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 p31msymmetry 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 structuresEg ~ 1 eV

Band Theory of Solidse.g. Silicon

E

adiabatic deformation

excited states

topological quantumcritical point

GapEG

Ground state E0

E

k

( ) : Bloch Hamiltonans Brillouin zone (torwith ener

usgy

) gap

H 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 …..

Many real materialsand experiments

Beyond Band Theory: Strongly correlated statesState with intrinsic topological order

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

- Symmetry protected topological states - Surface topological order ……

Much recent conceptual progress, but theory is

still far from the real electrons

Topological Superconductivity

Proximity induced topological superconductivity

Majorana bound states, quantum information

Tantalizing recent experimental progress

Topological 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 - 3D Quantum Hall Effect

- Topological Defects

- Weyl Semimetal

The Insulating State

The Integer Quantum Hall State

2D Cyclotron Motion, sxy = e2/h

E

Insulator vs Quantum Hall state

What’s the difference?

0 p/a-p/a

E

0 p/a-p/a

a

atomic insulator

atomic energylevels

Landaulevels

3p

4s

3s

2

3

1

Distinguished by Topological Invariant

g cE

/h e

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

Gauss Bonnet Theorem :

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

4 (1 )S

dA gk p -

1 2

1

r rk 2

10

rk 0k

0k

Band Theory of Solids

Bloch Theorem :

Band Structure :

E

kx p/a-p/a

Egap

=kx

ky

p/a

p/a

-p/a

-p/a

BZ

Lattice translation symmetry

Bloch Hamiltonian

A mapping

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

dT

k Brillouin Zone

Torus,

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

( )Hk k

( ) iT e k RR ( )ie u k r k

( ) Ηi iH e e- k r k rk

Berry Phase

Phase ambiguity of quantum mechanical wave function

Berry connection : like a vector potential

Berry phase : change in phase on a closed loop C

Berry curvature :

Famous example : eigenstates of 2 level Hamiltonian

C S

( )( ) ( )iu e u kk k

( ) ( )i u u- kA k k

( ) kA A k

C Cd A k

kF A 2C S

d k F

( ) ( ) z x y

x y z

d d idH

d id ds

-

- k d k

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

d̂

Topology in one dimension : Berry phase and electric polarization

Classical electric polarization :

+Q-Q

1D insulator

Proposition: The quantum polarization is a Berry phase

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

k

-p/a

p/a0

Bound charge density

End charge

BZ = 1D Brillouin Zone = S1

( )2 BZ

eP A k dk

p

dipole moment

lengthP

bound P

ˆendQ P n

( ) ( )i u u- kA k k

Quantum Polarization

Construct Localized Wannier Orbitals :

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

R

R

Bloch states are defined for periodic boundary conditions

( )( ) ( )2

ik R r

BZ

dkR e u k

p

- -

( ) ( )

( ) ( )2 kBZ

P e R r R R

ieu k u k

p

-

( )R r

( )R r

r

( ) ( )ikrk kr e u r

Gauge invariance and intrinsic ambiguity of P

• 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.

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

when with

k

-p/a p/a

l1

0

C

S

gauge invariant Berry curvature

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

( ) ( , ( ))u k u k tl

1 0 2 2C S

e eP P P dk dkdl l l

p p - A F

Q P eend mod

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

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 dt>0 and dt<0 are distinguished byan integer winding number. Without extra symmetry, all 1D band structures are topologically equivalent.

A,i

B,i

dt>0 : Berry phase 0

P = 0

dt<0 : Berry phase p

P = e/2

Gap 4|dt|

Peierls’ instability → dt

A,i+1

† †1( ) ( ) . .Ai Bi Ai Bi

i

H t t c c t t c c h cd d -

( ) ( )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

d d

d

-

-

0td

0td

“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:

• 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

( ), 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-

0

E

Many body ground state:

Single Particle Spectrum: zero mode

t – dt = 0 : decoupled E=0 state

t – dt >0 : protected by particle hole symmetry

charge fractionalization

A ABe/2 e/2

0

E

0

E

Occupied:charge +e/2

Empty:charge -e/2

Splitting the electron:

Ae

split

A BDomain Wall States

Jackiw Rebbi Problem

Low energy continuum theory for dt << t :

