# Symmetry, Topology and Phases of 2016-05-21¢ Topology and Band Theory I I. Introduction -...

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E

k=Λa k=Λb

E

k=Λa k=Λb

Symmetry, Topology and Phases of Matter

Topological Phases of Matter 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 …..

Topological superconductivity (BCS mean field theory) - Majorana bound states - Quantum information

Classical analogues: topological wave phenomena - photonic bands - phononic bands - isostatic lattices

Beyond Band Theory: Strongly correlated states State with intrinsic topological order (ie fractional quantum Hall effect)

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

- Symmetry protected topological states - Surface topological order ……

Many real materials and experiments

Much recent conceptual progress, but theory is

still far from the real electrons

Topological Band Theory Topological Band Theory I:

Introduction Topologically protected gapless states (without symmetry)

Topological Band Theory II: Time Reversal symmetry Crystal symmetry Topological superconductivity 10 fold way

Topological Mechanics

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 I. Introduction

- Insulating state, topology and band theory II. Band Topology in One Dimension

- Berry phase and electric polarization - Su Schrieffer Heeger model - Domain walls, Jackiw Rebbi problem - Thouless charge cump

III. Band Topology in Two Dimensions - Integer quantum Hall effect - TKNN invariant - Edge states, chiral Dirac fermions

IV. Generalizations - Higher dimensions - Topological defects - Weyl semimetal

The Insulating State

The Integer Quantum Hall State

g cE ω= h

2D Cyclotron Motion, σxy = e2/h

E

Insulator vs Quantum Hall state

What’s the difference? Distinguished by Topological Invariant

0 π/a−π/a

E

0 π/a−π/a/h eΦ =

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 κ that characterizes the local radii of curvature

4 (1 ) S dA gκ π= −∫

1 2

1 r r

κ =

Gauss Bonnet Theorem :

2

1 0 r

κ = > 0κ = 0κ <

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

Band Theory of Solids

Bloch Theorem :

( ) ( ) ( ) ( )n n nH u E u=k k k k

dT

∈

=

k Brillouin Zone Torus,

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

Band Structure :

( )Hk ka E

kx π/a−π/a

Egap

=kx

ky

π/a

π/a

−π/a

−π/a

BZ

( ) iT eψ ψ⋅= k RRLattice translation symmetry ( )ie uψ ⋅= k r k

Bloch Hamiltonian ( ) Ηi iH e e− ⋅ ⋅= k r k rk

A mapping

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 (tor with enerus gy) gapH k a

E

k

Berry Phase Phase ambiguity of quantum mechanical wave function

( )( ) ( )iu e uφ→ kk k Berry connection : like a vector potential ( ) ( )i u u= − ∇kA k k

( )φ→ +∇kA A k Berry phase : change in phase on a closed loop C C C

dγ = ⋅∫ A k— Berry curvature : =∇ ×kF A

2 C S

d kγ = ∫ F Famous example : eigenstates of 2 level Hamiltonian

( ) ( ) z x y x y z

d d id H

d id d σ

−⎛ ⎞ = ⋅ = ⎜ ⎟+ −⎝ ⎠

k d k r

( ) ( ) ( ) ( )H u u= +k k d k k ( )1 ˆSolid Angle swept out by ( )2Cγ = 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 π

= ∫—

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

k -π/a π/a

0

dipole moment length

P =

bound Pρ =∇⋅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 kφ→P P en→ + ( / ) ( / ) 2a a nφ π φ π π− − =

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

( ) ( , ( ))u k u k tλ→

1 0 2 2C S e eP P P dk dkdλ λ λπ π= =

Δ = − = =∫ ∫A F—

when with

k

-π/a π/a

λ 1

0

Q P e=end 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 π π

= = ∇∫ ∫— —

kr i ∇:

A more rigorous argument:

Construct Localized Wannier Orbitals :

( )( ) ( ) 2

ik R r

BZ

dkR e u kϕ π

− −= ∫—

( ) ( )

( ) ( ) 2 kBZ

P e R r R R ie u k u k

ϕ ϕ

π

= −

= ∇∫—

Wannier states are gauge dependent, but for a sufficiently smooth gauge, they are localized states associated with a Bravais Lattice point R

( )R rϕ

R

“ ”

R

( )R rϕ

r

Su Schrieffer Heeger Model model for polyacetylenesimplest “two band” model † †

1( ) ( ) . .Ai Bi Ai Bi i

H t t c c t t c c h cδ δ += + + − +∑

( ) ( )H k k σ= ⋅d r

( ) ( ) ( ) cos ( ) ( )sin

( ) 0

x

y

z

d k t t t t ka d k t t ka d k

δ δ

δ

= + + −

= −

=

0tδ >

0tδ <

a

d(k)

d(k)

dx

dy

dx

dy

E(k)

k

π/a−π/a

Provided symmetry requires dz(k)=0, the states with δt>0 and δt0 : Berry phase 0 P = 0

δt

“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 kσ σ σ= = −

iA iA

iB iB

c c c c

→

→−

z E z E z E EH Eσ ψ σ ψ σ ψ ψ−= − ⇒ =

( ) ( )x xH k H kσ σ− =

• 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 δt focus on low energy states with k~π/a xk q q ia

π → + →− ∂ ;

( )vF x x yH i m xσ σ= − ∂ +

0tδ > 0tδ <

Massive 1+1 D Dirac Hamiltonian

“Chiral” Symmetry :

2v ; F ta m tδ= =

{ , } 0 z z E EHσ σ ψ ψ−= → =

0

( ') '/ v

0

1 ( )

0

x

Fm x dx

x eψ −∫ ⎛ ⎞

= ⎜ ⎟ ⎝ ⎠

Egap=2|m| Domain wall

bound state ψ0

m>0

m

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 π

Δ = − =∫ ∫—

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