classiﬁcation of topological insulators · topological classes in high dimensions - perhaps not...

Classification of

Topological Insulators

Victor Gurarie

Windsor, August 2012

1

Sunday, August 19, 12

Classes of topologically distinct Hamiltonians 2

matrix (Hamiltonian) eigenvaluesEn

En

0

En

0

smoothly deform

Two matrices are topologically equivalentif one can be deformed into another without any of its energy levels ever becoming equal to 0.

Hij

H(1)ij H(2)

ij


Application of topological classes 3

H =X

ij

Hij a†i aj

En

0

En

0

smoothly deformH(1)ij H(2)

ij

filled states

empty states

gapless points

fill negative energy levels with fermions

negative = below the chemical potential


Topological invariants 4

Mathematical expressions which take integer values and change only if acquires a zero eigenvalue (acquires zero energy)

H

For example: N0 = # of levels below zero

very simple topological invariant

But there are many more less trivial invariants(more on that later)


Example: particle in 2D in a magnetic field 5

nx

ny Square lattice in 2D The spectrum consists of 1/q-bands (or “Landau levels”)

It is not possible to change the number of bands below 0 by smoothly changing the Hamiltonian (including by changing μ) without tuning through a point with zero energy single-particle states

(Thouless et al, 1982)

Bloch waves

topological invariant (Chern number)

En

(kx

, ky

)

H = tX

nx

,ny

ha†n

x

+1,ny

anx

,ny

+ e2⇡iqnx a†nx

,ny

+1anx

,ny

+ h.c.i�µ

X

nx

,ny

a†nx

,ny

anx

,ny


Chern number in terms of Green’s functions 6

Gnx

,ny

= [i! �H]�1

Gab2 the basis(kx, ky) =

X

n

x

,n

y

2Bravais lattice

e�i(nx

k

x

+n

y

k

y

) G(nx

, ny

)

Chern number (an alternative form)N2 =

1

24⇡2

X

↵��

✏↵��

Zd!dk

x

dky

tr⇥G�1@

↵

GG�1@�

GG�1@�

G⇤

G ! G+ �G �N2 = 0

The only way to change N2 is by making G singular.That requires to have zero energy eigenvalues. H

α, β, γ take values ω, kx, ky


Edge states 7

Particle hopping on a lattice with2π/3 magnetic flux through each plaquette

Periodic boundary conditions in the y-direction


Edge states 7

Particle hopping on a lattice with2π/3 magnetic flux through each plaquette

Hard wall boundary conditions in the y-direction


Edge states as a result of topology 8

NL2 NR

2

Zero energy states must live in the boundary

(x, y) ⇠ e

� |x|`

e

iky

y

E(ky) ⇠ ky � k0


Other topological classes? 9

For a long time 2D particle in a magnetic field was considered to be the only example of topological classes of single-particle Hamiltonians

Generalizations to 4D, 6D, generally even d, was known, however

Nd = ��d2

�!

(2⇡i)d2+1(d+ 1)!

X

↵0↵1...↵d

✏↵0↵1...↵d

Zd!ddk tr

⇥G�1@↵0GG�1@↵1G . . .G�1@↵dG

⇤

d must be even all α take values ω, k1, k2, ..., kd

Topological classes in high dimensions - perhaps not very physical

⇡d+1 (GL(N ,C) = Zexistence of this topological invariant reflects the homotopy classif d is even

Fortunately, it turns out these are not the only topological insulatorsSunday, August 19, 12

Chiral systems 10

What if we have Hamiltonians with a special symmetry

⌃⌃† = 1

⌃H⌃† = �H⌃2 = 1It follows

t1 t2 t1

⌃n,n0 = (�1)n�nn0

H =X

n

ht1 a

†2n+1a2n + t2 a

†2n+2a2n+1 + h.c.

i

Example: systems with sublattice symmetry

=X

n1,n2

Hn1n2 a†n1an2

Ek

H =

0

BBBBBBBB@

0 t1 0 . . . 0 0t1 0 t2 . . . 0 00 t2 0 . . . 0 00 0 t1 . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . 0 t10 0 0 . . . t1 0

1

CCCCCCCCA

E(k)

k

E(k) = ±q

t21 + t22 + 2t1t2 cos(k)

�⇡ ⇡Sunday, August 19, 12

Topological invariant for chiral systems 11

H =

✓0 VV † 0

◆ ✓aodd

aeven

◆basis

Nd = ��d�12

�!