Focus on low energy states with k~p/a

Massive 1+1 D Dirac Hamiltonian

Zero mode at boundary where m(x) changes sign :

Egap=2|m|Domain wall

bound state 0

m<0

m>0

Solve

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

xk q q ia

p - ;

( )vF x x yH i m xs s-

0td 0td

2v ; F ta m td

0

( ') '/ v

0

1( )

0

x

Fm x dx

x e-

2 2( ) ( )vFE q q m

0 0 0( / v ) 0 ( ) / vF x x z Fi H m xs s -

Thouless Charge Pump

t=0

t=T

P=0

P=e

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,

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.

The integer charge pumped across a 1D insulator in one period of an adiabatic cycle is a topological invariant that characterizes the cycle.

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

( , ) ( ,0)2

eP A k T dk A k dk ne

p -

( , )u k t

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

TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 82

Physical meaning: Quantized Hall conductivity

kx

ky

C1

C2

p/a

p/a

-p/a

-p/a

For 2D band structure, define

Laughlin argument: Thread flux = h/e

Alternative calculation: compute sxy via Kubo formula

E I

ne-ne

=

BZ

Thouless pump: Cylinder with circumference 1 lattice constant (a)

(Faraday’s law)

plays role of ky

21( )

2 BZ

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

/xy xyP Idt h es s

/ 2 /yh e k ap

/xy xyI Ea d dts s

/Ea d dt

P ne

Realizing a non trivial Chern number

Integer quantum Hall effect:

Landau levels

Chern insulator:

e.g. Haldane model

Band Inversion Paradigm

E

Eg

k

Chern BandC=1

k

E

k

E

conductionband

valenceband

g cE

Lattice model for Chern insulator

|Esp| > 4t : Uninverted Trivial Insulator

d(k)dz

dx

dy

dz

dx

dy

d(k)

Chern number 0

Chern number 1

Square lattice model with inversion of bands with s and px+ipy symmetry near G

|Esp| < 4t : Inverted Chern Insulator

Regularized continuum model for Chern insulator

Inverted near k=0 for m<0. Uninverted for

m = 4t0-Esp

a = t0 av = 2tsp a

k

E

conductionband tz = +1

valenceband tz = -1

𝜋𝑎 −

𝜋𝑎

p+ip

s

0( ) 2 cos cos

2 sin sin ( )

z x y sp

sp x x y y

H t k a k a E

t k a k a k

t

t t t

k

d

2( ) ( )z x x y yH m ak k k kt t t t k d

v

Edge StatesGapless states at the interface between topologically distinct phases

IQHE staten=1

Egap

Domain wallbound state 0

Vacuumn=0

Edge states ~ skipping orbitsLead 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.

Band inversion transition : Dirac Equation

ky

E0

x

y

Chiral Dirac Fermions

m<0

m>0

n=1m = -m0

n=0m = m0

in

|t|=1disorder

Fv ( ) ( )x x y y zH i k m xs s s -

F

0

( ') '/ v

0 ( ) ~

x

y

m x dxik yx e e

- 0 F( ) vy yE k k

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

kyKK’

EF

Haldane Model

E

kyKK’

EF

The boundary topological invariant N characterizing the gapless modes

Difference in the topological invariantsn characterizing the bulk on either side=

Chiral Anomaly :

In the presence of an electric field, the charge at the edge is not conserved

Single particle edge spectrum : “one way edge states”

Many body edge spectrum : “chiral Fermi liquid”

• Free Dirac fermion conformal field theory

• Quantized electrical conductance:

• Quantized thermal conductance:

Vacuum : n=0

QHE state : n=1

E

kx0

conduction band

valence band

p/a-p/a

EF

chiral centralcharge

2 2 xy

dQ e dk e eEE

dt dts

p p

2dI eG

dV h

22

3Q B

dI kc T

dT h

pk

1

1c

†v xH i -

Generalizations

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

d=4 : 4 dimensional generalization of IQHE

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

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

d F A A A

42

1[ ]

8Tr

Tn

p F F

3D Quantum Hall Effectz

More generally, the 3 independent Chern numbers (nx,ny,nz) define a reciprocal lattice vector G that characterizes a family of lattice planes.

kz

kx

ky

Chern number nz independent of kz

2

2ij ijk k

eG

hs

p

Topological DefectsConsider insulating Bloch Hamiltonians that vary slowly in real space

defect line

s

2nd Chern number :

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

Burgers’ vector

3D Chern number

(vector ┴ layers)

Are there other ways to engineer1D chiral dirac fermions?

