E

Topological Insulators and Topological Band Theory

E k= L a k= L b k= L a k= L b

The Quantum Spin Hall Effect and 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 Bi x Sb 1-x and Bi 2 Se 3 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

Characterized by energy gap: absence of low energy electronic excitations Covalent Insulator e.g. intrinsic semiconductor Atomic Insulator e.g. solid Ar The vacuum Silicon E gap ~ 1 eV 4s E gap ~ 10 eV 3p electron Dirac Vacuum E gap = 2 m e c 2 ~ 10 6 eV positron ~ hole

The Integer Quantum Hall State 2D Cyclotron Motion, Landau Levels

E

E gap

 

c

Energy gap, but

NOT

an insulator

Quantized Hall conductivity :

J y

 

xy E x

B J y E x 

xy



n e

2

h

Integer accurate to 10 -9

Graphene

Novoselov et al. ‘05 Low energy electronic structure: Two Massless Dirac Fermions  -  -  -   -  -   -  -  -  E www.univie.ac.at

k

E

  v |

k

| Haldane Model (PRL 1988) • • • Add a periodic magnetic field B(r) Band theory still applies Introduces energy gap Leads to Integer quantum Hall state 

xy



e

2

h E

  2 v |

k

| 2 

m

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

Topological Band Theory

The distinction between a conventional insulator and the quantum Hall state is a topological property of the manifold of occupied states |  ( ) Hilbert space Classified by the Chern (or TKNN) topological invariant (Thouless et al, 1982)

n

 1 2 

i



BZ d

2

k

 

k

u

 

k

u

 Integer Insulator : n = 0 IQHE state :  xy = n e 2 /h The TKNN invariant can only change at a quantum phase transition where the energy gap goes to zero Analogy: Genus of a surface : g = # holes g=0 g=1

Edge States

Gapless states must exist at the interface between different topological phases IQHE state n=1 Vacuum n=0 Edge states ~ skipping orbits Gapless Chiral Fermions : E = v k y x n=1 n=0 Smooth transition : gap must pass through zero E K’ Haldane Model K E gap k y Band inversion – Dirac Equation M>0 M<0 Domain wall bound state y 0 Jackiw, Rebbi (1976) Su, Schrieffer, Heeger (1980) E gap

Quantum Spin Hall Effect in Graphene

Kane and Mele PRL 2005 The intrinsic spin orbit interaction leads to a small (~10mK-1K) energy gap Simplest model: |Haldane| 2 (conserves S z )

H

  

H

 0 0

H



H

Haldane 0 0

H

* Haldane   J ↑ J ↓ E ↓

Bulk energy gap, but gapless edge states

Spin Filtered edge states vacuum ↑ Edge band structure ↓ ↑ QSH Insulator 0  /a k • • •

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

Topological Insulator : A New B=0 Phase

There are 2 classes of 2D time reversal invariant band structures Z 2 topological invariant: n = 0,1 n is a property of bulk bandstructure, but can be understood by considering the edge states

Edge States for 0



/a

n =0 : Conventional Insulator E n =1 : Topological Insulator E Kramers degenerate at

time reversal invariant momenta k* =

-

k* + G

k*=0 k*=  /a k*=0 k*=  /a

d

Quantum Spin Hall Insulator in HgTe quantum wells

Theory: Bernevig, Hughes and Zhang, Science 2006 HgTe Hg x Cd 1-x Te Hg x Cd 1-x Te Predict inversion of conduction and valence bands for d>6.3 nm → QSHI Expt: Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007 d< 6.3 nm normal band order conventional insulator Landauer Conductance G=2e 2 /h ↑