(2⇡i)d+12 d!

X

↵1...↵d

✏↵1...↵d

Zddk tr

⇥V �1@↵1V . . . V �1@↵dV

⇤

General topological invariant But here d is odd

⌃ =

✓1 00 �1

◆



H =

✓0 VV † 0

◆ ✓aodd

aeven

◆basis

Nd = ��d�12

�!

(2⇡i)d+12 d!

X

↵1...↵d

✏↵1...↵d

Zddk tr

⇥V �1@↵1V . . . V �1@↵dV

⇤

General topological invariant

H =

✓0 t1 + t2eik

t1 + t2e�ik 0

◆

N1 =

Z ⇡

�⇡

dk

2⇡i@k ln

�t1 + t2e

ik�

With some algebra, one can show

But here d is odd

⌃ =

✓1 00 �1

◆

Example: d=1



H =

✓0 VV † 0

◆ ✓aodd

aeven

◆basis

Nd = ��d�12

�!

(2⇡i)d+12 d!

X

↵1...↵d

✏↵1...↵d

Zddk tr

⇥V �1@↵1V . . . V �1@↵dV

⇤


t1 < t2N1 = 1

t1 + t2eik

H =

✓0 t1 + t2eik

t1 + t2e�ik 0

◆

N1 =

Z ⇡

�⇡

dk

2⇡i@k ln

�t1 + t2e

ik�


But here d is odd

⌃ =

✓1 00 �1

◆

Example: d=1



H =

✓0 VV † 0

◆ ✓aodd

aeven

◆basis

Nd = ��d�12

�!

(2⇡i)d+12 d!

X

↵1...↵d

✏↵1...↵d

Zddk tr

⇥V �1@↵1V . . . V �1@↵dV

⇤


t1 > t2N1 = 0

t1 + t2eik

H =

✓0 t1 + t2eik

t1 + t2e�ik 0

◆

N1 =

Z ⇡

�⇡

dk

2⇡i@k ln

�t1 + t2e

ik�


But here d is odd

⌃ =

✓1 00 �1

◆

Example: d=1


Edge states for 1D chiral systems 12

t1 t2 t1

H = 0

This works (decays for n>0) only if t1<t2

t1 t2

That is, if N1=1 (not when it is zero)

H =

0

BBBBBBBB@

0 t1 0 . . . 0 0t1 0 t2 . . . 0 00 t2 0 . . . 0 00 0 t1 . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . 0 t10 0 0 . . . t1 0

1

CCCCCCCCA

|ψ|

1

t1* t1*t2

t2* t1 t2*

t2

tt1 2n�1 + t2 2n+1 = 0

2n+1 ⇠✓� t1t2

◆n


Symmetry classes 13

Class A (no symmetry)

Class AIII (chiral symmetry) Z

Z

Z

Z

Z

Z

1 2 3 4 5 6space dimension

Altland-Zirnbauer nomenclature


Other relevant symmetries 14

Time reversal

U †TH⇤UT = H

U †TH⇤UT = H

U †TH⇤UT = H U⇤

TUT =

⇢either +1

or �1

Particle-hole conjugation U †

CH⇤UC = �H U⇤CUC =

⇢either +1

or �1

If both symmetries are present, chiral symmetryis automatically present, with ⌃ = U⇤

