Majorana Fermions and Topological Insulators Charles L. Kane, University of Pennsylvania Topological Band Theory - Integer Quantum Hall Effect - 2D Quantum Spin Hall Insulator - 3D Topological Insulator - Topological Superconductor Majorana Fermions - Superconducting Proximity Effect on Topological Insulators - A route to topological quantum computing? Thanks to Gene Mele, Liang Fu, Jeffrey Teo

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 Hall Conductance sxy = n e2/h IQHE without Landau Levels (Haldane PRL 1988) Graphene in a periodic magnetic field B(r) B(r) = 0 Zero gap, Dirac point B(r) ≠ 0 Energy gap sxy = e2/h Band structure + - k Egap

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 The set of occupied Bloch wavefunctions defines a U(N) vector bundle over the torus. Classified by the first Chern class (or TKNN invariant) (Thouless et al, 1984) Berry’s connection Berry’s curvature 1st Chern class Trivial Insulator: n = 0 Quantum Hall state: sxy = n e2/h The TKNN invariant can only change at a phase transition where the energy gap goes to zero

Edge States Bulk – Edge Correspondence : Dn = # Chiral Edge Modes Gapless states must exist at the interface between different topological phases IQHE state n=1 Vacuum n=0 n=1 n=0 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 K’ K ky Jackiw, Rebbi (1976) Su, Schrieffer, Heeger (1980) Haldane Model Bulk – Edge Correspondence : Dn = # Chiral Edge Modes

time reversal invariant momenta Time Reversal Invariant 2 Topological Insulator Time Reversal Symmetry : Kramers’ Theorem : All states doubly degenerate 2 topological invariant (n = 0,1) for 2D T-invariant band structures n=0 : Conventional Insulator n=1 : Topological Insulator E k*=0 k*=p/a Edge States E k*=0 k*=p/a Kramers degenerate at time reversal invariant momenta k* = -k* + G n is a property of bulk bandstructure. Easiest to compute if there is extra symmetry: 1. Sz conserved : independent spin Chern integers : (due to time reversal) J↑ J↓ E Quantum spin Hall Effect : 2. Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch states

2D Quantum Spin Hall Insulator ↑ I. Graphene Kane, Mele PRL ‘05 Eg ↓ Intrinsic spin orbit interaction  small (~10mK-1K) band gap Sz conserved : “| Haldane model |2” Edge states : G = 2 e2/h p/a 2p/a ↑ ↓ ↑ ↓ II. HgCdTe quantum wells HgTe Theory: Bernevig, Hughes and Zhang, Science ’06 Experiement: Konig et al. Science ‘07 HgxCd1-xTe d HgxCd1-xTe d < 6.3 nm Normal band order d > 6.3 nm: Inverted band order G ~ 2e2/h in QSHI E E Normal G6 ~ s G8 ~ p k Inverted G8 ~ p G6 ~ s Conventional Insulator QSH Insulator

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 2 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

Topological Superconductor, Majorana Fermions BCS mean field theory : Bogoliubov de Gennes Hamiltonian Particle-Hole symmetry : Quasiparticle redundancy : 1D 2 Topological Superconductor : n = 0,1 (Kitaev, 2000) Discrete end state spectrum : END n=0 “trivial” n=1 “topological” E Majorana Fermion bound state D D E E=0 -E -D -D “half a state”

Periodic Table of Topological Insulators and Superconductors Kitaev, 2008 Schnyder, Ryu, Furusaki, Ludwig 2008 Anti-Unitary Symmetries : - Time Reversal : - Particle - Hole : Unitary (chiral) symmetry : Complex K-theory Altland- Zirnbauer Random Matrix Classes Real K-theory Bott Periodicity

Majorana Fermion : spin 1/2 particle = antiparticle ( g = g† ) Potential Hosts : Particle Physics : Neutrino (maybe) Allows neutrinoless double b-decay. 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 Topological Quantum Computing Kitaev, 2003 2 Majorana bound states = 1 fermion bound state - 2 degenerate states (full/empty) = 1 qubit 2N separated Majoranas = N qubits Quantum Information is stored non locally - Immune from local sources of decoherence Adiabatic Braiding performs unitary operations - Non Abelian Statistics

Proximity effects : Engineering exotic gapped states on topological insulator surfaces m Dirac Surface States : Protected by Symmetry 1. Magnetic : (Broken Time Reversal Symmetry) Fu,Kane PRL 07 Qi, Hughes, Zhang PRB (08) Orbital Magnetic field : Zeeman magnetic field : Half Integer quantized Hall effect : M. ↑ T.I. 2. Superconducting : (Broken U(1) Gauge Symmetry) proximity induced superconductivity Fu,Kane PRL 08 S-wave superconductor Resembles spinless p+ip superconductor Supports Majorana fermion excitations S.C. T.I.

Majorana Bound States on Topological Insulators 1. h/2e vortex in 2D superconducting state E D h/2e SC -D TI Quasiparticle Bound state at E=0 Majorana Fermion g0 2. Superconductor-magnet interface at edge of 2D QSHI M S.C. m>0 Egap =2|m| QSHI m<0 Domain wall bound state g0

1D Majorana Fermions on Topological Insulators 1. 1D Chiral Majorana mode at superconductor-magnet interface E M SC kx TI : “Half” a 1D chiral Dirac fermion 2. S-TI-S Josephson Junction f = p f f  p SC SC TI Gapless non-chiral Majorana fermion for phase difference f = p

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

e e e h A Z2 Interferometer for Majorana Fermions g2 g1 g2 -g2 g1 A Signature for Neutral Majorana Fermions Probed with Charge Transport N even g2 e e g1 N odd Chiral electrons on magnetic domain wall split into a pair of chiral Majorana fermions “Z2 Aharonov Bohm phase” converts an electron into a hole dID/dVs changes sign when N is odd. g2 -g2 e h g1 Fu and Kane, PRL ‘09 Akhmerov, Nilsson, Beenakker, PRL ‘09

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, Bi2Te3 Superconductor/Topological Insulator structures host Majorana Fermions - A Platform for Topological Quantum Computation Experimental Challenges - Charge and Spin 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 - Further manifestations of Majorana fermions and non-Abelian states - Effects of disorder and interactions on surface states