V

↓

I

↓ ↑

0

d> 6.3nm

inverted band order QSH insulator

G=2e 2 /h

Measured conductance 2e 2 /h independent of W for short samples (L

3D Topological Insulators

There are 4 surface k y L 4 L 3 k x L 1 L 2 Dirac Points due to Kramers degeneracy E OR Surface Brillouin Zone 2D Dirac Point E k= L a k= L b k= L a k= L b How do the Dirac points connect? Determined by 4 bulk Z 2 topological invariants n 0 ; ( n 1 n 2 n 3 ) n 0 = 1 : Strong Topological Insulator E F Fermi circle encloses odd number of Dirac points Topological Metal : 1/4 graphene Robust to disorder: impossible to localize n 0 = 0 : Weak Topological Insulator Fermi circle encloses even number of Dirac points Related to layered 2D QSHI

Bi

1-x

Sb

x Theory: Predict Bi 1-x Sb x is a topological insulator by exploiting inversion symmetry of pure Bi, Sb (Fu,Kane PRL’07) Experiment: ARPES (Hsieh et al. Nature ’08) • Bi 1-x Sb x is a Strong Topological Insulator n 0 ;( n 1 , n 2 , n 3 ) = 1;(111) • 5 surface state bands cross E F between G and M

Bi

2

Se

3 Control E F on surface by exposing to NO 2 ARPES Experiment : Y. Xia et al., Nature Phys. (2009).

Band Theory : H. Zhang et. al, Nature Phys. (2009).

• • • n 0 ;( n 1 , n 2 , n 3 ) = 1;(000) : Band inversion at G Energy gap: D ~ .3 eV : A room temperature topological insulator E F Simple surface state structure : Similar to graphene, except only a single Dirac point

Superconducting Proximity Effect

Fu, Kane PRL 08 s wave superconductor Surface states acquire superconducting gap D due to Cooper pair tunneling Topological insulator -k ↓ BCS Superconductor : †

c c k

†  D

e i

 (s-wave, singlet pairing) Superconducting surface states † †

c c k

-

k

 D surface

e i

 (s-wave, singlet pairing) Half an ordinary superconductor Highly nontrivial ground state k ↑ -k

↓ ← → ↑

k Dirac point

Majorana Fermion at a vortex

Ordinary Superconductor : D E Andreev bound states in vortex core: E ↑ , ↓ Bogoliubov Quasi Particle-Hole redundancy : 0 -D -E ↑ , ↓  †

E

,   

E

 

h

/ 2

e

  2    0 Surface Superconductor : D 0 -D E Topological zero mode in core of h/2e vortex: E=0 • • • Majorana fermion : Particle = Anti-Particle “ Half a state”  † 0   0 Two separated vortices define one zero energy fermion state (occupied or empty)

Majorana Fermion

• • • Particle = Antiparticle :    † Real part of Dirac fermion :  =  † ;  Mod 2 number conservation  Z 2 =  1  i  2 “half” an ordinary fermion Gauge symmetry :  → ±  Potential Hosts : • Particle Physics : Neutrino (maybe) - Allows neutrinoless double b -decay. - Sudbury Neutrino Observatory • • • • Condensed matter physics : Possible due to pair condensation Quasiparticles in fractional Quantum Hall effect at n =5/2 h/4e vortices in p-wave superconductor Sr 2 RuO 4 s-wave superconductor/ Topological Insulator among others....

Current Status : NOT OBSERVED † 0

Majorana Fermions and Topological Quantum Computation

Kitaev, 2003 • 2 separated Majoranas = 1 fermion :  =  1  i  2 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 Non-Abelian Statistics y

a



U ab

y

b

Manipulation of Majorana Fermions

Control phases of S-TI-S Junctions Tri-Junction : A storage register for Majoranas f 2 0 f 1  Majorana present Create A pair of Majorana bound states can be created from the vacuum in a well defined state |0>.

Braid A single Majorana can be moved between junctions.

Allows braiding of multiple Majoranas Measure Fuse a pair of Majoranas.

• • States |0,1> distinguished by presence of quasiparticle.

supercurrent across line junction 0 E 0 0 0 E 0 f- 0 f- 0 E 0 1 0 f-

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 Bi 1-x Sb x , Bi 2 Se 3 and Bi 2 Te 3 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.