Topological Insulators and Superconductors

Lecture #1: Topology and Band Theory

Lecture #2: Topological Insulators in 2 and 3

dimensions

Lecture #3: Topological Superconductors, Majorana

Fermions an Topological quantum

compuation

General References :

M.Z. Hasan and C.L. Kane, RMP in press, arXiv:1002.3895

X.L. Qi and S.C. Zhang, Physics Today 63 33 (2010).

J.E. Moore, Nature 464, 194 (2010).

My collaborators :

Gene Mele, Liang Fu, Jeffrey Teo, Zahid Hasan

Topology and Band Theory

I. Introduction

- Insulating State, Topology and Band Theory

II. Band Topology in One Dimension

- Berry phase and electric polarization

- Su Schrieffer Heeger model :

domain wall states and Jackiw Rebbi problem

- Thouless Charge Pump

III. Band Topology in Two Dimensions

- Integer quantum Hall effect

- TKNN invariant

- Edge States, chiral Dirac fermions

IV. Generalizations

- Bulk-Boundary correspondence

- Higher dimensions

- Topological Defects

The Insulating State

Covalent Insulator

Characterized by energy gap: absence of low energy electronic excitations

The vacuum Atomic Insulator

e.g. solid Ar

Dirac

Vacuum Egap ~ 10 eV

e.g. intrinsic semiconductor

Egap ~ 1 eV

3p

4s

Silicon

Egap = 2 mec2

~ 106 eV

electron

positron ~ hole

The Integer Quantum Hall State

2D Cyclotron Motion, Landau Levels

Quantized Hall conductivity :

B

Jy

y xy xJ E

2

xyh

ne

Integer accurate to 10-9

Energy gap, but NOT an insulator

gap cE

E

gap cE

Ex

Topology The study of geometrical properties that are insensitive to smooth deformations

Example: 2D surfaces in 3D

A closed surface is characterized by its genus, g = # holes

g=0 g=1

g is an integer topological invariant that can be expressed in terms of the

gaussian curvature k that characterizes the local radii of curvature

4 (1 )S

dA gk

1 2

1

r rk

Gauss Bonnet Theorem :

2

10

rk 0k

0k

A good math book : Nakahara, ‘Geometry, Topology and Physics’

Band Theory of Solids

Bloch Theorem :

( ) ( ) ( ) ( )n n nH u E uk k k k

dT

k Brillouin Zone

Torus,

Topological Equivalence : adiabatic continuity

( ) ( )n nE uk k(or equivalently to and )

Band structures are equivalent if they can be continuously deformed

into one another without closing the energy gap

Band Structure :

( )Hk kE

kx /a /a

Egap

= kx

ky

/a

/a

/a

/a

BZ

( ) iT e k RRLattice translation symmetry ( )ie u k r

k

Bloch Hamiltonian ( ) Ηi iH e e k r k rk

A mapping

Berry Phase

Phase ambiguity of quantum mechanical wave function

( )( ) ( )iu e u kk k

Berry connection : like a vector potential ( ) ( )i u u k

A k k

( ) k

A A k

Berry phase : change in phase on a closed loop C CC

d A k

Berry curvature : k

F A 2

CS

d k F

Famous example : eigenstates of 2 level Hamiltonian

( ) ( )z x y

x y z

d d idH

d id d

k d k

( ) ( ) ( ) ( )H u u k k d k k 1 ˆSolid Angle swept out by ( )2

C d k

C

d̂

S

Topology in one dimension : Berry phase and electric polarization

Electric Polarization

+Q -Q 1D insulator

The end charge is not completely determined by the bulk polarization P

because integer charges can be added or removed from the ends :

Polarization as a Berry phase : ( )2

eP A k dk

P is not gauge invariant under “large” gauge transformations.

This reflects the end charge ambiguity

( )( ) ( )i ku k e u kP P en ( / ) ( / ) 2a a n

see, e.g. Resta, RMP 66, 899 (1994)

Changes in P, due to adiabatic variation are well defined and gauge invariant

( ) ( , ( ))u k u k t

1 02 2C S

e eP P P dk dkd

A F

when with

k

-/a

/a 0

k

-/a /a

1

0

dipole moment

lengthP

bP

mod Q P e

C

S

gauge invariant Berry curvature

Su Schrieffer Heeger Model model for polyacetalene

simplest “two band” model

† †

1( ) ( ) . .Ai Bi Ai Bi

i

H t t c c t t c c h c

( ) ( )H k k d

( ) ( ) ( )cos

( ) ( )sin

( ) 0

x

y

z

d k t t t t ka

d k t t ka

d k

0t

0t

a

d(k)

d(k)

dx

dy

dx

dy

E(k)

k

/a /a

Provided symmetry requires dz(k)=0, the states with t>0 and t<0 are topologically distinct.

Without the extra symmetry, all 1D band structures are topologically equivalent.

