
Multimer as a generic dimer
Covalent molecular orbitalc.f. Tamura ’06, Matsuda-Motome ’06, Katsufuji

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons
hopping

make energy band
metal

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons
hopping

make energy band
metal

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons
hopping

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons
hopping

Peierls instability
Opening gap
stabilize

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons
hoppingchemistform molecules first

Peierls instability
Opening gap
stabilize

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons
hoppingchemistform molecules first

Peierls instability
Opening gap
stabilize

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons
hoppingchemistform molecules first

Peierls instability
Opening gap
stabilize

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons
hoppingchemistform molecules first

Peierls instability
Opening gap
stabilize

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons

Peierls instability
stabilize

chemistform molecules first

non orthogonality
Dimer & Multimer

make bands ofmolecules

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons

Peierls instability
stabilize

chemistform molecules first

non orthogonality
Dimer & Multimer

make bands ofmolecules
Adiabatic process
EFInsulator

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons

Peierls instability
stabilize

chemistform molecules first

non orthogonality
Dimer & Multimer
quantum objects
to be respected

make bands ofmolecules
Adiabatic process
EFInsulator

Energy band & gap : physicist & chemist ? Sorry if I’m wrong
physicist

itinerant electrons

Peierls instability
stabilize

chemistform molecules first

non orthogonality

short range entanglement

Dimer & Multimer
quantum objects
to be respected

make bands ofmolecules
Adiabatic process
EFInsulator

Residual Entropy to form gapped
quantumliquidX

Local anti-ferromagnetism

H=

Hloc

local degeneracy

Local charge orderingResidual Entropy to form gapped
quantum
liquidX

Singet formation

H=

Hloc

local degeneracy

singlet pair

Dimer formation

Gap opening
by quantum effectStabilization to relax
local entropy

covalent bondMultimer as a generic dimer
Covalent molecular orbital

Residual Entropy to form gapped
quantumliquidX

Singet formationH=

Hloc

local degeneracy

singlet pair

Dimer formation

covalent bond

RVB: Anderson

Gap opening
by quantum effectStabilization to relax
local entropy

Multimer as a generic dimer
Covalent molecular orbital

Residual Entropy to form gapped
quantum
liquidX

Singet formation

H=

Hloc

local degeneracy

singlet pair

Dimer formation

covalent bond

RVB: Anderson

Gap opening
by quantum effectStabilization to relax
local entropy

Multimer as a generic dimer
Covalent molecular orbital

RVB: PaulingQuantum effects to relax classical frustration
Classical frustration
Quantum effects

gappedcN⇡# deg. e# low energy modes

finite low energy states(topological)(boundary condition dependent)

Quantum & local

GappedGapped quantum liquid(c.f. QHE, QSHE, top.super,“Topological insulators”

Haldane chain, ladders... )Gapped quantum liquids??

No low lying excitations
No Response against small perturbation

Nambu-Goldstone

Absence of fundamental symmetry breaking
Still Lots of variety

Zoogapless modes:
acoustic phonons
zero sounds
spin waves

Fermi surface
massless Dirac fermions
with/without doublingGapped quantum liquids??

No low lying excitations
No Response against small perturbation

Nambu-Goldstone

Absence of fundamental symmetry breaking
Still Lots of variety

gapless modes:
acoustic phonons
zero sounds
spin waves

Fermi surface

Zoomassless Dirac fermions
with/without doubling

Something for classification

Gapped quantum liquids??

No low lying excitations
No Response against small perturbation

Nambu-Goldstone

Absence of fundamental symmetry breaking
Still Lots of variety

gapless modes:
acoustic phonons
zero sounds
spin waves

Fermi surface

Zoomassless Dirac fermions
with/without doubling

Something for classification
Topological order
Berry connections
Edge states & entanglement

How to understand gapped quantum liquids ?
Bulk state
Bulk without boundary is boring

Edge state

Characteristic localized states with edges

Bulk-Edge correspondence

YH’93

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

Quantum Liquids & Entanglement
featurelessbulk

Gapped quantum liquids

Quantum Liquids & Entanglement
featurelessbulkA

Gapped quantum liquids

Divide into A & BB

Quantum Liquids & Entanglement
featurelessbulk

Gapped quantum liquids

Divide into A & B

How much the states are entangled
between A & B?
AB

Quantum Liquids & Entanglement
featurelessGapped quantum liquids

bulk

Divide into A & B

How much the states are entangled
between A & B?
AB

Entanglement entropy

SA =hlog ⇢A i

⇢AB = |AB ih

⇢A = TrB ⇢AB

AB |Characterize the gapped quantum liquids

Quantum Liquids & Entanglement
featurelessGapped quantum liquids

bulk

Divide into A & B

How much the states are entangled
between A & B?
AB

Entanglement entropy

SA =hlog ⇢A i

⇢AB = |AB ih

⇢A = TrB ⇢AB

AB |Characterize the gapped quantum liquids

Edge states near the boundary between A & B

Quantum Liquids & Entanglement
featurelessGapped quantum liquids

bulk

Divide into A & B

How much the states are entangled
between A & B?
AB

Entanglement entropy

SA =hlog ⇢A i

⇢AB = |AB ih

⇢A = TrB ⇢AB

AB |Characterize the gapped quantum liquids

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

local deg. freedom, entanglement and edge states
Quantum effects

with boundaries

How to relax local entropy
In-gap edge states# local degree of freedom
Entangled & gappedcN⇡# deg. e

Ve : volume (length) of edges
Ve /V0⇡# deg. e

local deg. freedom, entanglement and edge states
Quantum effects

with boundaries

How to relax local entropy
In-gap edge states# local degree of freedom
Entangled & gappedcN⇡# deg. e

Ve : volume (length) of edges
Ve /V0⇡# deg. e

Quantum entanglement EATS local degrees offreedom and makes the system gappedlocal deg. freedom, entanglement and edge states
Quantum effects

with boundaries

How to relax local entropy
In-gap edge states# local degree of freedom
Entangled & gappedcN⇡# deg. e

Ve : volume (length) of edges
Ve /V0⇡# deg. e

Quantum entanglement EATS local degrees offreedom and makes the system gapped
Superconductivity :Anderson mechanism
NG bosons are eaten by gauge field and gapped.

local deg. freedom, entanglement and edge states
Quantum effects

with boundaries

How to relax local entropy
In-gap edge states# local degree of freedom
Entangled & gappedcN⇡# deg. e

Ve : volume (length) of edges
Ve /V0⇡# deg. e

Quantum entanglement EATS local degrees of
freedom and makes the system gapped
Superconductivity :Anderson mechanism
NG bosons are eaten by gauge field and gapped.

They revive again as edge states near the boundaries

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

END