E

k=Λa k=Λb

E

k=Λa k=Λb

Symmetry, Topology andPhases of Matter

Topological Phases of MatterMany 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 statesState 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 materialsand experiments

Much recent conceptual progress, but theory is

still far from the real electrons

Topological Band TheoryTopological Band Theory I:

IntroductionTopologically protected gapless states (without symmetry)

Topological Band Theory II: Time Reversal symmetryCrystal symmetryTopological superconductivity10 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 II. Introduction

- Insulating state, topology and band theoryII. 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 energylevels

Landaulevels

3p

4s

3s

2

3

1

gE

Topology The study of geometrical properties that are insensitive to smooth deformationsExample: 2D surfaces in 3D

A closed surface is characterized by its genus, g = # holesg=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 )SdA gκ π= −∫

1 2

1r r

κ =

Gauss Bonnet Theorem :

2

1 0r

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 kaE

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 PhasesTopological 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 structuresEg ~ 1 eV

Band Theory of Solidse.g. Silicon

E

adiabatic deformation

excited states

topological quantumcritical point

GapEG

Ground state E0

( ) : Bloch Hamiltonans Brillouin zone (tor with enerus gy) gapH k a

E

k

Berry PhasePhase ambiguity of quantum mechanical wave function

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

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

dγ = ⋅∫ A k—Berry curvature : =∇ ×kF A 2

C Sd kγ = ∫ F

Famous example : eigenstates of 2 level Hamiltonian

( ) ( ) z x y

x y z

d d idH

d id dσ

−⎛ ⎞= ⋅ = ⎜ ⎟+ −⎝ ⎠

k d k r

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

Cd̂

S

Topology in one dimension : Berry phase and electric polarization

Classical electric polarization :

+Q-Q1D insulator

Proposition: The quantum polarization is a Berry phase

( )2 BZ

eP A k dkπ

= ∫—

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

k-π/aπ/a

0

dipole momentlength

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

λ1

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 kπ π

= = ∇∫ ∫— —

kr i ∇:

A more rigorous argument:

Construct Localized Wannier Orbitals :

( )( ) ( )2

ik R r

BZ

dkR e u kϕπ

− −= ∫—

( ) ( )

( ) ( )2 kBZ

P e R r R Rie 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

iH 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 kad k t t kad 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 δt<0 are distinguished byan integer winding number. Without extra symmetry, all 1D band structures are topologically equivalent.

A,iB,i

δt>0 : Berry phase 0P = 0

δt<0 : Berry phase πP = e/2

Gap 4|δt|

Peierls’ instability → δt

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 kσ σ σ= = −

iA iA

iB iB

c cc 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 StatesAn 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 i

aπ

→ + →− ∂ ;

( )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 wallbound state ψ0

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)2eP A k T dk A k dk neπ

Δ = − =∫ ∫— —k

-π/a π/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 bywhich must wrap around the sphere an integer n times.

ˆ ( , ),k td

2

12 T

n dkdtπ

= ∫ F

2

1 ˆ ˆ ˆ ( )4 k tT

n dkdtπ

= ⋅ ∂ ×∂∫ 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 Eπ σ=

0

T

xy xyd hQ dtdt e

σ σΦ

Δ = =∫

12

dER dtπ

Φ= +Q-Q

( 0) 0( ) /tt T h e

Φ = =

Φ = =

Just like a Thouless pump : 2

xyeQ ne nh

σΔ = → =

†( ) (0)H T U H U=

TKNN InvariantThouless, Kohmoto, Nightingale and den Nijs 82

View cylinder as 1D system with subbands labeled by0

1( )myk m

R φ⎛ ⎞Φ

Φ = +⎜ ⎟⎝ ⎠

( )0

0

, ( )2

mx x y

m

eQ d dk k k neφ

πΔ = Φ Φ =∑ ∫ ∫ F

kx

kyTKNN number = Chern number

21 1( )2 2 C

BZ

n d k dπ π

= = ⋅∫ ∫F k A k—

2

xyenh

σ =

kym(0)

kym(φ0)

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

Alternative calculation: compute σxy via Kubo formula

C

kx

( )( ) , ( )mm x x yE k E k k= Φ

TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 82

Physical meaning: Hall conductivity

Laughlin Argument: Thread magnetic flux φ0 = h/e through a 1D cylinderPolarization changes by σxy φ0

21 ( )2 BZ

d kπ

= ∫ F k

2

xyenh

σ =

1 2

1 12 2C C

n d dπ π

= ⋅ − ⋅ ∈∫ ∫A k A k ¢— —

( ) ( ) ( )i u u= − ∇kΑ k k k

kx

ky

C1

C2

π/a

π/a

−π/a

−π/a

For a 2D band structure, define

E IykΦ ∝ P neΔ =ne-ne