From the topological invariants to the edge states of ...

From the topological invariants to the edge states of topological insulators

Victor Gurarie

Euler Mathematical Institute, July 2013

1

work with A. Essin

Monday, July 15, 13

In this talk 2

• Bulk-boundary correspondence for the topological invariants of topological insulators with Green’s functions

• Integer quantum Hall effect

• Fractional quantum Hall effect

• 1D topological insulators with and without interactions, spin-1 spin chain

VG, PRB 2011A. Essin and VG, PRB 2012S. Manmana, A. Essin, R. Noack, VG, PRB 2012VG and A. Essin, JETP Lett., 2013

Monday, July 15, 13

3

Bulk-boundary correspondence oftopological invariants

Monday, July 15, 13

Topological insulators 4

1. Topological insulators insulate in the bulk,conduct at the boundary.

2. Topological insulators have topological invariants computed in the boundary-less geometry.

It is well known that

3. The boundary of the topological insulator is a topological (semi)metal, with its own topological invariant.

It is somewhat less known that

Monday, July 15, 13

Bulk-boundary correspondence 5

The bulk invariant and the edge invariant are equal.

Nd = Nd�2(⇤)�Nd�2(�⇤)

Bulk invariant in d dimensions Edge invariant

G. Volovik, 1980s; VG, A. Essin, PRB 2011

The edge is not an insulator; the edge invariant is of the type used to characterize the

topological (semi)metals.

Monday, July 15, 13

Edge topological invariant in d-dimensions 6

Monday, July 15, 13


1. Take a d-dim insulator with the bulk invariant Nd

Monday, July 15, 13



2. d-1 dimensional edge with d-1 momenta

Example: 2D edge of a 3D insulator (d=3)

Monday, July 15, 13



2. d-1 dimensional edge with d-1 momenta3. Edge is not an insulator


Fermi surface

px

py

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4. Fix pd-1=Λ some large value


Fermi surface

px

py⇤

Monday, July 15, 13





5. Now the edge is an d-2 dim insulator


Fermi surface

px

py⇤

H⇤(px) = H(px

, py

)|py=⇤

Effective 1D insulator

Monday, July 15, 13






6. Calculate its invariant Nd-2(Λ)


Fermi surface

px

py⇤

H⇤(px) = H(px

, py

)|py=⇤


Monday, July 15, 13







7. Claim: Nd =Nd-2(Λ)-Nd-2(-Λ)


Fermi surface

px

py

�⇤

⇤H⇤(px) = H(p

x

, py

)|py=⇤


The edge is a topological metalMonday, July 15, 13

Edge topological invariant 7

Example: 3D edge of a 4D insulator

Fermi surface






7. Claim: Nd =Nd-2(Λ)-Nd-2(-Λ)

Monday, July 15, 13

Example: an edge of a 4D TRI insulator 8

This edge is taken as

because it is1. linear in momenta2. time-reversal invariantH(p) = �yH

⇤(�p)�y

But does it have the right edge invariant?

H = vX

i=x,y,z

�i

pi

� µ

Monday, July 15, 13

Example: an edge of a 4D TRI insulator 8

This edge is taken as

because it is1. linear in momenta2. time-reversal invariantH(p) = �yH

⇤(�p)�y

But does it have the right edge invariant?

H = v�x

px

+ v�y

py

± v��z

� µ

Fix pz=+Λ or pz=-ΛEffectively 2D.

N2(�)�N2(��) = 1 Well known relation.Yes, it is an edge.

H = vX

i=x,y,z

�i

pi

� µ

LFSG, 1994Monday, July 15, 13

9

Integer quantum Hall effect

Monday, July 15, 13

Topological invariant for even dimensions type 10

Matsubara Green’s function

Z

Gab(!,p) = [i! �H]�1ab

Monday, July 15, 13



known numerical coefficient

Nd = Cd ��0...�d tr

Zd⇥ddpG�1⇤�0G . . .G�1⇤�dG

Summation over each � = ⇥, p1, . . . , pd is implied

topological invariant

Z

Works only if d is even

Gab(!,p) = [i! �H]�1ab

Monday, July 15, 13


If d=2 this coincides with the TKNN invariant (“Chern number”). Niu, Thouless, Wu (1985)


known numerical coefficient

Nd = Cd ��0...�d tr

Zd⇥ddpG�1⇤�0G . . .G�1⇤�dG


topological invariant

Z

Works only if d is even

Gab(!,p) = [i! �H]�1ab

⇡d+1 (GL(N ,C)) = Z ensures this is a topological invariant

Monday, July 15, 13

Integer QHE 11

p

B ✏

p�⇤

⇤

Edge spectrum

Edge invariant

!

p

tr

Idsµ

2⇡iG�1@µG = tr

Z 1

�1

d!

