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）

Bulk-edge correspondence
in real world

Institute of Physics
University of Tsukuba

JAPAN
Yasuhiro Hatsugai

Outline
Are insulators boring ?
Zoo of insulators : variety to universality
Classification of the insulators

Zoo of boundary states
From the textbook to the research
Bulk-Edge correspondence

A lucky example: Bulk-Edge correspondence
Quantum Hall states
Graphene

Applications
Topological quantum phase transitions
How to observe 1/2 Hall conductance of Dirac fermions
Edge state device ?

Are insulators boring ??
Metal is useful.

copper, silver, gold: good conductors
doped semiconductors

Gapless excitations above the ground state

Lots of applications

Insulators : Gapped

Definition (today)

Energy gap above the ground state

Band insulators
Superconductors ?
Integer & Fractional Quantum Hall States
Integer spin chains (Haldane)
Dimer Models (Shastry-Sutherland)
Valence bond solid (VBS) states“Topological insulators”
Half filled Kondo LatticeSpin Hall insulators (TI in a narrow sense)
Kitaev model & string net

Are insulators boring ??
Insulators : Gapped
Band insulators
Superconductors
Integer & Fractional Quantum Hall States
Integer spin chains (Haldane)
Dimer Models (Shastry-Sutherland)
Valence bond solid (VBS) states
Half filled Kondo LatticeSpin Hall insulators
Kitaev model & string net

Absence of low energy excitations
Energy gap above the ground state

Lots of variety
Absence of fundamental symmetry breaking (mostly)
Quantum/spin liquids (gapped)

Are insulators boring ??
Insulators : Non metal, gapped
Band insulators
Gapped: Nothing in the gap : cf. Nambu-Goldstone boson
Superconductors
No lowInteger
lying excitations& Fractional Quantum Hall States
No Response
againstperturbation
Integer spin
chains small(Haldane)gapless modes:
acoustic phonons
Dimer Models (Shastry-Sutherland)zero sounds？？？
Valence bond solid (VBS)
statesspin waves??
Half filled Kondo LatticeSpin Hall insulators

Absence of low energy excitations
Energy gap above the ground state

Lots of variety
Absence of fundamental symmetry breaking (mostly)
No responses against for small perturbation

Are insulators boring ??

Quantum(gapped)Insulators: Nonliquids
metal, gapped
Band insulators
Superconductors
Integer & Fractional Quantum Hall States
Integer spin chains (Haldane)
Dimer Models (Shastry-Sutherland)
Valence bond solid (VBS) states
Half filled Kondo LatticeSpin Hall insulators
Kitaev model & string net
Zoo

Are insulators boring ??

Quantum(gapped)Insulators: Nonliquids
metal, gapped
Band insulators
Superconductors
Integer & Fractional Quantum Hall States
Integer spin chains (Haldane)
Dimer Models (Shastry-Sutherland)
Valence bond solid (VBS) states
Half filled Kondo LatticeSpin Hall insulators
Kitaev model & string net
Zoo

Something for classification

Are insulators boring ??

Quantum(gapped)Insulators: Nonliquids
metal, gapped
Band insulators
Superconductors
Integer
Integer && Fractional
Fractional Quantum
Quantum Hall
Hall States
StatesTopological Order
Integer spin chains (Haldane)
X.G.Wen ’89
Dimer Models (Shastry-Sutherland)
Valence bond solid (VBS) states
Half filled Kondo LatticeSpin Hall insulators
Kitaev model & string net
Zoo

Something for classification

Are insulators boring ??

