Organizing Principles for Understanding Matter

Symmetry

Topology

Interplay between symmetry and topology has led to a new understanding of

electronic phases of matter.

• Conceptual simplification

• Conservation laws

• Distinguish phases of matter by pattern of broken symmetries

• Properties insensitive to

smooth deformation • Quantized topological numbers

• Distinguish topological phases

of matter

genus = 0 genus = 1

symmetry group p31m symmetry group p4

Topology and Quantum Phases

Topological Equivalence : Principle of Adiabatic Continuity

Quantum phases with an energy gap are topologically

equivalent if they can be smoothly deformed into one

another without closing the gap.

Topologically distinct phases are separated by

quantum phase transition.

Topological Band Theory

Describe states that are adiabatically connected to

non interacting fermions

Classify single particle Bloch band structures

Eg ~ 1 eV

Band Theory of Solids

e.g. Silicon

E

adiabatic deformation

excited states

topological quantum

critical point

Gap

EG

Ground state E0

( ) :Bloch Hamiltonans

Brillouin zone (torwith ener

usgy

) gap

H k

E

k

Topological Electronic Phases

Many examples of topological band phenomena

States adiabatically connected to independent electrons:

- Quantum Hall (Chern) insulators - Topological insulators - Weak topological insulators

- Topological crystalline insulators - Topological (Fermi, Weyl and Dirac) semimetals …..

Beyond Band Theory: Strongly correlated states

State with intrinsic topological order

- fractional quantum numbers - topological ground state degeneracy - quantum information

- Symmetry protected topological states - Surface topological order ……

Many real materials

and experiments

Topological Superconductivity

Much recent conceptual

progress, but theory is

still far from the real electrons

Proximity induced topological superconductivity

Majorana bound states, quantum information

Tantalizing recent

experimental progress

Topological Band Theory

Lecture #1: Topology and Band Theory

Lecture #2: Topological Insulators in 2 and 3 dimensions

Topological Semimetals

Lecture #3: Topological Superconductivity

Majorana Fermions

Topological quantum computation

General References :

“Colloquium: Topological Insulators”

M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)

“Topological Band Theory and the Z2 Invariant,”

C. L. Kane in “Topological insulators”

edited by M. Franz and L. Molenkamp, Elsevier, 2013.

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

- Weyl semimetal

- Higher dimensions

- Topological Defects

The Insulating State

The Integer Quantum Hall State

g cE

2D Cyclotron Motion, sxy = e2/h

E

Insulator vs Quantum Hall state

What’s the difference? Distinguished by Topological Invariant

0 p/a -p/a

E

0 p/a -p/a /h e

a

atomic insulator

atomic energy

levels

Landau

levels

3p

4s

3s

2

3

1

gE

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 p -

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 p/a -p/a

Egap

= kx

ky

p/a

p/a

-p/a

-p/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 - kA k k

( ) kA A k

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

d A k

Berry curvature : kF A 2

CS

d k F

Famous example : eigenstates of 2 level Hamiltonian

( ) ( )z x y

x y z

d d idH

d id ds

-

- 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

Classical electric polarization :

+Q -Q 1D insulator

Proposition: The quantum polarization is a Berry phase

( )2 BZ

eP A k dk

p

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

k

-p/a

p/a 0

dipole moment

lengthP

bound P Bound charge density

End charge ˆendQ P n

( ) ( )i u u - kA k k

BZ = 1D Brillouin Zone = S1

Circumstantial evidence #1 :

• The end charge is not completely determined by the bulk

polarization P because integer charges can be added or

removed from the ends :

• The Berry phase is gauge invariant under continuous gauge transformations,

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

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

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

p p - A F

when with

k

-p/a p/a

1

0

Q P eend mod

C

S

gauge invariant Berry curvature

The polarization and the Berry phase share the same ambiguity:

They are both only defined modulo an integer.