TUC

7

Table 1. Listed are the ten generic symmetry classes of single-particleHamiltonians H, classified according to their behavior under time-reversalsymmetry (T ), charge-conjugation (or particle–hole) symmetry (C), as wellas ‘sublattice’ (or ‘chiral’) symmetry (S). The labels T, C and S representthe presence/absence of time-reversal, particle–hole and chiral symmetries,respectively, as well as the types of these symmetries. The column entitled‘Hamiltonian’ lists, for each of the ten symmetry classes, the symmetric space ofwhich the quantum mechanical time-evolution operator exp(itH) is an element.The column ‘Cartan label’ is the name given to the corresponding symmetricspace listed in the column ‘Hamiltonian’ in Élie Cartan’s classification scheme(dating back to the year 1926). The last column entitled ‘G/H (ferm. NL�M)’lists the (compact sectors of the) target space of the NL�M describing Andersonlocalization physics at long wavelength in this given symmetry class.Cartan label T C S Hamiltonian G/H (ferm. NL�M)

A (unitary) 0 0 0 U(N ) U(2n)/U(n) ⇥ U(n)

AI (orthogonal) +1 0 0 U(N )/O(N ) Sp(2n)/Sp(n) ⇥ Sp(n)

AII (symplectic) �1 0 0 U(2N )/Sp(2N ) O(2n)/O(n) ⇥ O(n)

AIII (ch. unit.) 0 0 1 U(N + M)/U(N ) ⇥ U(M) U(n)

BDI (ch. orth.) +1 +1 1 O(N + M)/O(N ) ⇥ O(M) U(2n)/Sp(2n)

CII (ch. sympl.) �1 �1 1 Sp(N + M)/Sp(N ) ⇥ Sp(M) U(2n)/O(2n)

D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)

C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)

DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)

CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)

The only case when the behavior under the combined transformation S = T · C is not determinedby the behavior under T and C is the case where T = 0 and C = 0. In this case, either S = 0or S = 1 is possible. This then yields (3 ⇥ 3 � 1) + 2 = 10 possible types of behavior of theHamiltonian.

The list of ten possible types of behavior of the first quantized Hamiltonian under T , Cand S is given in table 1. These are the ten generic symmetry classes (the ‘tenfold way’),which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.

Let us first point out a very general structure seen in table 1. This is listed in the columnentitled ‘Hamiltonian’. When the first quantized Hamiltonian H is ‘regularized’ (or ‘put on’)a finite lattice, it becomes an N ⇥ N matrix (as discussed above). The entries in the column‘Hamiltonian’ specify the type of N ⇥ N matrix that the quantum mechanical time-evolutionoperator exp(itH) is. For example, for systems that have no time-reversal or charge-conjugationsymmetry properties at all, i.e. for which T = 0, C = 0, S = 0, which are listed in the first rowof the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus, H is ageneric Hermitian matrix and the time-evolution operator is a generic unitary matrix, so thatexp(itH) is an element of the unitary group U(N ) of unitary N ⇥ N matrices. By imposingtime-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N ⇥ N matrix. This,in turn, can be expressed as saying that the time-evolution operator is an element of the coset

New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)

From: Ryu, Schnyder,

Furusaki, Ludwig, 2010

This leads to 10 “symmetry classes”,

introduced by Altland and Zirnbauer


Classes with time reversal invariance only 15

Time reversal U⇤TUT =

⇢either +1

or �1

Class AI: time reversal for spinless particles or spin rotation invariant Hamiltonians

U †TH⇤(�k)UT = H(k)

Class AII: time reversal for spin-dependent spin-1/2 Hamiltonians (usually implies spin-orbit coupling)

Example: H↵�(k) =k2

2m�↵� ↵,� =", #

H↵�(k) =k2

2m�↵� + gSO

X

µ

kµ�µ↵,�

�yH⇤↵�(�k)�y = H(k) UT = �y UTU

⇤T = �1


Only time-reversal is present 16

These are classes AI, AIIG = [i! �H]�1

U †TG

TUT = G

Nd = ��d2

�!

(2⇡i)d2+1(d+ 1)!