Quantum(gapped)Insulators: Nonliquids
metal, gapped
Band insulators
Superconductors
Integer
Integer && Fractional
Fractional Quantum
Quantum Hall
Hall States
StatesTopological Order
Integer spin chains (Haldane)
X.G.Wen ’89
Dimer Models (Shastry-Sutherland)
Valence bond solid (VBS) states
Half filled Kondo LatticeSpin Hall insulators
Kitaev model & string net
Zoo

Something for classification
Topological order
Berry connections
Edge states

Zoo of Boundary (Edge) States in Cond. Mat.
From textbook examples to new discoveries

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals# bound states

Spin Ladder with ring exchanges

phase shiftSurface states of Semiconductors & polarization

Levinson’s theorem to the Friedel’s sum rule
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals

Surface states of Semiconductors & polarization
GaAs

Si 7×7

Levinson’s theorem to the Friedel’s sum rule
Solitons in polyacetylene
Edge states in quantum Hall effects

Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way
edge
modes
in
gyromagnetic
photonic
crystals
Spin Ladder+ --+ ---+ with---+ ---+ring---+ ---+exchanges---+ ---+ ---+ ---Surface states of Semiconductors & polarization
GaAs

Si 7×7

Levinson’s theorem to the Friedel’s sum rule
Solitons in polyacetylene
Edge states in quantum Hall effects

Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states

Berry
Edge states in 2D cold
atomsphasein optical lattice Polarization
One-way
edge
modes
in
gyromagnetic
photonic
crystals
Spin Ladder+ --+ ---+ with---+ ---+ring---+ ---+exchanges---+ ---+ ---+ ---Solitons in polyacetylene

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Solitons in polyacetylene

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals

polyacetylene

Spin Ladder with graphenering exchanges

diamondEdge states in quantum Hall effects

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Edge states in quantum Hall effects

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Edge states in quantum Hall effects

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene

Everything started from here
Local moments in integer discussspin chainslaternear the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Local moments in integer spin chains near the impurities

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects

Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Local moments in integer spin chains near the impurities

Edge states of the Haldane phase

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects

Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Spin Ladder with ring exchanges

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals

Spin Ladder with ring exchanges

Spin edge states in gapped spin systems
Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals

Zero bias conductance peaks of the d-wave superconductors

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities

Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Zero bias conductance peaks of the d-wave superconductors

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities

Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Zero bias conductance peaks of the d-wave superconductors

Levinson’s theorem to the Friedel’s sum rule

des
(11
0)
exi
bou
st
nda
ry

Surface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects

Zer
o

mo

Local moments in integer spin chains near the impurities

Andreev
bound states

Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice

y
r
a
d
s
n
e
u
One-way edge modes in gyromagnetic
photonic
crystals
o
d
b
o
)
m
0
0
o
1
r
(
e
z
Spin Ladder with ring exchangesNoZero energy localized states of graphene

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors

Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Zero energy localized states of graphenearmchairzigzagLevinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
s
e
d
o es
m
g
states in quantum Hall effects
zero air edEdge

No Armch

Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors

g
Zi
g

za
od
m

ro
Ze

ed
s

ge

Quantum Spin Hall Edge states

es

ist
ex

Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Kobayashi et al,
STM imageSpin Ladder
with
ring
exchanges
Phys. Rev. B71, 193406 (2005)

Quantum Spin Hall Edge states

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Quantum Spin Hall Edge states

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Quantum Spin Hall Edge states

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

One more thing :Ternary

Quantum Spin Hall Edge states

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

One more thing :Ternary

Quantum Spin Hall Edge states

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Edge states in 2D cold atoms in optical lattice

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Edge states in 2D cold atoms in optical lattice

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Edge states in 2D cold atoms in optical latticeOne-way edge modes in gyromagnetic photonic crystals

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical latticeSpin Ladder with ring exchanges

One-way edge modes in gyromagnetic photonic crystals

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene
Edge states in quantum Hall effects
Local moments in integer spin chains near the impurities
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical latticeSpin Ladder with ring exchanges

Why do we care edge states?
Why the Edge States are there??
Accidental ?
NO !