Teo, Kane ‘10

( , )H H s k

3 12

1[ ]

8Tr

T Sn

p F F

1

2 cnp

G B

Weyl Semimetal

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

A sphere in momentum space can have a Chern number:

S

nS=+1: S must enclose a degenerate Weyl point: Magnetic monopole for Berry flux

k

E

kx,y,z

Total magnetic charge in Brillouin zone must be zero: Weyl pointsmust come in +/- pairs.

2 ( )S Sn d k F k

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

Surface Fermi Arc

n1=n2=0

n0=1

k0

k1

k2kz

kx

ky

E

ky

EF

kz=k1, k2

E

ky

EF

kz=k0

Surface BZ

kz

ky

Chiral Anomaly

In the presence of E and B, the charge at one (or the other) Weylpoint is not conserved:

2

2

dn dn e

dt dt h -- E B

Topological Band Theory II: Time reversal symmetry

I. Graphene - Haldane model

- Time reversal symmetry and Kramers’ theorem

II. 2D quantum spin Hall insulator - Z2 topological invariant

- Edge states

- HgCdTe quantum wells, expts

III. Topological Insulators in 3D - Weak vs strong

- Topological invariants from band structure

IV. The surface of a topological insulator - Dirac Fermions

- Absence of backscattering and localization

- Quantum Hall effect

- q term and topological magnetoelectric effect

Graphene E

k

Two band model

Novoselov et al. ‘05www.univie.ac.at

Inversion and Time reversal symmetry require

2D Dirac points at : point vortices in

Massless Dirac Hamiltonian

-K +K

Berry’s phase p around Dirac point

A

B

( ) ( )H s k d k

( ) | ( ) |E k d k

ˆ ( , )x yk kd

( ) 0zd k

( , )x yd d

( ) vH s K q qk K

†Ai Bj

ij

H t c c

-

3

1

ˆ ˆ( ) cos sinj jj

t x y

- d k k r k r

Topological gapped phases in Graphene

1. Broken P : eg Boron Nitride

Break P or T symmetry :

Chern number n=0 : Trivial Insulator

2. Broken T : Haldane Model ’88

+K & -K

Chern number n=1 : Quantum Hall state

+K

-K

- - -

- -

- - -

( ) v zH ms s K q q

ˆ# ( )n d ktimes wraps around sphere

m m -

2 2 2( ) | |vE m q q

m m --

2ˆ ( ) Sd k

2ˆ ( ) Sd k

- - -

- -

- - -

Broken Inversion Symmetry

Broken Time Reversal Symmetry

Quantized Hall Effect

Respects ALL symmetries

Quantum Spin-Hall Effect

1. Staggered Sublattice Potential (e.g. BN)

2. Periodic Magnetic Field with no net flux (Haldane PRL ’88)

3. Intrinsic Spin Orbit Potential

Energy gaps in graphene:

B

2

2 2 2F( ) vE p p

zCDWV s

Haldanez zV s t

z z zSOV ss t

vFH p Vs ~

~

~

z

z

zs

s

t

sublattice

valley

spin

2

sgnxy

e

hs

Quantum Spin Hall Effect in Graphene

The intrinsic spin orbit interaction leads to a small (~10mK-1K) energy gap

Simplest model:|Haldane|2

(conserves Sz)

Edge states form a unique 1D electronic conductor• HALF an ordinary 1D electron gas

• Protected by Time Reversal Symmetry

J↑ J↓

E

Bulk energy gap, but gapless edge states Edge band structure

↑↓

0 p/a k

“Spin Filtered” or “helical” edge states

↓ ↑QSH Insulator

vacuum

Haldane*Haldane

0 0

0 0

H HH

H H

Quantum Spin Hall Effect in HgTe quantum wellsTheory: Bernevig, Hughes and Zhang, Science ‘06