X

↵0↵1...↵d

✏↵0↵1...↵d

Zd!ddk tr

⇥G�1@↵0GG�1@↵1G . . .G�1@↵dG

⇤Applying the symmetry to G, we can show that the invariant is identically zero if d = 2 + 4n

Consequence: no topological band structure for time reversal invariant systems in 2D.This is not quite true, however - there is a different topological invariant we haven’t yet looked at.


Class AI (time reversal) Z

Z

Z

Z ZZ

1 2 3 4 5 6 7 8 space dimension

Class AII (time reversal withspin-1/2) Z Z

Green’s function transposed

GTab(k) = Gba(�k)


Only particle-hole is present 17

These are classes D, CU †CH⇤UC = �HG = [i! �H]�1

Nd = ��d2

�!

(2⇡i)d2+1(d+ 1)!

X

↵0↵1...↵d

✏↵0↵1...↵d

Zd!ddk tr

⇥G�1@↵0GG�1@↵1G . . .G�1@↵dG

⇤Applying the symmetry to G, we can show that the invariant is identically zero if

U †CG

T (!)UC = �G(�!)

d = 4n


Class D (p-h, ) Z

Z

Z

Z ZZ

1 2 3 4 5 6 7 8 space dimension

Class C (p-h, ) Z Z

UCU⇤C = 1

UCU⇤C = �1


The origin of p-h symmetry 18

H =X

ij

h2hij a

†i aj +�ij a

†i a

†j +�†

ij aiaj

i=

X

ij

�a†i ai

�✓hij �ij

�†ij �hT

ij

◆✓aia†j

◆

H =

✓hij �ij

�†ij �hT

ij

◆UC

⌘ �x

This is class D

BCS superconductor

This describes Bogoliubov quasiparticles

Famous example: px+i py spin-polarized superconductor (important: it breaks time reversal)

H =

p

2

2m � µ �(px

+ ipy

)

�(px

� ipy

) � p

2

2m + µ

!

N2 = 1

This superconductor has edge states, just like a particle in a magnetic field

�x

H⇤�x

= �H

� = ��T h = h†

Ep = ±

s✓p2

2m� µ

◆2

+�2p2

H =X

µ

nµ�µ

N2 =1

8⇡

X

↵��

X

µ⌫

✏µ⌫�✏↵�

Zd2p

nµ@↵n⌫@�n�

n3


Class C 19

BCS spin-singlet superconductor

H =X

ij

2

4X

�=",#2hij a

†i�aj� +�ij(a

†i"a

†j,# � a†i,#a

†j") +�†

ij (ai,#aj," � aj,#ai,")

3

5

� = �T h = h†

= 2X

ij

⇣a†i," aj,#

⌘✓hij �ij

�†ij �hT

ij

◆✓aj,"a†j,#

◆

H =

✓hij �ij

�†ij �hT

ij

◆�yH⇤�y = �H UC = �y

UCU⇤C = �1

Example: d-wave superconductor with the order parameter dx

2�y

2 + idxy


Classes with both TR and PH 20

Automatically have chiral symmetry. Topological invariant in odd dimensional space.

Nd = ��d�12

�!

(2⇡i)d+12 d!

X

↵1...↵d

✏↵1...↵d

Zddk tr

⇥V �1@↵1V . . . V �1@↵dV

⇤

Nd = �1

2

�d�12

�!

(2⇡i)d+12 d!