Y.Hatsugai, PRL 71,3697 (1993)

Inevitable reasonsPhysical Structures behind:“Bulk determines the edges”“Edge determines the bulk”
Bulk-Edge Correspondence
Protected by Topological constraints

Right / left to the symmetry
unnormalizable

Extended states

1 ikr! 0 (V ! 1)
e (r) ⇠ p e
V
V: Volume

8

6

4

2

5

10

15

20

normalizable
Bound states / Edge states1
r/a0
b (r) ⇠ p 3 ea0a0 :size of the bound state
1.2
1.0
0.8
0.6
0.4

Edge state is topological

0.2

1

2

3

4

5

6

Clear difference only in the infinite system with boundaries

Bulk-Edge correspondence
Universality
Bulk state(scattering state)
Bulk Gap
Non trivial VacuumControlwitheach other

Edge state(Bound state)

Particles in the gap

Y.Hatsugai, PRL 71,3697 (1993)

Bulk-Edge correspondence
Universality
Bulk state(scattering state)
Bulk Gap
Non trivial VacuumControlwitheach other

Edge state(Bound state)

Particles in the gap

Y.Hatsugai, PRL 71,3697 (1993)

Let’s discuss 2 lucky examples
QHE & Dirac fermionsBulk-Edge correspondence in QHE
QHE

Bulk-Edge correspondence
Common property of topological ordered statesclassically featurelessBulk

Thouless-Kohmoto-1-st Chern number for QHE
YH

Edge

Nightingale-den Nijs (TKNN)
Niu-Thouless-Wu

low energy localized modes in the gap
edge states for QHE

Edge states

Laughlin, Halperin, Wen, YH

Hall Conductance has a Topological meaning
When EF is in the j-th gap
Two topological quantities
2e
Sum of the First Chern numbers below EF
bulkC⌅Thouless-Kohmoto-Nightingale-den Nijs ‘82
xy =h⌅: ⇥ (k)<EF2eWinding number of the edge stateedgej⇥xy = I( j , C )on the complex energy surface
YH ‘93ahBulk ---- Edge Correspondence
Y.Hatsugai, PRL 71,3697 (1993)

xy

bulk

=

Cj = Ij
Chern # of j-th band

xy

edge

Ij 1

Manybody

As for the Chern # of
the single band

Difference in edge states
just above and below the band

Bulk - Edge Correspondence

xy

bulk

=

xy

edge

Manybody

Bulk --- Edge Correspondencebulkedge Graphene
= xy
xy
F=1/5

(a)

F=1/21

(b)
3

3
2

2
1

E

E

1
0

0

-1
-1

-2
-2

-3
0

0.2

0.4

ky

0.6

0.8

1

-3
0

0.2

0.4

k

0.6

0.8

1

Bulk --- Edge Correspondencebulkedge Graphene= xy
xy
Near Zero(a)

F=1/51(d)

F=1/21
F=1/5

(c)

1

1

F=1/21

(b)
3

3
2

0.5

2
1

E

1

0

E

0

E

E

0.5

0

-0.5-1

-0.5

0

-1

-2
-1

-1

-3

0

0.2

0

0.2

0.4 k 0.6
0.4

y

ky

0.6

0.8
0.8

1

-2

0
1

0.2

-3
0

0.2

0.4

ky

0.4

0.6

k

0.8
0.6

1
0.8

1

Universality in the zero modes
2D CuOof Dirac Fermions

2

2D Dirac fermions :Edge States
Zero mode localized states ??
Graphene

Fractional charge:
Jackiw ’84
YH, ’09 (review)d-wave superconductorAnalogy between graphene & d-wave superconductorUniversality of Zero Energy Edge States‘02---’04 S. Ryu & YH

Universality of Zero Energy Edge States‘02---’04 S. Ryu & YH

1.Zero energy edge states of graphene
Boundary Magnetic moments of grapheneUniversality of Zero Energy Edge States‘02---’04 S. Ryu & YH

1.Zero energy edge states of graphene
Boundary Magnetic moments of graphene
2.Andreev bound states of d-wave superconductors
Zero bias conductance peakUniversality of Zero Energy Edge States‘02---’04 S. Ryu & YH

1.Zero energy edge states of graphene
Boundary Magnetic moments of graphene
2.Andreev bound states of d-wave superconductors
Zero bias conductance peak

These 2 systems are
topologically equivalentUniversality of Zero Energy Edge States‘02---’04 S. Ryu & YH

