Characterization of Topological Phases

(1)

Focus lecture: Kyushu Univ., July17-19, 2018

Y. Hatsugai

Department of Physics, Univ. Tsukuba

Topological Order parameters

Characterization of Topological Phases

(2)

Y. Hatsugai

Department of Physics, Univ. Tsukuba

Topological Order parameters

Characterization of Topological Phases

(3)

Plan

Gap nodes

Dimension & co-dimension

Anisotropic superconductivity/fluidity Graphene & Chiral symmetry

Adiabatic invariants

Gapped Gapless

Gapped Bulk-edge correspondence

Gapped Entanglement entropy

Gapless

(4)

Gapless Topological !

Nodes structures

protected by symmetry

point nodes, line nodes,...

gapless : generic 2 levels near the gap

H(k) = R(k) · =

✓ R_z R_x iR_y R_y + iR_y R_z

◆

(R_x, R_y, R_z)

3 parameters

expanded by Pauli matrices von Neumann-Wigner ’29

Berry ’84

To be gapless: 3 parameters to be tuned

co-dimension=3 (3 conditions)

2 1

2-D closed surface in 3D

T ² R(T ²)

2D Brillouin zone :periodic in kx & ky

2D Torus

map ex.

R = 0

gapless point

Single particle problem (mean field)

E = ±|R(k)|

(5)

2D examples

2 1

2D Brillouin zone d-wave superconductivity

H(k) = R(k) · =

◆

YH-Ryu, ’02

p-wave superconductivity

Chern #=+2/ -2

Chern #=+1/ -1

(6)

ABM states & Dirac mono pole

2 1

2D Brillouin zone

3rd momentum: time line

co-dimension 3

In 3D, 3--3=0 : point nodes topological stability

Anderson-Brinkman-Morel (ABM) phase of He

H(k) = R(k) · =

◆

YH-Ryu-Kohmoto, ’04

(7)

ABM states & Dirac mono pole

2 1

2D Brillouin zone

3rd momentum: time line

co-dimension 3

In 3D, 3--3=0 : point nodes topological stability

Anderson-Brinkman-Morel (ABM) phase of He

H(k) = R(k) · =

◆

(8)

Chiral symmetry ?

Wallace ’47 Semenoff ’85

Haldane ’88 Nielsen-Ninomiya ‘81

chiral symmetric

{H, } = 0, ^doubling² = 1

Hopping between

Chiral Symmetry

honeycomb lattice: Bipartite

(9)

Chiral symmetry ?

Wallace ’47 Semenoff ’85

Haldane ’88 Nielsen-Ninomiya ‘81

chiral symmetric

{H, } = 0, ^doubling² = 1

Fermion doubling

K K’

2D analogue of

Nielsen-Ninomiya theorem in 4D lattice Gauge theory

Topological ! Hopping between

Chiral Symmetry

honeycomb lattice: Bipartite

(10)

Dirac Cones are Stable!

Doubled Dirac Cones

Dirac Cones are stable for  

small but finite perturbation It can be gapped, if it’s large.

The Dirac Cornes are not accidental

It has topological stability Hatsugai, Fukui, Aoki, ‘06

extended BZ

chiral symmetric perturbation ^t’

t’

t t

t

Chiral Symmetry

{H, } = 0, ² = 1

respect chiral symmetry

(11)

Dirac Cones are Stable!

Doubled Dirac Cones

Dirac Cones are stable for  

small but finite perturbation It can be gapped, if it’s large.

The Dirac Cornes are not accidental

It has topological stability Hatsugai, Fukui, Aoki, ‘06

extended BZ

chiral symmetric perturbation ^t’

t’

t t

t

Chiral Symmetry

{H, } = 0, ² = 1

respect chiral symmetry

(12)

Topological Stability of the Dirac Cornes

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(13)

Topological Stability of the Dirac Cornes

General zeros of Dirac Cones

(k

_x

, k

_y

)

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(14)

Topological Stability of the Dirac Cornes

(k

_x

, k

_y

)

(k_x, k_y), k_x : 0 2 : loop C(k_y) in C

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(15)

Topological Stability of the Dirac Cornes

(k

_x

, k

_y

)