HgTe

HgxCd1-xTe

HgxCd1-xTed

d < 6.3 nm : Normal band order d > 6.3 nm : Inverted band order

Conventional InsulatorQuantum spin Hall Insulatorwith topological edge states

G6 ~ s

G8 ~ pk

E

G6 ~ sG8 ~ p k

E

Egap~10 meVBand inversion transition:

Switch parity at k=0

BHZ Model : 4 band T-invariant band inversion model

2 ( ) 1n a 2 ( ) 1n a -

2( ) z x x x y x yH m ak k kt t s t s k v

Expt: Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007

Measured conductance 2e2/h independent of W for short samples (L<Lin)

d< 6.3 nmnormal band orderconventional insulator

d> 6.3nminverted band orderQSH insulator

Experiments on HgCdTe quantum wells

G=2e2/h

↑↓

↑ ↓V 0I

Landauer Conductance G=2e2/h

Time Reversal Symmetry :

Kramers’ Theorem: for spin ½ all eigenstates are at least 2 fold degenerate

Anti Unitary time reversal operator :

Spin ½ :

Proof : for a non degenerate eigenstate

Consequences for edge states :

States at “time reversal invariant momenta” k*=0 and k*=p/a (=-p/a) are degenerate.

The crossing of the edge states is protected, even if spin conservation is volated.

Absence of backscattering, even for strong disorder. No Anderson localization

1D “Dirac point”

k*

in

r=0 |t|=1T invariant disorder

/ *yi Se p

2 1 - *

*

-

2 2| |

c

c

2 2| | 1c -

[ , ] 0H

Time Reversal Invariant 2 Topological Insulator

=0 : Conventional Insulator =1 : Topological Insulator

Kramers degenerate at

time reversal

invariant momenta

k* = -k* + G

k*=0 k*=p/a k*=0 k*=p/a

Even number of bandscrossing Fermi energy

Odd number of bandscrossing Fermi energy

Understand via Bulk-Boundary correspondence : Edge States for 0<k<p/a

2D Bloch Hamiltonians subject to the T constraint

with 2-1 are classified by a 2 topological invariant ( = 0,1)

1 ( )H H- -k k

Physical Meaning of 2 Invariant

0 = h/e

Q = N e

Flux 0 Quantized change in Electron Number at the end.

=N IQHE on cylinder: Laughlin Argument

Quantum Spin Hall Effect on cylinder

0 / 2

Flux 0 /2 Change in

Electron Number Parityat the end, signaling changein Kramers degeneracy.

KramersDegeneracy

No Kramers Degeneracy

Sensitivity to boundary conditions in a multiply connected geometry

Formula for the 2 invariant

• Bloch wavefunctions :

• T - Reversal Matrix :

• Antisymmetry property :

• T - invariant momenta :

4

1 2

3

kx

ky

Bulk 2D Brillouin Zone

• Pfaffian :

• Z2 invariant :

Gauge invariant, but requires continuous gauge

(N occupied bands)

• Fixed point parity :

• Gauge dependent product :

“time reversal polarization” analogous to

( ) ( ) ( ) ( )mn m nw u u U - k k k N

2 1 ( ) ( )Tw w - - -k k

( ) ( )Ta a a aw w - - k

2

det[ ( )] Pf[ ( )]a aw w 20e.g. det

- 0

zz

z

4

1

( 1) ( ) 1aa

d

-

( )nu k

Pf[ ( )]( ) 1

det[ ( )]a

a

a

w

wd

( ) ( )a bd d

( )2

eA k dk

p

1. Sz conserved : independent spin Chern integers :

(due to time reversal)

is easier to determine if there is extra symmetry:

2. Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch states

Quantum spin Hall Effect :J↑ J↓

E

Allows a straightforward determination of from band structurecalculations.