1.Zero energy edge states of graphene
Boundary Magnetic moments of graphene
2.Andreev bound states of d-wave superconductors
Zero bias conductance peakd-wave superconductor
graphene
These 2 systems are
topologically equivalent
2 Dirac cones

4 Dirac cones

Universality of Zero Energy Edge States‘02---’04 S. Ryu & YH

1.Zero energy edge states of graphene
Boundary Magnetic moments of graphene
2.Andreev bound states of d-wave superconductors
Zero bias conductance peakgraphened-wave superconductor
These 2 systems are
topologically equivalent
2 Dirac cones

Symmetry protected
Zero modes of Dirac fermions:1D Flat Band of edge stateschiral symmetry{ , H} = H + H =0 ,2

=1

4 Dirac cones

Universality of Zero Energy Edge States‘02---’04 S. Ryu & YH

1.Zero energy edge states of graphene
Boundary Magnetic moments of graphene
2.Andreev bound states of d-wave superconductors
Zero bias conductance peakgraphened-wave superconductor
These 2 systems are
topologically equivalent
2 Dirac cones:Bipartite(A-B sublatticesymmetry)

Symmetry protected
Zero modes of Dirac fermions:1D Flat Band of edge stateschiral symmetry{ , H} = H + H =0 ,2

=1

4 Dirac cones

Universality of Zero Energy Edge States‘02---’04 S. Ryu & YH

1.Zero energy edge states of graphene
Boundary Magnetic moments of graphene
2.Andreev bound states of d-wave superconductors
Zero bias conductance peakgraphened-wave superconductor
These 2 systems are
topologically equivalent
2 Dirac cones:Bipartite(A-B sublatticesymmetry)

Symmetry protected
Zero modes of Dirac fermions:1D Flat Band of edge stateschiral symmetry{ , H} = H + H =0 ,2

=1

4 Dirac cones:Time Reversal(Real
Order parameter)

Universality of Zero Energy Edge States‘02---’04 S. Ryu & YH

Spontaneous breaking ofthese chiral symmetries: Peierls instabilities ofFlat (edge) bands
d-wave superconductor

graphene

These 2 systems are
topologically equivalent
2 Dirac cones:Bipartite(A-B sublatticesymmetry)

Symmetry protected
Zero modes of Dirac fermions:1D Flat Band of edge stateschiral symmetry{ , H} = H + H =0 ,2

=1

4 Dirac cones:Time Reversal(Real
Order parameter)

Universality of Zero Energy Edge States‘02---’04 S. Ryu & YH

Spontaneous breaking ofthese chiral symmetries: Peierls instabilities ofFlat (edge) bands
d-wave superconductor

Boundarymagnetic
moments
graphene

These 2 systems are
topologically equivalent
2 Dirac cones:Bipartite(A-B sublatticesymmetry)

Symmetry protected
Zero modes of Dirac fermions:1D Flat Band of edge stateschiral symmetry{ , H} = H + H =0 ,2

=1

4 Dirac cones:Time Reversal(Real
Order parameter)

Universality of Zero Energy Edge States‘02---’04 S. Ryu & YH

Spontaneous breaking of
Spontaneous local
these chiral symmetries
flux generation: Peierls instabilities ofnear defects
Flat (edge) bands
d-wave superconductor

Boundarymagnetic
moments
graphene

These 2 systems are
topologically equivalent
2 Dirac cones:Bipartite(A-B sublatticesymmetry)

Symmetry protected
Zero modes of Dirac fermions:1D Flat Band of edge stateschiral symmetry{ , H} = H + H =0 ,2

=1

4 Dirac cones:Time Reversal(Real
Order parameter)

When the zero modes exist ?
S.Ryu & Y.Hatsugai, Phys. Rev. Lett. 89, 077002 (2002)
Y.Hatsugai., J. Phys. Soc. Jpn. 75 123601 (2006)
Kuge, Maruyama, Y. Hatsugai, arXiv:0802.2425

Lattice analogue ofWitten’s SUSY QM

Edge states with boundaries
Determined by the Berry phase of the bulk (without boundaries)
Zak