(k_x, k_y), k_x : 0 2 : loop C(k_y) in C loop C(k_y) moves : k_y : 0 2

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(16)

Topological Stability of the Dirac Cornes

(k

_x

, k

_y

)

The loop cut the origin Dirac Cones

Gapless

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(17)

Topological Stability of the Dirac Cornes

(k

_x

, k

_y

)

Topological Stability of

the doubled Dirac Cones

Gapless

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(18)

Topological Stability of the Dirac Cornes

(k

_x

, k

_y

)

Gapped if the perturbation is too large

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(19)

Topological Stability of the Dirac Cornes

(k

_x

, k

_y

)

Gapless

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(20)

Topological Stability of the Dirac Cornes

(k

_x

, k

_y

)

Gapless

chiral symmetric

H(k₁, k₂) =

✓ 0

0

◆

E(k₁, k₂) = ±| | {H, } = 0 =

✓ 1 0 0 1

◆

= ₃

(21)

Geometrical meaning of Chiral symmetry

{ H

_e↵

,

⁹

} = H

_e↵

+ H

_e↵

= 0

: real : Time reversal & Inversion

=

⇢

z y

: bipartite lattice & hopping between them

H_e↵

R_z = 0 R_y = 0

= n · _{ ^H

e↵

, } = 0 n ? R

R(k)

n

(X, Y , n ) =

⇢ right handed = +1 left handed = 1 chirality

X, Y m^⇤

X = @_k₁R, Y = @_k₂R

X Y

H_e↵ ! 0, k ! k₀

Zero gap condition expand by k = k k₀

H_e↵ = R(k) · ⇡ (X · ) k_x + (Y · ) k_y

2 = 1 E = ±|R(k)|

Generically

H(k) = R(k) · =

◆

(R_x, R_y, R_z)

3D

(22)

Geometrical meaning of Chiral symmetry

{ H

_e↵

,

⁹

} = H

_e↵

+ H

_e↵

= 0

=

⇢

z y

H_e↵

R_z = 0 R_y = 0

= n · _{ ^H

e↵

, } = 0 n ? R

R(k)

n

(X, Y , n ) =

X, Y m^⇤

X = @_k₁R, Y = @_k₂R

X Y

H_e↵ ! 0, k ! k₀

H_e↵ = R(k) · ⇡ (X · ) k_x + (Y · ) k_y

2 = 1 E = ±|R(k)|

Generically

H(k) = R(k) · =

◆

(R_x, R_y, R_z) (n · )(R · ) = n · R + i(n ⇥ R) ·3D

(R · )(n · ) = n · R i(n ⇥ R) · {R · , n · } = 2n · R

(23)

Geometrical meaning of Chiral symmetry

{ H

_e↵

,

⁹

} = H

_e↵

+ H

_e↵

= 0

=

⇢

z y

H_e↵

R_z = 0 R_y = 0

= n · _{ ^H

e↵

, } = 0 n ? R

R(k)

n

(X, Y , n ) =

X, Y m^⇤

X = @_k₁R, Y = @_k₂R

X Y

H_e↵ ! 0, k ! k₀

H_e↵ = R(k) · ⇡ (X · ) k_x + (Y · ) k_y

2 = 1 E = ±|R(k)|

Generically

H(k) = R(k) · =

◆

(R_x, R_y, R_z)

3D

(24)

Geometrical meaning of Chiral symmetry

{ H

_e↵

,

⁹

} = H

_e↵

+ H

_e↵

= 0

=

⇢

z y

H_e↵

R_z = 0 R_y = 0

= n · _{ ^H

e↵

, } = 0 n ? R

R(k)

n X

Y

H_e↵ ! 0, k ! k₀

Zero gap condition: Dirac dispersion

2 = 1 E = ±|R(k)|

Generically

H(k) = R(k) · =

◆

(R_x, R_y, R_z)

3D

Chiral Symmetry

{H, } = 0, ² = 1

co-dimension of Dirac cones=2

graphene, d-wave superconductor in 2D

Chiral symmetry

(25)

Topological stability of the Doubled Dirac cones

nγ nγ

R R

2 1

2-D closed surface in 3D

T ²

Generically

R(T ²) {H, } = 0

c.f. 4D graphene & chiral symmetry, M. Creutz ’08

H(k) = R(k) · =

◆

2D Brillouin zone :periodic in k^x & ky

2D Torus map

(R_x, R_y, R_z)

