Topological Insulators and Topological Band Theory E E

Topological Insulators and Topological Band Theory E E k=La k=Lb

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 Hg. Cd. Te quantum wells III. Three Dimensions : Topological Insulator - Topological Insulator & Surface States - Experiment: Photoemission on Bix. 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 Atomic Insulator e. g. intrinsic semiconductor e. g. solid Ar The vacuum electron 4 s Egap ~ 10 e. V 3 p Dirac Vacuum Egap = 2 mec 2 ~ 106 e. V Egap ~ 1 e. V Silicon positron ~ hole

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

Graphene E +- + - + +- + - + Novoselov et al. ‘ 05 www. univie. ac. at 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. k

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) Insulator IQHE state : n=0 : sxy = 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 y n=1 n=0 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 M<0 K’ Haldane Model K ky Domain wall bound state y 0 Jackiw, Rebbi (1976) Su, Schrieffer, Heeger (1980)

Quantum Spin Hall Effect in Graphene Kane and Mele PRL 2005 The intrinsic spin orbit interaction leads to a small (~10 m. K-1 K) energy gap Simplest model: |Haldane|2 (conserves Sz) J↓ J↑ E Bulk energy gap, but gapless edge states Edge band structure Spin Filtered edge states vacuum ↓ ↑ ↑ ↓ QSH Insulator 0 p/a k Edge states form a unique 1 D electronic conductor • • • HALF an ordinary 1 D electron gas Protected by Time Reversal Symmetry Elastic Backscattering is forbidden. No 1 D Anderson localization

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

Quantum Spin Hall Insulator in Hg. Te quantum wells Theory: Bernevig, Hughes and Zhang, Science 2006 d Hgx. Cd 1 -x. Te Hg. 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=2 e 2/h ↑ V d> 6. 3 nm inverted band order QSH insulator ↓ I ↓ 0 ↑ G=2 e 2/h Measured conductance 2 e 2/h independent of W for short samples (L<Lin)

3 D Topological Insulators There are 4 surface Dirac Points due to Kramers degeneracy ky L 4 L 1 L 3 E E kx OR L 2 2 D Dirac Point Surface Brillouin Zone k=La k=Lb 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 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 2 D QSHI EF

Bi 1 -x. Sbx Theory: Predict Bi 1 -x. Sbx 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 Sbx is a Strong Topological Insulator n 0; (n 1, n 2, n 3) = 1; (111) • 5 surface state bands cross EF between G and M Bi 2 Se 3 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 e. V : A room temperature topological insulator • Simple surface state structure : Control EF on surface by exposing to NO 2 Similar to graphene, except only a single Dirac point EF

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

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

Majorana Fermion • Particle = Antiparticle : g = g† • Real part of Dirac fermion : g = Y+Y†; Y = g 1+i g 2 “half” an ordinary fermion • Mod 2 number conservation Z 2 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/4 e vortices in p-wave superconductor Sr 2 Ru. O 4 s-wave superconductor/ Topological Insulator among others. . Current Status : NOT OBSERVED

Majorana Fermions and Topological Quantum Computation Kitaev, 2003 • 2 separated Majoranas = 1 fermion : Y = g 1+i g 2 2 degenerate states (full or empty) 1 qubit • 2 N separated Majoranas = N qubits • Quantum information stored non locally Immune to local sources decoherence • Adiabatic “braiding” performs unitary operations Non-Abelian Statistics

Manipulation of Majorana Fermions Control phases of S-TI-S Junctions f 1 f 2 Tri-Junction : A storage register for Majoranas Majorana present + - 0 Create Braid Measure A pair of Majorana bound states can be created from the vacuum in a well defined state |0>. A single Majorana can be moved between junctions. Allows braiding of multiple Majoranas Fuse a pair of Majoranas. States |0, 1> distinguished by • presence of quasiparticle. • supercurrent across line junction E E E 0 0 f-p

Conclusion • • A new electronic phase of matter has been predicted and observed - 2 D : Quantum spin Hall insulator in Hg. Cd. Te QW’s - 3 D : Strong topological insulator in Bi 1 -x. Sbx , 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.