=

A=

⇧
d⇧k · A

Require Local Chiral Symmetry(ex. bipartite ){ , H} = H + H =0=⇥

⇤ =⇥ (k)|⇧⇤ k (k)⇤A
Quantized

=

⇥

A=

⇥
0

Zero energy localized states EXIST: There exists odd number of zero modes

Bulk-edge correspondence: “Bulk determines the edges”

Close look at E=0
Graphene
3



2





1





0



-1



-2









-3








= 1/4
strong





0

0.2

0.4

0.6

0.8

1





= 1/7





= 1/21
weak





Close look at E=0
n=0 Landau Level
= 1/4

= 1/7

= 1/21

Bulk Landau Level and the zero mode edge states coexist

Armchairx1
I(x) =
2CEC

dky | (x, ky , E)|2

1.5

φ=1/21
φ=1/31
φ=1/41dE

I(x)/I0

Boundary

LDOS around E=0 Ewith Landau Level

1
0.5

0.5

Suppression near the edge

1

1.5

2

2.5

x/lB

Standard behavior due to edge potential3

Zigzagx

1
I(x) =
2

Cdky | (x, ky , E)|2dE
EC

M. Arikawa, H. Aoki & YH

2 Rev. B79, 075429 (2009),arXiv: & 0806.2429
Phys.
φ=1/21
φ=1/31
φ=1/41

1.5

I(x)/I0

Boundary

LDOS around E=0 Ewith Landau Level

scaled by the magnetic length

1

0.5

0.5

1

1.5

STM observable
x/lB
Strong enhancement near the edge2

2.5

Characteristic feature of the Graphene Zigzag edges!

3

Bulk-Edge correspondence: Dirac fermions
2D Dirac fermions
Graphened-wave superconductor

surfaces of 3D TI ...

Edge states: quantum Hall edge state
zero mode localized states

B 6= 0
B=0

Universality in the zero modesof Dirac FermionsApplications
Use of the edge states
Edge states in the topological quantum phasetransition

Edge states to see 1/2 Hall conductance ofDirac fermions

Adiabatic transport among edge states

Zero modes as critical edge states
Quantum phase transition: Bulk

Gapped phase A

Gapped phase A

Critical phase A/B

Gapped phase B

Gapped phase B

Dirac fermions doublingZero modes as critical edge states : Zigzag type
Topological quantum phase transition: Edge
Nontrivial phase A

Nontrivial phase A

Critical phase A/B
Zigzag type

L
R

R

L

L

Trivial phase B

L

Trivial phase B

L

Trivial

R

L

R

L

Trivial phase B

L
R
L

L

Zero modes as critical edge states : Zigzag type
Topological quantum phase transition: Edge
Nontrivial phase A

Nontrivial phase A

Critical phase A/B
Zigzag type

L
R

R

L

L

Trivial phase B

L

Trivial phase B

L

Trivial

R

L

R

L

Trivial phase B

L
R
L

L

Zero modes as critical edge states : Armchair type
Topological quantum phase transition: Edge
Nontrivial phase A

Nontrivial phase A

Critical phase A/B
Armchair type

Non trivial
R

L

Trivial phase B

TrivialR

L

Trivial phase B

R

L

Trivial phase B

Zero modes as critical edge states : Armchair type
Topological quantum phase transition: Edge
Nontrivial phase A

Nontrivial phase A

Critical phase A/B
Armchair type

Non trivial
R

L

Trivial phase B

TrivialR

L

Trivial phase B

R

L

Trivial phase B

How to see 1/2 Hall conductance of Dirac fermions1e2(n + ), n = 0, 1, 2, · · ·
xy =
h
2

with inevitable (topological) fermion doubling1e2
e2(n + ) = (2n + 1)
xy = 2h
2h: integer

It can not be odd/2 since the Chern number is integer
surfaces of the TI
This is also doublingc.f. domain wall fermions in lattice gauge theory

How to see 1/2 Hall conductance of Dirac fermions1e2(n + ), n = 0, 1, 2, · · ·
xy =
h
2

with inevitable (topological) fermion doubling1e2
e2(n + ) = (2n + 1)
xy = 2h
2h: integer

It can not be odd/2 since the Chern number is integer
surfaces of the TI
This is also doublingc.f. domain wall fermions in lattice gauge theory

How to observe ??