Circumstantial evidence #2 :

( ) ( ) ( ) ( )2 2

kBZ BZ

dk ieP e u k r u k u k u k

p p

kr i

A slightly more rigorous argument:

Construct Localized Wannier Orbitals :

( )( ) ( )2

ik R r

BZ

dkR e u k

p

- -

( ) ( )

( ) ( )2

kBZ

P e R r R R

ieu k u k

p

-

Wannier states are gauge dependent, but for a sufficiently smooth gauge,

they are localized states associated with a Bravais Lattice point R

( )R r

R

“ ”

R

( )R r

r

Su Schrieffer Heeger Model model for polyacetylene

simplest “two band” model

† †

1( ) ( ) . .Ai Bi Ai Bi

i

H t t c c t t c c h c -

( ) ( )H k k s 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

p/a -p/a

Provided symmetry requires dz(k)=0, the states with t>0 and t<0 are distinguished by

an integer winding number. Without 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

P = e/2

Gap 4|t|

Peierls’ instability → t

A,i+1

“Chiral” Symmetry :

Reflection Symmetry :

Symmetries of the SSH model

• Artificial symmetry of polyacetylene. Consequence

of bipartite lattice with only A-B hopping:

• Requires dz(k)=0 : integer winding number

• Leads to particle-hole symmetric spectrum:

( ), 0 ( ) ( ) (or )z z zH k H k H ks s s -

iA iA

iB iB

c c

c c

-

z E z E z E EH Es s s - -

( ) ( )x xH k H ks s-

• Real symmetry of polyacetylene.

• Allows dz(k)≠0, but constrains dx(-k)= dx(k), dy,z(-k)= -dy,z(k)

• No p-h symmetry, but polarization is quantized: Z2 invariant

P = 0 or e/2 mod e

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~p/a xk q q ia

p - ;

( )vF x x yH i m xs s -

0t 0t

Massive 1+1 D Dirac Hamiltonian

“Chiral” Symmetry :

2v ; F ta m t

{ , } 0 z z E EHs s -

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

p -

k

-p/a p/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

p F

2

1 ˆ ˆ ˆ ( )4

k tT

n dkdtp

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 Ep s

0

T

xy xy

d hQ dt

dt es s

1

2

dE

R dtp

+Q -Q ( 0) 0

( ) /

t

t T h e

Just like a Thouless pump :

2

xy

eQ ne n

hs

†( ) (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

p F

kx

ky TKNN number = Chern number

21 1( )

2 2 CBZ

n d k dp p

F k A k

2

xy

en

hs

kym(0)

kym(0)

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

Alternative calculation: compute sxy via Kubo formula

C

kx

( ) , ( )m

m x x yE k E k k

TKNN Invariant Thouless, Kohmoto,

Nightingale and den Nijs 82

Physical meaning: Hall conductivity

Laughlin Argument: Thread magnetic flux 0 = h/e through a 1D cylinder

Polarization changes by sxy 0

21( )

2BZ

d kp

F k

2

xy

en

hs

1 2

1 1

2 2C Cn d d

p p - A k A k

( ) ( ) ( )i u u - kΑ k k k

kx

ky

C1

C2

p/a

p/a

-p/a

-p/a

For a 2D band structure, define

E I yk P ne

ne -ne

Graphene E

k

Two band model

Novoselov et al. ‘05 www.univie.ac.at

( ) ( )H s k d k

( ) | ( ) |E k d k

ˆ ( , )x yk kd

Inversion and Time reversal symmetry require ( ) 0zd k

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

( ) vH s K q q Massless Dirac Hamiltonian

k K

-K +K

Berry’s phase p around Dirac point

A

B

†

Ai Bj

ij

H t c c

-

3

1

ˆ ˆ( ) cos sinj j

j

t x y

- d k k r k r

Topological gapped phases in Graphene

1. Broken P : eg Boron Nitride

( ) v zH ms s 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 k m xs s s -

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 =

Weyl Semimetal

Gapless “Weyl points” in momentum space are topologically protected in 3D

A sphere in momentum space can have a Chern number:

2 ( )SS

n d k F kS

nS=+1: S must enclose a degenerate Weyl point:

Magnetic monopole for Berry flux

0( ) )v( x x y y z zH k q q q qs s s

[ ] 0( or v with det v )ia i a iaqs

k

E

kx,y,z

Total magnetic charge in Brillouin zone must be zero: Weyl points

must come in +/- pairs.

Surface Fermi Arc

n1=n2=0

n0=1

k0

k1

k2

kz

kx

ky

E

ky

EF

kz=k1, k2

E

ky

EF

kz=k0

Surface BZ

kz

ky

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

p 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

p 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

p G B

Burgers’ vector

3D Chern number

(vector ┴ layers)

Are there other ways to engineer

1D chiral dirac fermions?

Teo, Kane ‘10