In a special gauge:

n n -

, mod 2n

4

21

( 1) ( )n aa n

- ( ) ( ) ( )

( ) 1n a n a n a

n a

P

( ) ( )a n an

d

3D Topological InsulatorsThere are 4 surface Dirac Points due to Kramers degeneracy

Surface Brillouin Zone

2D Dirac Point

E

k=a k=b

E

k=a k=b

0 = 1 : Strong Topological Insulator

Fermi circle encloses odd number of Dirac points

Topological Metal : 1/4 graphene Berry’s phase p Robust to disorder: impossible to localize

0 = 0 : Weak Topological Insulator

Related to layered 2D QSHI ; (123) ~ Miller indicesFermi surface encloses even number of Dirac points

OR

4

1 2

3

EF

How do the Dirac points connect? Determined by 4 bulk Z2 topological invariants 0 ; (123)

kx

ky

kx

ky

kx

ky

Topological Invariants in 3D

1. 2D → 3D : Time reversal invariant planes

The 2D invariant

kx

ky

kz

Weak Topological Invariants (vector):

ki=0plane

Strong Topological Invariant (scalar)

a

p/ap/a

p/a

Each of the time reversal invariant planes in the 3D Brillouin zone is characterized by a 2D invariant.

“mod 2” reciprocal lattice vector indexes lattice planes for layered 2D QSHI

G

4

1

( 1) ( )aa

d

- Pf[ ( )]

( )det[ ( )]

aa

a

w

wd

4

1

( 1) ( )ia

a

d

-

8

1

( 1) ( )oa

a

d

-

1 2 3

2, ,

a

p G

Topological Invariants in 3D2. 4D → 3D : Dimensional Reduction

Add an extra parameter, k4, that smoothly connects the topological insulatorto a trivial insulator (while breaking time reversal symmetry)

H(k,k4) is characterized by its second Chern number

n depends on how H(k) is connected to H0, butdue to time reversal, the difference must be even.

(Trivial insulator)

k4

Express in terms of Chern Simons 3-form :

Gauge invariant up to an even integer.

42

1[ ]

8Trn d k

p F F ( ,0) ( )H Hk k

14 4( , ) ( , )H k H k -- k k

0( ,1)H Hk

0 2 mod n

3[ ]Tr dQ F F

0 32

1( ) 2

43d mod kQ

p k 3

2( ) [ ]

3TrQ d k A A A A A

Topological Superconductivity

0. … from last time: The surface of a topological insulator

1. Bogoliubov de Gennes Theory

2. Majorana bound states, Kitaev model

3. Topological superconductor

4. Periodic Table of topological insulators and superconductors

5. Topological quantum computation

6. Proximity effect devices

Bi1-xSbx

Theory: Predict Bi1-xSbx is a topological insulator by exploiting inversion symmetry of pure Bi, Sb (Fu,Kane PRL’07)

Experiment: ARPES (Hsieh et al. Nature ’08)

• Bi1-x Sbx is a Strong Topological Insulator 0;(1,2,3) = 1;(111)

• 5 surface state bands cross EF

between G and M

ARPES Experiment : Y. Xia et al., Nature Phys. (2009).Band Theory : H. Zhang et. al, Nature Phys. (2009).Bi2 Se3

• 0;(1,2,3) = 1;(000) : Band inversion at G

• Energy gap: ~ .3 eV : A room temperature topological insulator

• Simple surface state structure : Similar to graphene, except only a single Dirac point

EF

Control EF on surface byexposing to NO2

Unique Properties of Topological Insulator Surface States

“Half” an ordinary 2DEG ; ¼ Graphene

Spin polarized Fermi surface

• Charge Current ~ Spin Density• Spin Current ~ Charge Density

p Berry’s phase

• Robust to disorder• Weak Antilocalization• Impossible to localize, Klein paradox

Broken symmetry can lead to surface energy gap:

• Quantum Hall state, topological magnetoelectric effect (broken Time Reversal) Fu, Kane ’07; Qi, Hughes, Zhang ’08, Essin, Moore, Vanderbilt ‘09

• Superconducting state (broken gauge symmetry) Fu, Kane ‘08

EF

Surface Quantum Hall Effect

B

=1 chiral edge state

Orbital QHE :

M↑ M↓

E=0 Landau Level for Dirac fermions. “Fractional” IQHE

Anomalous QHE : Induce a surface gap by depositing magnetic material

Chiral Edge State at Domain Wall : M ↔ -M

Mass due to Exchange field

Egap = 2|M|EF

0

1

2

-2

-1

TI

2 1

2xy

en

hs

2

2xy

e

hs

2

2xy

e

hs

†0 ( v )zMH i s s - -

2

sgn( )2xy M

e

hs

2

2

e

h

2

2

e

h-

Topological Magnetoelectric Effect

Consider a solid cylinder of TI with a magnetically gapped surface

Magnetoelectric Polarizability

EJ

M

The fractional part of the magnetoelectric polarizability is determinedby the bulk, and independent of the surface (provided there is a gap)Analogous to the electric polarization, P, in 1D.