Continuously shifts the Dirac cones !
H. Watanabe, YH, H.Aoki, Phys.Rev.B82, 24140(R), (2010)

Continuously shifts the Dirac cones !
H. Watanabe, YH, H.Aoki, Phys.Rev.B82, 24140(R), (2010)

Continuously shifts the Dirac cones !
H. Watanabe, YH, H.Aoki, Phys.Rev.B82, 24140(R), (2010)

without shift

with shift

Integers to integers
transitions
asgovernedpby the Dirac fermion’sn rule of the Landau levels

Continuously shifts the Dirac cones !
H. Watanabe, YH, H.Aoki, Phys.Rev.B82, 24140(R), (2010)

without shift

with shift

sequence of quantum phase transitions as the shift varies

Integers to integers
transitions
asgovernedpby the Dirac fermion’sn rule of the Landau levels

Continuously shifts the Dirac cones !
H. Watanabe, YH, H.Aoki, Phys.Rev.B82, 24140(R), (2010)

without shift

with shift

It can be applicable for surface Dirac fermions of TI

sequence of quantum phase transitions as the shift varies

Integers to integers
transitions
asgovernedpby the Dirac fermion’sn rule of the Landau levels

Edge states are also governed by the rule
H. Watanabe, YH, H.Aoki, Phys.Rev.B82, 24140(R), (2010)
graphene

Our model with continuous shift

Integers to integers transitions
p as governedn ruleby the Dirac fermion’sAlso some experiments in cold atoms
in a non-Abelian optical latticeUse of the edge states
Edge states
localized particles in the gap

Novel quantum degrees
with topological protection by bulk

Adiabatic transports among edge states
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ex.

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Edge is topological

Right / left to the symmetry
With translation invariance

Bloch theorem[H, T ] = 0
ik
T (r) ⌘ (r + t) = e (r)10| (r)| = | (r + t)| = | (r + 2t)| = · · · = | (r + 10 t)| = · · ·Extended over the whole space
Energy bands : energy of the extended states

With boundaries/ impurities
As for extended states, effects of edges can be negligible !
dimension is less ! 0D impurities/1D boundaries in 2DStates in the energy gap are localized !
since they can not be extended

Bound states / Edge states

Edge is topological

Right / left to the symmetry
unnormalizable

Extended states

1 ikr! 0 (V ! 1)
e (r) ⇠ p e
V
V: Volume

8

6

4

2

5

10

15

20

normalizable
Bound states / Edge states1
r/a0
b (r) ⇠ p 3 ea0a0 :size of the bound state
1.2
1.0
0.8
0.6
0.4
0.2

1

2

3

4

5

6

Clear difference only in the infinite system

Edge is topological

Right / left to the symmetry

On cylinder

L

Finite system

R|± = |L± |R

Right / left to the symmetry
only in the infinite system,
since the Boundaries are far away !

Edge is topological

Right / left to the symmetry

On cylinder

L

Finite system

R|± = |L± |R

Right / left to the symmetry
only in the infinite system,
since the Boundaries are far away !
Bulk-edge correspondence：Emergent principle

Edge is topological

Right / left to the symmetry

On cylinder

L

Finite system

R|± = |L± |R

Right / left to the symmetry
only in the infinite system,
since the Boundaries are far away !
Bulk-edge correspondence：Emergent principle
c.f. More is different

Edge is topologicalWhy do we care edge states?
Why the Edge States are there??
Accidental ?
NO !
Inevitable reasonsPhysical Structures behind:“Bulk determines the edges”“Edge determines the bulk”
Bulk-Edge Correspondence
Protected by Topological constraints

Conclusion

Conclusion
Edge states are everywhereincondensed matter physics

Conclusion
Edge states are everywhereincondensed matter physics
Edge states are useful for
applicationsin quantum physics /devices

Thank you
Edge states are everywhereincondensed matter physics
Edge states are useful for
applicationsin quantum physics /devices