k= L a

1. E E k=La k=Lb k=La k=Lb Topological Insulatorsand Topological Band Theory

2. 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)

3. The Insulating State Characterized by energy gap: absence of low energy electronic excitations Covalent Insulator Atomic Insulator The vacuum e.g. intrinsic semiconductor e.g. solid Ar electron Dirac Vacuum 4s Egap ~ 10 eV Egap = 2 mec2 ~ 106 eV 3p Egap ~ 1 eV positron ~ hole Silicon

4. The Integer Quantum Hall State 2D Cyclotron Motion, Landau Levels E Energy gap, but NOT an insulator Quantized Hall conductivity : Jy Ex B Integer accurate to 10-9

5. + - + - + - + + - + - + + - + - + - + Graphene E k www.univie.ac.at Novoselov et al. ‘05 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 The band structure of the IQHE state looks just like an ordinary insulator.

6. Topological Band Theory The distinction between a conventional insulator and the quantum Hall state is a topological property of the manifold of occupied states Classified by the Chern (or TKNN) topological invariant (Thouless et al, 1982) The TKNN invariant can only change at a quantum phase transition where the energy gap goes to zero Insulator : n = 0 IQHE state : sxy = n e2/h Analogy: Genus of a surface : g = # holes g=0 g=1

7. Edge States Gapless states must exist at the interface between different topological phases IQHE state n=1 Vacuum n=0 n=0 n=1 y x Smooth transition : gap must pass through zero Edge states ~ skipping orbits Gapless Chiral Fermions : E = v k Band inversion – Dirac Equation E M>0 Egap Egap M<0 Domain wall bound state y0 ky K’ K Jackiw, Rebbi (1976) Su, Schrieffer, Heeger (1980) Haldane Model

8. Edge band structure ↑ ↓ k 0 p/a 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 Sz) J↓ J↑ E Bulk energy gap, but gapless edge states Spin Filtered edge states vacuum ↑ ↓ QSH Insulator • 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

9. Topological Insulator : A New B=0 Phase There are 2 classes of 2D time reversal invariant band structures Z2 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<k<p/a n=1 : Topological Insulator n=0 : Conventional Insulator E E Kramers degenerate at time reversal invariant momenta k* = -k* + G k*=p/a k*=0 k*=p/a k*=0

10. I ↑ ↓ V 0 ↓ ↑ HgTe HgxCd1-xTe d HgxCd1-xTe Quantum Spin Hall Insulator in HgTe quantum wells Theory: Bernevig, Hughes and Zhang, Science 2006 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=2e2/h d> 6.3nm inverted band order QSH insulator G=2e2/h Measured conductance 2e2/h independent of W for short samples (L<Lin)

11. L3 L4 E E L1 L2 k=La k=Lb k=La k=Lb 3D Topological Insulators There are 4 surfaceDirac Points due to Kramers degeneracy ky kx OR 2D Dirac Point How do the Dirac points connect? Determined by 4 bulk Z2 topological invariantsn0 ; (n1n2n3) Surface Brillouin Zone n0 = 1 : Strong Topological Insulator EF Fermi circle encloses odd number of Dirac points Topological Metal : 1/4 graphene Robust to disorder: impossible to localize n0 = 0 : Weak Topological Insulator Fermi circle encloses even number of Dirac points Related to layered 2D QSHI

12. 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-xSbx • Bi1-x Sbx is a Strong Topological • Insulator n0;(n1,n2,n3) = 1;(111) • 5 surface state bands cross EF • between G and M Bi2 Se3 ARPES Experiment : Y. Xia et al., Nature Phys. (2009). Band Theory : H. Zhang et. al, Nature Phys. (2009). • n0;(n1,n2,n3) = 1;(000) : Band inversion at G • Energy gap: D ~ .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 by exposing to NO2

13. Superconducting Proximity Effect Fu, Kane PRL 08 Surface states acquire superconducting gap D due to Cooper pair tunneling s wave superconductor Topological insulator -k↓ BCS Superconductor : (s-wave, singlet pairing) k↑ Superconducting surface states -k ← Dirac point ↑ ↓ (s-wave, singlet pairing) → Half an ordinary superconductor Highly nontrivial ground state k

14. Majorana Fermion at a vortex Ordinary Superconductor : Andreev bound states in vortex core: E Bogoliubov Quasi Particle-Hole redundancy : D E ↑,↓ 0 -E ↑,↓ -D Surface Superconductor : Topological zero mode in core of h/2e vortex: E • Majorana fermion : • Particle = Anti-Particle • “Half a state” • Two separated vortices define one zero energy • fermion state (occupied or empty) D 0 E=0 -D

15. Majorana Fermion • Particle = Antiparticle :g = g† • Real part of Dirac fermion :g= Y+Y†; Y = g1+i g2 “half” an ordinary fermion • Mod 2 number conservation  Z2 Gauge symmetry :g→± g • 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 Sr2RuO4 • s-wave superconductor/ Topological Insulator • among others.... • Current Status : NOT OBSERVED

16. Kitaev, 2003 Majorana Fermions and Topological Quantum Computation • 2 separated Majoranas = 1 fermion : Y = g1+i g2 • 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

17. f1 f2 + - 0 Manipulation of Majorana Fermions Control phases of S-TI-S Junctions Majorana present Tri-Junction : A storage register for Majoranas Create A pair of Majorana bound states can be created from the vacuum in a well defined state |0>. Measure Fuse a pair of Majoranas. States |0,1> distinguished by • presence of quasiparticle. • supercurrent across line junction Braid A single Majorana can be moved between junctions. Allows braiding of multiple Majoranas E E E 0 0 0 f-p f-p f-p 0 0 0

18. 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.