Topological Insulators and Topological Band Theory k=La k=Lb 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 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 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 4s Dirac Vacuum Egap ~ 10 eV Egap = 2 mec2 ~ 106 eV 3p Egap ~ 1 eV positron ~ hole Silicon

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

Graphene Haldane Model (PRL 1988) + - 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.

is a topological property of the manifold of occupied states 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

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

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 Edge band structure ↑ ↓ p/a k 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

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=0 : Conventional Insulator n=1 : Topological Insulator E E Kramers degenerate at time reversal invariant momenta k* = -k* + G k*=0 k*=p/a k*=0 k*=p/a

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

3D Topological Insulators There are 4 surface Dirac Points due to Kramers degeneracy ky L4 L1 L2 L3 E k=La k=Lb E k=La k=Lb kx OR 2D Dirac Point How do the Dirac points connect? Determined by 4 bulk Z2 topological invariants n0 ; (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

Bi1-xSbx Bi2 Se3 Insulator n0;(n1,n2,n3) = 1;(111) between G and M 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

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

Majorana Fermion at a vortex Ordinary Superconductor : Andreev bound states in vortex core: E D Bogoliubov Quasi Particle-Hole redundancy : E ↑,↓ -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 E=0 -D

Majorana Fermion Potential Hosts : Current Status : NOT OBSERVED 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

Majorana Fermions and Topological Quantum Computation Kitaev, 2003 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

Manipulation of Majorana Fermions Control phases of S-TI-S Junctions f1 f2 + - 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>. 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 E E E f-p f-p f-p

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.