2⇡iG�1@!G

��p=⇤

p=�⇤

⌘ N0(⇤)�N0(�⇤)

�⇤ ⇤

Monday, July 15, 13

Topological invariant in zero dimensions 12

N0 = tr

Zd!

2⇡iG�1@!G =

Z 1

�1

d!

2⇡i@! ln detG

G = [i! �H]�1

N0 = �X

n

Z 1

�1

d!

2⇡i@! ln (i! � ✏n)

energy levels H n = ✏n n

topological invariant = excess of the number of positive energy states over the number of negative energy states

N0 =1

2

X

n

sign ✏n

Nd = Cd ��0...�d tr

Zd⇥ddpG�1⇤�0G . . .G�1⇤�dG

Monday, July 15, 13

Integer QHE 13

p

B

N0(⇤)�N0(�⇤) =1

2

X

n

[sign ✏n(⇤)� sign ✏n(�⇤)]

Monday, July 15, 13

Integer QHE 13

N0(⇤)�N0(�⇤) = 1p

✏

p�⇤

⇤

There has to be a level such thatThis is the chiral edge state!

✏m(⇤) > 0, ✏m(�⇤) < 0

B

N0(⇤)�N0(�⇤) =1

2

X

n

[sign ✏n(⇤)� sign ✏n(�⇤)]

Monday, July 15, 13

Integer quantum Hall effect 14

Magnetic fieldthrough this 2D rectangle

�xy

=e2

h

B

Suppose

Monday, July 15, 13

Integer quantum Hall effect 14

Magnetic fieldthrough this 2D rectangle

�xy

=e2

h

B

Suppose

h

e2�xy

= N2 =1

24⇡2

X

↵��

✏↵��

Zd!d2p tr

⇥G�1@

↵

GG�1@�

GG�1@�

G⇤

TKNN invariant constructed out of the Green’s function G(!, p

x

, py

) = [i! �H(px

, py

)]�1

h

e2�xy

= N2 = 1In this example

Monday, July 15, 13

Derivation of the bulk-boundary correspondence 15

B

pG(x0

, x

00) ! G(x, k)

tr

Zd!G

�1@

!

G

��p=⇤

p=�⇤

= tr1

2i

X

a,b=x,k

Zd!dxdk G

�1@

a

GG

�1@

b

GG

�1@

!

G ✏

ab

��

p=⇤

p=�⇤

G

�1(x0, x

00) ! G

�1(x, k) +1

2i

X

a,b=x,k

G

�1@

a

GG

�1@

b

G

�1✏

ab

+ . . .

x

= N2|x=L

x=�L

BoundaryBulk

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Integer quantum Hall effect with disorder 16

Disorder breaks translational invariance, so no momentum.

Monday, July 15, 13


Old idea of Thouless, Wu, Niu: impose phases across the system

✓x

✓y

Gij

(!, ✓x

, ✓y

. . . )

Summation over each ↵ = !, ✓1, . . . , ✓d is implied

Nd = Cd ✏↵0...↵d tr

Zd!dd✓G�1@↵0G . . .G�1@↵dG


Monday, July 15, 13


Old idea of Thouless, Wu, Niu: impose phases across the system

✓x

✓y

Gij

(!, ✓x

, ✓y

. . . )

Summation over each ↵ = !, ✓1, . . . , ✓d is implied

Nd = Cd ✏↵0...↵d tr

Zd!dd✓G�1@↵0G . . .G�1@↵dG

N0(⇤)�N0(�⇤) = 1

✏

�⇤⇤ ✓

d = 2 : This edge level must be delocalized!


Monday, July 15, 13

17

Fractional quantum Hall effect

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Bulk-boundary valid with interactions 18

Nd = Cd ��0...�d tr

Zd⇥ddpG�1⇤�0G . . .G�1⇤�dG


Nd = Nd�2(⇤)�Nd�2(�⇤)

Bulk invariant in d dimensions Edge invariant

⇡d+1 (GL(N ,C)) = Z

Bulk-boundary is valid regardless of the origin of G

ensures this is a topological invariant

Monday, July 15, 13

0D invariant with interactions 19

N0 =

Z 1

�1

d!

2⇡i@! ln detG

detG =Y

n

1

i! � ✏n no interactions

detG =

Qm (i! � rn)Qn (i! � ✏n)

with interactions

poles (states)

zeros

VG, PRB 2011

Monday, July 15, 13

0D invariant with interactions 19

N0 =

Z 1

�1

d!