X

↵1...↵d

Zddk tr

⇥⌃H�1@↵1H . . .H�1@↵dH

⇤

UCU⇤C = ✏C UTU

⇤T = ✏T

Nd = 0 if✏C✏T = 1 d = 3 + 4n✏C✏T = �1 d = 1 + 4n


Example: class DIII 21

7








D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)

C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)

DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)

CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)





UTU⇤T = �1

UCU⇤C = 1



7








D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)

C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)

DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)

CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)





UTU⇤T = �1

1. can be a superconductorUCU

⇤C = 1



7








D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)

C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)

DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)

CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)





UTU⇤T = �1

1. can be a superconductor2. has to be a spin-triplet (p-wave)superconductor

UCU⇤C = 1



7








D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)

C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)

DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)

CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)





UTU⇤T = �1

1. can be a superconductor2. has to be a spin-triplet (p-wave)superconductor3. has to have spin-orbit coupling

UCU⇤C = 1



7








D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)

C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)

DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)

CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)





UTU⇤T = �1

1. can be a superconductor2. has to be a spin-triplet (p-wave)superconductor3. has to have spin-orbit coupling

UCU⇤C = 1

This is 3He phase B.H =

X

p,↵=",#

✓p2

2m� µ

◆a†p↵ap↵+�

X

p,↵,�,�

pµ �y↵��

µ�� ap↵a�p�+�

X

p,↵,�,�

pµ �µ↵��

y�� a

†�p↵a

†p�

H =

0

@12

⇣p2

2m � µ⌘�↵� �

P�µ pµ�

µ↵��

y��

�P

�µ pµ�y↵��

µ�� 1

2

⇣p2

2m � µ⌘�↵�

1

Athis must be chirally symmetric.Its invariant is N3=1.

3He is topological and has edge states(discovered only in ~2008 by Ludwig et al)


Another example of a 3D DIII insulator 22

t+ tB

�x

t� tB

�x

t+ tB

�x

t�tB�

y

t�tB�

y t+ tB�

y

t+tB�

y

t+t B

�z

t�t B

�zt+t B

�z

t+t B

�z

sublattice Asublattice B

N3 = 1


Example: Class CI 23

UTU⇤T = 1

UCU⇤C = �1

Spin-singlet time-reversal invariant superconductor

This is a conventional s-wave spin-singlet superconductor.

Can be topological in 3D

Conventional superconductors are not topological,but an example of a 3D CI topological superconductor is known (Ludwig et al)


Full classification table 24

11

Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.

dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .

Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .

Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .

dimensions if and only if the target space of the NL�M on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when ⇡d(G/H) = ⇡d�1(G/H) = Z2, or(ii) a WZW term, which is the case when ⇡d(G/H) = ⇡d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.

A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.

24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).


space dimensionality

symmetryclasses Kitaev, 2009;

Ludwig, Ryu, Schnyder, Furusaki, 2009.

Table from Ryu, Schnyder, Furusaki, Ludwig, 2010

White - nonchiralGrey - chiral



11


dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .







IQHE








11


dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .







IQHESu,

Schrieffer,Heeger








11


dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .







IQHESu,

Schrieffer,Heeger

2D p-wave supercond

uctor








11


dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .







IQHESu,

Schrieffer,Heeger

2D p-wave supercond

uctor

3He, phase B








11


dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .







IQHESu,

Schrieffer,Heeger

2D p-wave supercond

uctor

3He, phase B





New Kane-Mele topological insulators



New crucial feature - Z2 invariant 25

Take class AII: time reversal invariance with spin-1/2 (with spin-orbit coupling)

In 4D it can be topological

Its 3D boundary has gapless excitations. These generally form a Fermi spheres.

U †TG

T (!,�k)UT = G(!,k)

px

pyq

3D boundary of a 4D insulator

TR invariancerelates p,q and -p,-q

Fermi surfaces at the edge

Declare q “unphysical” and reduce dimensions to 3D insulator with

2D boundary

Gphys(!, px, py) = G(!, px

, py

, q)|q=0

If the number of Fermi spheres wasodd, the physical 3D insulator has

gapless excitations.Otherwise, it does not.

Z2Sunday, August 19, 12


11


dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .








symmetryclasses

New Kane-Mele topological insulators



Edge excitations 27

Take nonchiral insulator in d-dimensions (d even)It has a d-1 dimensional edge with gapless excitations

Here α go over !, k1, . . . , kd�1

n↵0 = ��d2 � 1

�!

(2⇡i)d2 (d� 1)!