Qi, Hughes, Zhang ’08; Essin, Moore, Vanderbilt ‘09

d=1 : Polarization P

d=3 : Magnetoelectric poliarizability a

formula “uncertainty quantum”

(extra end electron)

(extra surface quantum Hall layer)

L

topological “q term”

2 1

2xy

eJ E n E M

hs

M Ea2 1

2

en

ha

[ ]2

TrBZ

e

p A

2

2

2[ ]

4 3Tr

BZ

ed

hp A A A A A

e2 /e h

3S d xdta E B

a E B

PE

2

2

e

ha q

p

0 2TR sym. : or mod q p p

Topological Insulator coupled to compact U(1) gauge field, A

Low energy theory for A : q term

• Time reversal symmetry : q 0 or p mod 2p. q p mod 2p for electron TI

• Witten Effect: Magnetic monopoles are charged

• Monopoles (or dyons) are bosons with half integer charge

Qi, Hughes and Zhang ‘08

For a compact gauge field, magnetic monopoles are excitations in the theory.Useful diagnostic for strongly interacting theories.

Strong Interactions

32

1

32 S i N N d xd F Fl lq t

p

2Q e

q

p

1

2Q e n

Can the surface of a TI be gapped without breaking symmetry ?

Pass monopole from inside to outside of TI :

Q=e/2boson

Q=0 boson

Charge e/2 must bedeposited at surface

Break T at surface:

Superconductor at surface:

Keep U(1) and T at surface :

sxy = e2/2h : charge e/2 flows away on surface

Charge conservation is violated at surface

Charge e/2 stays at surface : Requires a topologically ordered surface state with e/2 quasiparticle

TI

Requirements for a Topological Surface Phase on TI

It should be impossible in 2D if symmetry is preserved, but if symmetry is broken there should be a 2D state with the same topological order

Broken T: Topo – M slab

TI

M

Topo M↑

M↓Topo

= 1c = 1

= 1/2c = 1/2

A 2D Non-Abelian quantum Hall state with • Hall conductance e2/h; = 1/2 • Thermal Hall cond. c p2 kB

2/6h; c = 1/2 Topo – M slab

= 1/2c = 1/2

Theories of symmetry preserving gapped state:

Related to Moore-Read (Pfaffian) state of FQHE at =1/2

• “T-Pfaffian state” Bonderson, Nayak, Qi ’13 ; Chen, Fidkowski, Vishwanath ‘13

• “Moore-Read/antisemion state” Metlitski, Kane, Fisher ’13 ; Wang, Potter, Senthil ‘13

BCS Theory of Superconductivity

Intrinsic anti-unitary particle – hole symmetry

Bogoliubov de GennesHamiltonian

mean field theory :

Bloch - BdG Hamiltonians satisfy

Topological classification problem similar to time reversal symmetry

Particle – hole redundancy=

E

k

E

k

same state

††

1

2 BdGH H

k

1BdG BdGH H- -

†E E E E - -

0

*0

BdG

HH

H

-

† † † † *

*x t

2 1

0 1

1 0xt

1( ) ( )BdG BdGH H- - -k k

Bulk-Boundary correspondence : Discrete end state spectrum

0

-

E

E

-E0

-

E=0

=0 “trivial” =1 “topological”

Majorana fermionbound state

1D 2 Topological Superconductor : = 0,1END

(Kitaev, 2000)

Majorana Fermion : Particle = Antiparticle

Real part of a Dirac fermion :

“Half a state” Two Majorana fermions define a single two level system

Zero mode

0empty

occupied

†0 0E E G G

†E E-G G

EG

†1 1 2

† †2 1 2

;