3D

Chiral symmetry n ? R

= n · R(k) is on a plane normal to n

R(T²) is collapsed on the plane

Topologically stable

Gepped :

perturbation is too large

=(0,0,0)

doubled Dirac cones

also with TR inv. 5D YH, ’10

“balloon”

“collapsed balloon” “collapsed balloon”

2D Nielsen-Ninomiya theorem

YH-Fukui-Aoki, ’06

(26)

Graphene with deformation

2 1

2D Brillouin zone

deformation of the system: time line

d-wave superconductor In 2D with chiral symmetry, 2--2=0

co-dimension 2

Dirac cones of graphene topological stability in 2D

(27)

Graphene with deformation

2 1

2D Brillouin zone

deformation of the system: time line

d-wave superconductor In 2D with chiral symmetry, 2--2=0

co-dimension 2

Dirac cones of graphene topological stability in 2D

(28)

c.f. Blount’85

(29)

Gapless Topological !

Nodes characterize the phase topologically

co-dimension 3

d-wave superconductor In 3D, 3--3=0 : point nodes :ABM state of He

Weyl semi-metal

In 3D with TR invariance, 3--2=1 : line nodes super^Blount’85 In 2D with chiral symmety

with TR invariance

with TR invariance/chiral symmetry co-dimension 2

: Dirac cones of graphene Generic

Volovik ’97

YH-Ryu-Kohmoto ’04 YH-Ryu’02

YH-Ryu & Ryu-YH ’02

topological stabile Dirac point Burkov-Balents ’11

2--2=0

Wallace’47

(30)

Hall Conductance has double Topological meanings

Thouless-Kohmoto-Nightingale-den Nijs ‘82 Sum of the First Chern numbers below EF

xybulk = e²

h ⌅: _⇥(k)<E_F

C_⌅

⇥_xy^edge = e²

h I( _j, C^j)

When EF is in the j-th gap

Winding number of the edge state  

on the complex energy surface YH ‘93a YH ‘93b

Two topological quantities

Bulk ---- Edge Correspondence

xy bulk = _xy ^edge

Physically

Bulk C

_j

= I

_j

I

_j ₁

Edge

YH, PRL 71, 3697 (1993)

(31)

Another type of Edge states in Graphene

Quantum Hall edge states

Topologically protected edge states

Symmetry protected edge states  

( with topological origin & stability )

S. Ryu (now KITP) & YH

Dirac Dispersion : Chiral Symmetry!

without magnetic field

with magnetic field

M. Arikawa, H. Aoki & YH arXiv:0805.3240 & 0806.2429

New feature : topological compensation can be observed by STM experiments

(32)

Graphene on a Cylinder

now called as Graphene

(33)

(34)

d-wave superconductivity

(35)

These 2 systems are

topologically equivalent with each other

Localized zero modes of topological ordered states

cf. Witten’s SUSY QM

(36)

1D systems with parameter ky : momentum along the ribbon

(37)

Zigzag

Bearded

Armchair

(38)

(39)

As for a1D system parametrized by ky

(40)

S.Ryu & Y.Hatsugai, Phys. Rev. Lett. 89, 077002 (2002)

(41)

Zero energy localized states EXIST

(42)

Zero energy localized states EXIST

Kuge, Maruyama, YH, arXiv:0802.2425

Topological Aspects of Surface States in Semiconductors

= ⇥

A = ⇥

d⌃k · A⌃ = ⇥

0 quantized

due to chiral symmetry

A⇤ =⇥ (k)|⇧⇤ ^k (k)⇤

= ⇥ : There exists odd number of zero modes

{ , H} = H + H =0

Berry phase

(43)

(44)

(45)

(46)

(47)

(48)

When the zero modes exist?

