E

k=a k=b

E

k=a k=b

Topological Insulatorsand Topological Band Theory

The Quantum Spin Hall Effectand Topological Band Theory

I. Introduction - Topological band theory

II. Two Dimensions : Quantum Spin Hall Insulator - Time reversal symmetry & Edge States - Experiment: Transport in HgCdTe quantum wells

III. Three Dimensions : Topological Insulator - Topological Insulator & Surface States - Experiment: Photoemission on BixSb1-x and Bi2Se3

IV. Superconducting proximity effect - Majorana fermion bound states - A platform for topological quantum computing?

Thanks to Gene Mele, Liang Fu, Jeffrey Teo, Zahid Hasan + group (expt)

The Insulating State

Covalent Insulator

Characterized by energy gap: absence of low energy electronic excitations

The vacuumAtomic Insulator

e.g. solid Ar

Dirac Vacuum

Egap ~ 10 eV

e.g. intrinsic semiconductor

Egap ~ 1 eV3p

4s

Silicon

Egap = 2 mec2 ~ 106 eV

electron

positron ~ hole

The Integer Quantum Hall State

2D Cyclotron Motion, Landau Levels

gap cE E

Quantized Hall conductivity :

ExB

Jy

y xy xJ E2

xy hne

Integer accurate to 10-9

Energy gap, but NOT an insulator

Graphene

2

xy

e

h

E

k

Low energy electronic structure:

Two Massless Dirac Fermions

Haldane Model (PRL 1988)

Add a periodic magnetic field B(r)

• Band theory still applies• Introduces energy gap

• Leads to Integer quantum Hall state

v | |E k

2 2 2v | |E m k

The band structure of the IQHE state looks just like an ordinary insulator.

Novoselov et al. ‘05

www.univie.ac.at

Topological Band Theory

g=0 g=1

21( ) ( ) Integer

2 BZn d u u

i k kk k k

The distinction between a conventional insulator and the quantum Hall state is a topological property of the manifold of occupied states

Analogy: Genus of a surface : g = # holes

| ( ) : ( )k

Brillouin zone a torus Hilbert space

Insulator : n = 0IQHE state : xy = n e2/h

The TKNN invariant can only changeat a quantum phase transition where theenergy gap goes to zero

Classified by the Chern (or TKNN) topological invariant (Thouless et al, 1982)

Edge StatesGapless states must exist at the interface between different topological phases

IQHE staten=1

Egap

Domain wall bound state 0

Vacuumn=0

Edge states ~ skipping orbits

n=1 n=0

Band inversion – Dirac Equation

x

y

M<0

M>0

Gapless Chiral Fermions : E = v k

E

Haldane Modelky

Egap

KK’

Smooth transition : gap must pass through zero

Jackiw, Rebbi (1976)Su, Schrieffer, Heeger (1980)

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)

Haldane*Haldane

0 0

0 0

H HH

H H

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

• Protected by Time Reversal Symmetry

• Elastic Backscattering is forbidden. No 1D Anderson localization

Kane and Mele PRL 2005

J↑ J↓

E

Bulk energy gap, but gapless edge states Edge band structure

↑↓

0 /a k

Spin Filtered edge states

↓ ↑QSH Insulator

vacuum

Topological Insulator : A New B=0 Phase There are 2 classes of 2D time reversal invariant band structures

Z2 topological invariant: = 0,1

is a property of bulk bandstructure, but can be understood byconsidering the edge states

=0 : Conventional Insulator =1 : Topological Insulator

Kramers degenerate at

time reversal

invariant momenta

k* = k* + G

E

k*=0 k*=/a

E

k*=0 k*=/a

Edge States for 0<k</a

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

Quantum Spin Hall Insulator in HgTe quantum wells

Theory: Bernevig, Hughes and Zhang, Science 2006

HgTe

HgxCd1-xTe

HgxCd1-xTed

Predict inversion of conduction and valence bands for d>6.3 nm → QSHI

G=2e2/h

↑↓

↑ ↓V 0I

Landauer Conductance G=2e2/h

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

Robust to disorder: impossible to localize

0 = 0 : Weak Topological Insulator

Fermi circle encloses even number of Dirac pointsRelated to layered 2D QSHI

OR

4

1 2

3

EF

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

kx

ky

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 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

• 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

Superconducting Proximity Effect

s wave superconductor

Topological insulator

Fu, Kane PRL 08

BCS Superconductor :† † ik kc c e

k↑

-k↓

Superconducting surface states

† †surface

ik kc c e

Surface states acquiresuperconducting gap due to Cooper pair tunneling

Half an ordinary superconductorHighly nontrivial ground state

-k

k

↑

←

↓

→

Dirac point

(s-wave, singlet pairing)

(s-wave, singlet pairing)

Majorana fermion : • Particle = Anti-Particle

• “Half a state”

• Two separated vortices define one zero energy

fermion state (occupied or empty)

Majorana Fermion at a vortex

Ordinary Superconductor :

Andreev bound states in vortex core:

0

E

E ↑,↓

-E ↑,↓

Bogoliubov Quasi Particle-Hole redundancy :

†, ,E E

Surface Superconductor :

Topological zero mode in core of h/2e vortex:

0

E †0 0

E=0

0 2

/ 2h e

Majorana Fermion• Particle = Antiparticle : †

• Real part of Dirac fermion : = †; = i“half” an ordinary fermion

• Mod 2 number conservation Z2 Gauge symmetry : → ±

Potential Hosts :

Particle Physics :

• Neutrino (maybe)

- Allows neutrinoless double -decay. - Sudbury Neutrino Observatory

Condensed matter physics : Possible due to pair condensation

• Quasiparticles in fractional Quantum Hall effect at =5/2• h/4e vortices in p-wave superconductor Sr2RuO4

• s-wave superconductor/ Topological Insulator• among others....

Current Status : NOT OBSERVED

† † 0

Majorana Fermions and Topological Quantum Computation

• 2 separated Majoranas = 1 fermion : = i 2 degenerate states (full or empty)

1 qubit

• 2N separated Majoranas = N qubits

• Quantum information stored non locally Immune to local sources decoherence

• Adiabatic “braiding” performs unitary operations

Kitaev, 2003

a ab bU

Non-Abelian Statistics

Manipulation of Majorana FermionsControl phases of S-TI-S Junctions

12

0

Majorana present

Tri-Junction : A storage register for Majoranas

CreateA pair of Majorana boundstates can be created from the vacuum in a well definedstate |0>.

BraidA single Majorana can bemoved between junctions.Allows braiding of multipleMajoranas

MeasureFuse a pair of Majoranas.

States |0,1> distinguished by• presence of quasiparticle.• supercurrent across line junction

E

00

1E

E

0

0

0

0

0

0

0

Conclusion

• A new electronic phase of matter has been predicted and observed - 2D : Quantum spin Hall insulator in HgCdTe QW’s - 3D : Strong topological insulator in Bi1-xSbx , Bi2Se3 and Bi2Te3

• Superconductor/Topological Insulator structures host Majorana Fermions - A Platform for Topological Quantum Computation

• Experimental Challenges - Transport Measurements on topological insulators

- Superconducting structures : - Create, Detect Majorana bound states - Magnetic structures : - Create chiral edge states, chiral Majorana edge states - Majorana interferometer

• Theoretical Challenges

- Effects of disorder on surface states and critical phenomena - Protocols for manipulating and measureing Majorana fermions.