2⇡i@! ln detG

detG =Y

n

1

i! � ✏n no interactions

detG =

Qm (i! � rn)Qn (i! � ✏n)

with interactions

poles (states)

zeros

VG, PRB 2011

N0(⇤)�N0(�⇤) =1

2

X

n

(sign ✏n(⇤)� sign ✏n(�⇤))�1

2

X

n

(sign rn(⇤)� sign rn(�⇤))

relevant formula

Monday, July 15, 13

FQHE at simple fractions 20

⌫ =1

2m+ 1G(x, t) ⇠ 1

(x� vt)2m+1

electron Green’s function

G(!, p) =(i! + vp)2m

i! � vp

Wen, 1995

✏(p) = vp, r(p) = �vp

B

pppole

zero

Monday, July 15, 13

FQHE at simple fractions 20

⌫ =1

2m+ 1G(x, t) ⇠ 1

(x� vt)2m+1

electron Green’s function

G(!, p) =(i! + vp)2m

i! � vp

Wen, 1995

✏(p) = vp, r(p) = �vp

B

p

N0(⇤)�N0(�⇤) =1

2

X

n

(sign ✏n(⇤)� sign ✏n(�⇤))�1

2

X

n

(sign rn(⇤)� sign rn(�⇤))

N0(⇤)�N0(�⇤) = 2m+ 1 = N2

N2 = ✏↵0↵1↵2 tr

Zd!dp

x

dpy

24⇡2G�1@

↵0GG�1@↵1GG�1@

↵2G = 2m+ 1

�xy

=1

2m+ 16= N2 = 2m+ 1

ppole

zero

Monday, July 15, 13

Issues with the winding 21

G =X

n

⇢ni! � ✏n

Can’t have arbitrary winding

Large windings probably are a low energy statement

Z 1

�1

d!

2⇡i@! lnG =

1

2sign

X

n

⇢n✏n

Monday, July 15, 13

Significance of N2 for FQHE 22

• We don’t know G in the bulk of quantum Hall system analytically

• We do know the low energy edge structure analytically, and we used it to calculate the invariant.

• Thus we calculated a quantity in the bulk from knowing the theory structure at the edge

• Numerically we typically calculate the bulk properties, but not the edge properties.

• We could determine N2 in the bulk and compare with the value calculated analytically at the edge

Monday, July 15, 13

Generic Abelian FQH states 23

S =1

4⇡

X

ab

Zdxdt [K

ab

@

t

�

a

@

x

�

b

� V

ab

@

a

�@

b

�]

K-matrix (universal) V-matrix (non-universal)

a = eiP

Kab�b electron operators

Gab(x, t) ⇠Y

c

�ab

(x� vct)↵a

c can be found in terms of K, Vvc, ↵a

c

N0(⇤)�N0(�⇤) =1

2⇡

X

a

Im lnGaa(!, p)|!=1!=�1

��

p=⇤

p=�⇤

= trK

N2 = trKMonday, July 15, 13

Non-Abelian QH states 24

Read-Rezayi (parafermion) stateslevelN

parameterM , always odd

⌫ =N

MN + 2Filling fraction

Edge Green’s function calculated with the conformal field theory

G(x, t) ⇠ 1

(x� vnt)2� 2

N (x� vct)M+ 2

N

what matters is the sum of these two

N2 = M + 2

N2 = 3For a Pfaffian at ν=5/2, M=1, and For an anti-Pfaffian at ν=5/2, N2 = 1

Monday, July 15, 13

FQHE: conclusions 25

• Chern number-like topological invariant labels FQH states

• The correspondence between the invariant and the states is not one to one: changing the invariant changes the state, but two different states can have the same invariant

• Nevertheless, it should, at least, serve as an additional way to distinguish FQH states numerically

Monday, July 15, 13

26

1D interacting topological insulators

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1D: Peierls-Hubbard model 27

ax,�

! (�1)xa†x,�

a†x,�

! (�1)xax,�

Chiral transformations

This model is chirally invariant at half filling

t1, t2 real

H =X

x, �=",#

ht1 a

†2x+1,�a2x,� + t2 a

†2x+2,�a2x+1,�

i+h. c.+U

X

x

a†x"a

†x#ax#ax"

H = ⌃H⌃⌃

Monday, July 15, 13


ax,�

! (�1)xa†x,�

a†x,�

! (�1)xax,�



t1, t2 real

U > 0 Mott insulator -> AF spin chain

H ⇡X

x

t21U

S2x · S2x+1 +t22U

S2x+1 · S2x+2

�

H =X

x, �=",#

ht1 a

†2x+1,�a2x,� + t2 a

†2x+2,�a2x+1,�

i+h. c.+U

X

x

a†x"a

†x#ax#ax"

