Topological Insulators and Topological Band Theory E E k=La k=Lb 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 e.g. intrinsic semiconductor e.g. solid Ar The vacuum electron 4s Egap ~ 10 eV 3p Dirac Vacuum Egap = 2 mec2 ~ 106 eV Egap ~ 1 eV Silicon positron ~ hole The Integer Quantum Hall State 2D Cyclotron Motion, Landau Levels Egap c E Energy gap, but NOT an insulator Quantized Hall conductivity : 2 Jy B J y xy Ex Ex e xy n h Integer accurate to 10-9 Graphene E - - - - - - - - www.univie.ac.at Novoselov et al. ‘05 Low energy electronic structure: Two Massless Dirac Fermions k E v | k | Haldane Model (PRL 1988) E v 2 | k |2 m 2 Add a periodic magnetic field B(r) • • • Band theory still applies Introduces energy gap Leads to Integer quantum Hall state xy e2 h 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 | (k ) : Brillouin zone (a torus) Hilbert space Classified by the Chern (or TKNN) topological invariant (Thouless et al, 1982) n Insulator IQHE state 1 2 d k k u (k ) k u (k ) Integer 2 i BZ : n=0 : xy = n e2/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 Egap M<0 K’ Haldane Model K ky Domain wall bound state y0 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 (~10mK-1K) energy gap Simplest model: |Haldane|2 (conserves Sz) H H 0 0 H Haldane H 0 J↓ J↑ * H Haldane 0 E Bulk energy gap, but gapless edge states Edge band structure Spin Filtered edge states vacuum ↓ ↑ ↑ ↓ 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 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</a n=1 : Topological Insulator n=0 : Conventional Insulator E E Kramers degenerate at time reversal invariant momenta k* = -k* + G k*=0 k*=/a k*=0 k*=/a Quantum Spin Hall Insulator in HgTe quantum wells Theory: Bernevig, Hughes and Zhang, Science 2006 d HgxCd1-xTe HgxCd1-xTe HgTe 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 d> 6.3nm inverted band order QSH insulator ↓ I G=2e2/h Measured conductance 2e2/h independent of W for short samples (L<Lin) ↓ 0 ↑ 3D Topological Insulators There are 4 surface Dirac Points due to Kramers degeneracy ky L4 L1 L3 E E kx OR L2 2D Dirac Point Surface Brillouin Zone k=La k=Lb k=La k=Lb How do the Dirac points connect? Determined by 4 bulk Z2 topological invariants n0 ; (n1n2n3) n0 = 1 : Strong Topological Insulator 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 EF 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 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 : Control EF on surface by exposing to NO2 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↓ ck† c-† k Dei (s-wave, singlet pairing) k↑ Superconducting surface states † † k -k cc D surface e i (s-wave, singlet pairing) Half an ordinary superconductor Highly nontrivial ground state -k ← Dirac point ↑ ↓ → k h / 2e Majorana Fermion at a vortex Ordinary Superconductor : Andreev bound states in vortex core: E D 0 -D E ↑,↓ 2 0 Bogoliubov Quasi Particle-Hole redundancy : † E , - E , -E ↑,↓ Surface Superconductor : Topological zero mode in core of h/2e vortex: E D 0 -D E=0 Majorana fermion : 0 0 • Particle = Anti-Particle • “Half a state” • Two separated vortices define one zero energy † fermion state (occupied or empty) Majorana Fermion • Particle = Antiparticle : † • Real part of Dirac fermion : = †; = 1i 2 “half” an ordinary fermion • Mod 2 number conservation Z2 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 Sr2RuO4 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 : = 1i 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 Uab y b Manipulation of Majorana Fermions Control phases of S-TI-S Junctions f1 f2 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 0 E 1 0 0 0 f- 0 0 0 0 f- 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 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.