( ) ;

i

i i

- - - , 2i j ij d

†0 1 2 02H i

2 1i

†

Kitaev Model for 1D p wave superconductor

||>2t : Strong pairing phase trivial superconductor

d(k)

d(k)

dz

dx

dz

dx

||<2t : Weak pairing phase topological superconductor

Similar to SSH model, except different symmetry :

- t

† † † † †1 1 1 1( ) ( )i i i i i i i i i i

i

H N t c c c c c c c c c c - -

( ) (2 cos ) sin ( )BdG z xH k t k k kt t t - d

††( ) k

k k BdGk k

cc c H k

c-

-

( , , ) ( , , )x y z x y zk kd d d d d d

- - -

Majorana Chain

t1>t2

trivial SC

t1<t2

topological SC

t1 t2

1i 2i

Unpaired Majorana Fermion at end

For =t : nearest neighbor Majorana chain

1 2i i ic i

†1 2

† †1 1 1 2 1 2 1 1

† †1 1 1 2 1 2 1 1

2

2

2

i i i i

i i i i i i i i

i i i i i i i i

c c i

t c c c c it

c c c c i

-

1 1 2 2 2 1 12 i i i ii

H i t t

1 2, 2t t t

2D topological superconductor (broken T symmetry)

Bulk-Boundary correspondence: n = # Chiral Majorana Fermion edge states

k

E

Examples

• Spinless px+ipy superconductor (n=1)

• Chiral triplet p wave superconductor (eg Sr2RuO4) (n=2)

Read Green model :

Lattice BdG model :

||>4t : Strong pairing phase trivial superconductor

d(k)dz

dx

||<4t : Weak pairing phase topological superconductor

dy

dz

dx

dy

d(k)

Chern number 0

Chern number 1

T-SC

†k k -

k

2

† ( ) . .2

H c c c c c cm

-

-

k k k kk

kk 0( ) x yk ik k

( ) 2 cos cos sin sin ( )BdG z x y x x y yH t k k k k kt t t t - k d

Majorana zero mode at a vortex

Boundary condition on fermion wavefunction

0

LA

p even p odd

zero mode

Hole in a topological superconductor threaded by flux

EE

q q

Without the hole : Caroli, de Gennes, Matricon theory (’64)

2

hp

e

1( ) ( 1) (0)pL -

( ) ; 2 1miq xmx e q m p

L

p

2 v~

L

p

2

~FE

Periodic Table of Topological Insulators and SuperconductorsAnti-Unitary Symmetries :

- Time Reversal :

- Particle - Hole :

Unitary (chiral) symmetry :

RealK-theory

ComplexK-theory

Bott Periodicity d→d+8

Altland-ZirnbauerRandom MatrixClasses

Kitaev, 2008Schnyder, Ryu, Furusaki, Ludwig 2008

8 antiunitary symmetry classes

1( ) ( ) 12 ; H H- - - k k

1( ) ( ) 12 ; H H- - k k

1( ) ( )H H- - k k ;

• 2 Majorana separated bound states = 1 fermion - 2 degenerate states (full/empty) = 1 qubit

• 2N separated Majoranas = N qubits• Quantum Information is stored non locally - Immune from local decoherence

Create

Braid

Measure

t

Majorana Fermions and Topological Quantum Computing

The degenerate states associated with Majorana zero modesdefine a topologically protected quantum memory

(Kitaev ’03)

Braiding performs unitary operations

Non-Abelian statistics

Interchange rule (Ivanov 03)

These operations, however, are not sufficientto make a universal quantum computer

1 2i

12 34 12 340 0 1 1 / 2

12 340 0

i j

j i

-

Potential condensed matter hosts for topological superconductivity

• Quasiparticles in fractional Quantum Hall effect at =5/2 Moore Read ‘91

• Unconventional superconductors

- Sr2RuO4 Das Sarma, Nayak, Tewari ‘06

- Fermionic atoms near feshbach resonance Gurarie ‘05

- CuxBi2Se3 ?