A,i B,i

t>0 : Berry phase 0

P = 0

t<0 : Berry phase

P = e/2

Gap 4|t|

Peierl’s instability → t

A,i+1

Domain Wall States An interface between different topological states has topologically protected midgap states

Low energy continuum theory :

For small t focus on low energy states with k~/a xk q q ia

;

( )vF x x yH i m x

0t 0t

Massive 1+1 D Dirac Hamiltonian

“Chiral” Symmetry :

2v ; F ta m t

{ , } 0 z z E EH

0

( ') '/ v

0

1( )

0

x

Fm x dx

x e

Egap=2|m| Domain wall

bound state 0

m>0

m<0

2 2( ) ( )vFE q q m

Zero mode : topologically protected eigenstate at E=0

(Jackiw and Rebbi 76, Su Schrieffer, Heeger 79)

Any eigenstate at +E

has a partner at -E

Thouless Charge Pump

t=0

t=T

P=0

P=e

( , ) ( , )H k t T H k t

( , ) ( ,0)2

eP A k T dk A k dk ne

k

-/a /a

t=T

t=0 =

The integral of the Berry curvature defines the first Chern number, n, an integer

topological invariant characterizing the occupied Bloch states, ( , )u k t

In the 2 band model, the Chern number is related to the solid angle swept out by

which must wrap around the sphere an integer n times.

ˆ ( , ),k td

2

1

2 Tn dkdt

F

2

1 ˆ ˆ ˆ ( )4

k tT

n dkdt

d d dˆ ( , )k td

The integer charge pumped across a 1D insulator in one period of an adiabatic cycle

is a topological invariant that characterizes the cycle.

Integer Quantum Hall Effect : Laughlin Argument

Adiabatically thread a quantum of magnetic flux through cylinder.

2 xyI R E

0

T

xy xy

d hQ dt

dt e

1

2

dE

R dt

+Q -Q ( 0) 0

( ) /

t

t T h e

Just like a Thouless pump :

2

xy

eQ ne n

h

†( ) (0)H T U H U

TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 82

View cylinder as 1D system with subbands labeled by 0

1( )m

yk mR

0

0

, ( )2

m

x x y

m

eQ d dk k k ne

F

kx

ky TKNN number = Chern number

21 1( )

2 2 CBZ

n d k d

F k A k

2

xy

en

h

kym(0)

kym(0)

Distinguishes topologically distinct 2D band structures. Analogous to Gauss-Bonnet thm.

Alternative calculation: compute xy via Kubo formula

C

kx

( ) , ( )m

m x x yE k E k k

Graphene E

k

Two band model

Novoselov et al. ‘05 www.univie.ac.at

( ) ( )H k d k

( ) | ( ) |E k d k

ˆ ( , )x yk kd

Inversion and Time reversal symmetry require ( ) 0zd k

2D Dirac points at : point zeros in ( , )x yd d

( ) vH K q q Massless Dirac Hamiltonian

k K

-K +K

Berry’s phase around Dirac point

A

B

†

Ai Bj

ij

H t c c

Topological gapped phases in Graphene

1. Broken P : eg Boron Nitride

( ) v zH m K q q

ˆ# ( )n d ktimes wraps around sphere

m m

Break P or T symmetry :

2 2 2( ) | |vE m q q

Chern number n=0 : Trivial Insulator

2. Broken T : Haldane Model ’88

+K & -K

m m

Chern number n=1 : Quantum Hall state

+K

-K

2ˆ ( ) Sd k

2ˆ ( ) Sd k

Edge States Gapless states at the interface between topologically distinct phases

IQHE state

n=1

Egap

Domain wall

bound state 0

Fv ( ) ( )x x y y zH i m x

F

0

( ') '/ v

0 ( ) ~

x

y

m x dxik y

x e e

Vacuum

n=0

Edge states ~ skipping orbits

Lead to quantized transport

Chiral Dirac fermions are unique 1D states : “One way” ballistic transport, responsible for quantized

conductance. Insensitive to disorder, impossible to localize

Fermion Doubling Theorem :

Chiral Dirac Fermions can not exist in a purely 1D system.

0 F( ) vy yE k k

Band inversion transition : Dirac Equation

ky

E0

x

y

Chiral Dirac Fermions

m<0

m>0

n=1

m= m+

n=0

m= m+

in

|t|=1 disorder

Bulk - Boundary Correspondence

Bulk – Boundary Correspondence :

NR (NL) = # Right (Left) moving chiral fermion branches intersecting EF

N = NR - NL is a topological invariant characterizing the boundary.

N = 1 – 0 = 1

N = 2 – 1 = 1

E

ky K K’

EF

Haldane Model

E

ky K K’

EF

The boundary topological invariant

N characterizing the gapless modes Difference in the topological invariants

n characterizing the bulk on either side =

Generalizations

Higher Dimensions : “Bott periodicity” d → d+2

d=4 : 4 dimensional generalization of IQHE

( ) ( )ij i ju u d kA k k k

d F A A A

42

1[ ]

8Tr

Tn

F F

Boundary states : 3+1D Chiral Dirac fermions

Non-Abelian Berry connection 1-form

Non-Abelian Berry curvature 2-form

2nd Chern number = integral of 4-form over 4D BZ

no symmetry

chiral symmetry

Zhang, Hu ‘01

Topological Defects Consider insulating Bloch Hamiltonians that vary slowly in real space

defect line

s

( , )H H s k

2nd Chern number : 3 12

1[ ]

8Tr

T Sn

F F

Generalized bulk-boundary correspondence :

n specifies the number of chiral Dirac fermion modes bound to defect line

1 parameter family of 3D Bloch Hamiltonians

Example : dislocation in 3D layered IQHE

Gc 1

2cn

G B

Burgers’ vector

3D Chern number

(vector ┴ layers)

Are there other ways to engineer

1D chiral dirac fermions?

Teo, Kane ‘10