A⇤ =⇥ (k)|⇧⇤ ^k (k)⇤

Consider Berry phase of the bulk (without boundaries)

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

1D two site problem with boundaries

= A = d⇧k · A⇧

Zak

{ , H} = H + H =0

Require Local Chiral Symmetry (ex. bipartite )

Quantized

= ⇥

A = ⇥ 0

: There exists odd number of zero modes

Zero energy localized states EXIST

= ⇥

Symmetry protected zero modes: bulk-edge correspondence

(49)

1D systems with parameter ky : momentum along the ribbon

(50)

Zigzag

Bearded

Armchair

(51)

Universality of Zero Energy Edge States

‘02---’04 S. Ryu & YH

(52)

1.Zero energy edge states of graphene

Boundary Magnetic moments of graphene

‘02---’04 S. Ryu & YH

(53)

2.Andreev bound states of d-wave superconductors Zero bias conductance peak

‘02---’04 S. Ryu & YH

(54)

These 2 systems are topologically equivalent

‘02---’04 S. Ryu & YH

(55)

d-wave superconductor

4 Dirac cones

graphene

2 Dirac cones

‘02---’04 S. Ryu & YH

(56)

{ , H} = H + H =0 , ² = 1 chiral symmetry

4 Dirac cones

graphene

2 Dirac cones

‘02---’04 S. Ryu & YH

Symmetry protected

Zero modes of Dirac fermions :1D Flat Band of edge states

(57)

(A-B sublattice symmetry)

:Bipartite

4 Dirac cones

graphene

2 Dirac cones

‘02---’04 S. Ryu & YH

(58)

:Bipartite

(Real

Order parameter)

:Time Reversal

4 Dirac cones

graphene

2 Dirac cones

‘02---’04 S. Ryu & YH

(59)

:Bipartite

(Real

Order parameter)

:Time Reversal

4 Dirac cones

graphene

2 Dirac cones

Spontaneous breaking of these chiral symmetries

: Peierls instabilities of Flat (edge) bands

‘02---’04 S. Ryu & YH

(60)

:Bipartite

(Real

Order parameter)

:Time Reversal

4 Dirac cones

graphene

2 Dirac cones

Boundary magnetic

moments

‘02---’04 S. Ryu & YH

(61)

:Bipartite

(Real

Order parameter)

:Time Reversal

4 Dirac cones

graphene

2 Dirac cones

Boundary magnetic

moments

Spontaneous local flux generation

near defects

‘02---’04 S. Ryu & YH

(62)

Hubbard model Mean-field

U>UC

U<UC

Chiral symmetry breaking (graphene)

Chiral symmetry is broken only in the boundaries

Chiral symmetry is broken

both in the bulk and boundaries

DFT, Okada-Oshiyama, ’01

flatband: unstable for infinitesimal U (Stoner)

(63)

Characterization of Topological Phases

Topological Order parameters

Characterization of Topological Phases

Topological Order parameters

Characterization of Topological Phases

Plan

Gapless Topological !

Nodes structures

protected by symmetry

gapless : generic 2 levels near the gap

2D examples

ABM states & Dirac mono pole

ABM states & Dirac mono pole

Chiral symmetry ?

honeycomb lattice: Bipartite

Chiral symmetry ?

honeycomb lattice: Bipartite

Dirac Cones are Stable!

Dirac Cones are Stable!

Topological Stability of the Dirac Cornes

Topological Stability of the Dirac Cornes

(k

, k

)

Topological Stability of the Dirac Cornes

(k

, k

)

Topological Stability of the Dirac Cornes

(k

, k

)

Topological Stability of the Dirac Cornes

(k

, k

)

Topological Stability of the Dirac Cornes

(k

, k

)

Topological Stability of the Dirac Cornes

(k

, k

)

Topological Stability of the Dirac Cornes

(k

, k

)

Topological Stability of the Dirac Cornes

(k

, k

)

Geometrical meaning of Chiral symmetry

{ H

,

} = H

+ H

= 0

= n · { H

, } = 0 n ? R

Geometrical meaning of Chiral symmetry

{ H

,

} = H

+ H

= 0

= n · { H

, } = 0 n ? R

Geometrical meaning of Chiral symmetry

{ H

,

} = H

+ H

= 0

= n · { H

, } = 0 n ? R

Geometrical meaning of Chiral symmetry

{ H

,

} = H

= n · _{ ^H

= n · _{ ^H

= n · _{ ^H

= n · _{ ^H

xy bulk = _xy ^edge