• Proximity Effect Devices using ordinary superconductors

- Topological Insulator devices Fu, Kane ‘08

- 2D Semiconductor/Magnet devices Sau, Lutchyn, Tewari, Das Sarma ’09, Lee ’09

- 1D Semiconductor devices: eg In As quantum wires Oreg, von Oppen, Alicea, Fisher ’10 Lutchyn, Sau, Das Sarma ‘10

Expt : Maurik et al. (Kouwenhoven) ‘12

- 1D Ferromagnetic atomic chains on superconductors Expt : Nadj-Perg et al. (Yazdani) ‘14

Topological Superconductors

Ordinary Superconductor :

k↑

-k↓

Surface of topological insulator

Half an ordinary superconductorNontrivial ground state supports Majorana fermions at vortices

-k

k

↑

←

↓

→

Dirac point

(s-wave, singlet pairing)

(s-wave, singlet pairing)

-k

k

Spinless p-wave superconductor:

† † ik k

c c e

-

† †surface

ik kc c e

-

† † ( )ik k x yc c e k ik

-

Majorana Bound States on Topological Insulators

SC

h/2e

1. h/2e vortex in 2D superconducting state

2. Superconductor-magnet interface at edge of 2D QSHI

Quasiparticle Bound state at E=0 Majorana Fermion 0 “Half a State”

TI

0

-

E

S.C.M

QSHIEgap =2|m|

Domain wall bound state 0

m<0

m>0

†0 0

†E

†E E -

| | | |S Mm -

1D Majorana Fermions on Topological Insulators

2. S-TI-S Josephson Junction

SC

TI

SC

p

p

SCM

1. 1D Chiral Majorana mode at superconductor-magnet interface

TI

kx

E

: “Half” a 1D chiral Dirac fermion

0

Gapless non-chiral Majorana fermion for phase difference p

†k k -

Fv xH i -

cos( / 2)Fv L x L R x R L RH i i - -

Another route to the 2D p+ip superconductor

Semiconductor - Magnet - Superconductor structureSau, Lutchyn, Tewari, Das Sarma ‘09

Magnetic Insulator

Superconductor

Semiconductor

Rashba split2DEG bands

Zeeman splitting

E

k

EF

• Single Fermi circle with Berry phase p-

• Topological superconductor with Majorana edge states and Majorana bound states at vortices.

• Variants : - use applied magnetic field to lift Kramers degeneracy (Alicea ’10)

- Use 1D quantum wire (eg InSb ). A route to 1D p wave superconductor with Majorana end states. (Oreg, von Oppen, Alicea, Fisher ’10

Experiments semiconductor nanowires

supercondutor semiconductornanowire

Majorana bound state

B

Observe zero bias peak in tunneling conductance:

Attributed to Majorana end state.

First experiment: 1D Semiconductor quantum wire on superconductor

• Superconductor + Semiconductor nanowire Sau, Lutchyn, Tewari, Das Sarma ’09; Alicea ’10; Oreg, Refael, von Oppen ’10

Mourik, …, Kouwenhoven, et al. ‘12

InSb nanowire zero bias peak

Vortex at superconductor TI interfaceJ-P Xu, et al. PRL 114, 017001 (2015)HH Sun, et al. PRL 116, 257003 (2016)

Vortex in Iron based superconductor Wang et al. Science 362, 333 (2018).

STMtopography

Majorana end mode

Nadj-Perg, Science ’14 STM Spectroscopy E=0

Ferromagnetic Atomic Chains on Superconductors (Iron on lead)

4p periodic Josephson effect in HgTeSC – 2D TI – SC junctions

Wiedenmann, et al., Nature Comm. 7:10303 (2016).

“TI with SC on insided”

Fractional Josephson Effect

e

o

Kitaev ’01Kwon, Sengupta, Yakovenko ’04Fu, Kane ’08

1 2

43

• 4p perioidicity of E() protected by local conservation of fermion parity.

• AC Josephson effect with half the usual frequency: f = eV/h

12 340 0

12 341 1

A Z2 Interferometer for Majorana Fermions

A signature for neutral Majorana fermions probed with charge transport

e e

e h

1

2

1

2 -2

N even

N odd• Chiral electrons on magnetic domain wall split into a pair of chiral Majorana fermions

• “Z2 Aharonov Bohm phase” converts an electron into a hole

Akhmerov, Nilsson, Beenakker, PRL ’09Fu and Kane, PRL ‘09

N

G

2

hN

e

†1 2

1 2

c i

c i

-