X

↵1...↵d�1

✏↵0↵1...↵d�1

Zd!dd�2k tr

⇥G�1@↵1GG�1@↵2G . . .G�1@↵d�1G

⇤

X

↵

@↵n↵ = 0 except where G is singular or where there are zero-energy excitations

Nd =

Ids↵n↵

this integral must be equal to the bulk topological invariant of the insulator

similarly for d odd and chiral insulators


Example: quantum hall edge 28

Bn↵0 =

1

2⇡i

X

↵1

✏↵0↵1 tr⇥G�1@↵1G

⇤

kk

!

Ids↵n↵ = N0(⇤)�N0(�⇤)

N0(k) = tr

Zd!

2⇡iG�1@!G =

X

n

Zd!

2⇡i@! ln

✓1

i! � ✏n(k)

◆=

1

2

X

n

sign ✏n(k)

⇤�⇤

�⇤⇤ k

chiral edge state

✏n


Example: an edge of a 3D DIII insulator 29

This is 3He

H = �xkx

+ �yky

�x

H⇤(�k)�x

= �H(k) p.h.�yH⇤(�k)�y = H(k) t.r.

Z X

↵=x,y

ds↵ n↵

= 1.

�zH(k)�z = H(k) chiral


Example: an edge of a 3D DIII insulator 29

This is 3He

H = �xkx

+ �yky

�x

H⇤(�k)�x

= �H(k) p.h.�yH⇤(�k)�y = H(k) t.r.

Z X

↵=x,y

ds↵ n↵

= 1.

�zH(k)�z = H(k) chiral

H =

✓0 k

x

� iky

kx

+ iky

0

◆ Zds↵n

↵

=1

2⇡i

Idkµ@

kµ ln (kx

� iky

)

V = kx

� iky


Edge theory of AII 3D topological insulators 30

Literature states that its 2D edge is a Dirac fermion theory

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

+ �y

py

)� µ





⇤(�p)�y


H = v (�x

px

+ �y

py

)� µ

H = v (�x

px

+ �y

py

+ �z

q)� µEnlarge dimensions to 4D (3D edge)





⇤(�p)�y


H = v�x

px

+ v�y

py

± v��z

� µ

Fix q=+Λ or q=-Λ

Effectively 2D.

H = v (�x

px

+ �y

py

)� µ

q

H = v (�x

px

+ �y

py

+ �z






⇤(�p)�y


H = v�x

px

+ v�y

py

± v��z

� µ

Fix q=+Λ or q=-Λ

Effectively 2D.N2(�)�N2(��) = 1 Well known relation

in this context.LFSG, 1994

H = v (�x

px

+ �y

py

)� µ

q

H = v (�x

px

+ �y

py

+ �z






⇤(�p)�y


H = v�x

px

+ v�y

py

± v��z

� µ

Fix q=+Λ or q=-Λ

Effectively 2D.N2(�)�N2(��) = 1 Well known relation

in this context.LFSG, 1994

H = v (�x

px

+ �y

py

)� µ

q

H = v (�x

px

+ �y

py

+ �z


Yes, it does have the right edge invariant.


Disorder 31

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

It follows that an edge of a topological insulator does not localize in the presence of disorder


Disorder 31

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

It follows that an edge of a topological insulator does not localize in the presence of disorder


Sigma models with “topological” terms 32

Describe lack of localization at the boundary of an insulator

S ⇠ �

Zd

dx (@µQ)

2+ topological term

Topological term = either WZW term or “Z2” term.

Can be added if either ⇡d(T ) = Z2 ⇡d+1(T ) = Zor

It is believed that these sigma models with topological terms result in the absence of localization

T - target space

10 classes of sigma modelsfirst derived byAltland & Zirnbauerto describe localization properties

d = d� 1

Justifies 10 symmetryclasses of topological insulators

From: Ryu, Schnyder,

Furusaki, Ludwig, 2010


classiﬁcation of topological insulators · topological classes in high dimensions - perhaps not...

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