H = ⌃H⌃⌃

Monday, July 15, 13


ax,�

! (�1)xa†x,�

a†x,�

! (�1)xax,�



t1, t2 real

U > 0 Mott insulator -> AF spin chain

H ⇡X

x

t21U

S2x · S2x+1 +t22U

S2x+1 · S2x+2

�

H =X

x, �=",#

ht1 a

†2x+1,�a2x,� + t2 a

†2x+2,�a2x+1,�

i+h. c.+U

X

x

a†x"a

†x#ax#ax"

U

free fermions

Mott insulator withgapless spin excitations

Mott insulator withgapless charge

excitations

U=0 nontopological gapped phase

U=0 topological gapped phase

�

t=

t1 � t2t1 + t2

What about the rest of the

phase diagram?

H = ⌃H⌃⌃

Monday, July 15, 13

Calculating the 1D invariant with DMRG 28

g =

✓0 vv⇤ 0

◆g = �⌃g⌃

N1 = tr

Zdp

4⇡i⌃ g�1@pg = tr

Zdp

2⇡iv�1@pv

winding of v around the complex plane origin

� = arg v

�(pj) = �0 ! N1 =X

j

sign

d�(p)

dp

��p=pj

!

Discussions with A. Kitaev

⌃ =

✓1 00 �1

◆g = G(!, p)|!=0

Monday, July 15, 13

Numerical phase diagram 29

δ/t

U/t

-1 1-0.3 0.3

U/t=10

U/t=0

U/t=2

U/t=10

U/t=0

U/t=2

��

��/2

0

�/2

�

�� /2 0 �/2 �

k

L = 22, � = 0.3, OBC; � = i

Winding number of Green’s function frequency � = i

U = 0U = 2

U = 10U = 10, L = 250

��

��/2

0

�/2

�

�� /2 0 �/2 �

k

L = 22, � = -0.3, OBC; � = i


U = 0U = 2

U = 10

A. Essin, S. Manmana, R. Noack, VGPRB 2012

Monday, July 15, 13

Numerical phase diagram 29

δ/t

U/t

-1 1-0.3 0.3

U/t=10

U/t=0

U/t=2

U/t=10

U/t=0

U/t=2

��

��/2

0

�/2

�

�� /2 0 �/2 �

k

L = 22, � = 0.3, OBC; � = i


U = 0U = 2

U = 10U = 10, L = 250

��

��/2

0

�/2

�

�� /2 0 �/2 �

k

L = 22, � = -0.3, OBC; � = i


U = 0U = 2

U = 10


U

free fermions

Mott insulator withgapless spin excitations

Mott insulator withgapless charge

excitations

Nontopological gapped phase with

N1=0

Topological gapped phase with

N1=2�

t=

t1 � t2t1 + t2

N1 = tr

Zdp

4⇡i⌃ g�1@pg

⌃ =

✓1 00 �1

◆

Edge states only if δ<0

N1 =

⇢0, �t > 02, �t < 0

Monday, July 15, 13

Spin-1 chain 30

H =X

x even,�=",#

ht1a

†x+1,�ax,� + t2a

†x+2,�ax+1,�

i+ h.c.+U

X

x

a†x"a

†x#ax#ax"

�JX

x

S2x · S2x+1 Large U Mott insulator

H ⇡X

x

⇢t21U

� J

�S2x · S2x+1 +

t22U

S2x+1 · S2x+2

�

Sufficiently large J -> Haldane’s spin-1 Heisenberg chain

{ { { {S = 1S = 1

S = 1S = 1

Monday, July 15, 13

Application 6: Haldane spin chain 31

H =X

x even,�=",#

ht1a

†x+1,�ax,� + t2a

†x+2,�ax+1,�

i+ h.c.+U

X

x

a†x"a

†x#ax#ax"

�JX

x

S2x · S2x+1

{ { { {S = 1S = 1

S = 1S = 1

DMRG confirms that N1=2(large positive J)Spin-1/2 edge states are due to the nonzero value of the topological invariant.


−π

−π/2

0

π/2

π

−π −π/2 0 π/2 π

k

V(k,ω = i), U/t = 10, δt = 0J = 2 (AFM)J = -2 (FM)

Monday, July 15, 13

Conclusions 32

• Topological invariants constructed out of single-particle Green’s functions tell us something about the edge states, even when there are interactions, in 1D and in 2D.

• Yet they are not as powerful as when there are no interactions. In particular, there is no one to one correspondence between the invariant and the phase. The invariant is not equal to Hall conductance in the FQH phase.

• They are still useful to distinguish different topological states, if used with care.

• Generalizations: many particle Green’s function, Thouless’s many body invariant?

Monday, July 15, 13

The end

33

Monday, July 15, 13

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