Majorana Fermions and Topological Insulators Charles Kane, - PowerPoint PPT Presentation

Majorana Fermions and Topological Insulators Charles Kane, University of Pennsylvania I. Introduction: Topological insulators in 2D, 3D Exotic surface phases : - “Fractional” Integer Quantized Hall Effect - Superconducting Proximity Effect II. Majorana Fermions on Topological Insulators - A route to Topological Quantum Computing? - A Z 2 Interferometer for Majorana Fermions III. Non-Abelian Statistics in Three Dimensions Thanks to Gene Mele, Liang Fu, Jeffrey Teo,

Topological Insulators Two dimensions: Quantum Spin Hall Insulator Graphene Kane, Mele ’05 HgCdTe quantum well Bernevig, Hughes, Zhang ’06 G=2e 2 /h Edge state transport experiments Konig, et al. ‘07 Three dimensions: Strong Topological Insulator Theory: Fu, Kane, Mele ’06, Moore, Balents ’06, Roy ‘06 Surface States probed by ARPES: Bi 1-x Sb x Fu, Kane ’07 (Th) Hsieh, et al ’07 (Exp) Bi 2 Se 3 , Bi 2 Te 3 Xia, et al ’09 (Exp+Th) Zhang, et al ’09 (Th) Hsieh, et al ’09 (Exp) Chen et al. ’09 (Exp) Bi 1-x Sb x

Time Reversal Invariant � 2 Topological Insulator ( ) − Θ Θ = − Θ ψ = σ ψ 1 y i * H k H ( k ) Time Reversal Symmetry : � Θ = − 2 1 All states doubly degenerate Kramers’ Theorem : � 2 : two ways to connect Kramers E E OR pairs on surface k= Λ a k= Λ b k= Λ a k= Λ b Bulk - Boundary Correspondence Equivalence classes of surface/edge: even or odd number ∈ d ( ) of enclosed Dirac points H k k T � � E F k � d=2 k F 2 k y E F ⊕ � � 3 d=3 2 2 k x (weak Topo. Ins.)

Unique Properties of Surface States “Half” an ordinary 2DEG ; ¼ Graphene E F Spin polarized Fermi surface • Charge Current ~ Spin Density • Spin Current ~ Charge Density π Berry’s phase • Robust to disorder Isolated surface Dirac point on • Weak Antilocalization Bi 2 Se 3 D Hsieh, et al. Nature ‘09 • Impossible to localize, Klein paradox Exotic States when broken symmetry leads to surface energy gap: • Quantum Hall state, magnetoelectric effect Fu, Kane ’07; Qi, Hughes, Zhang ’08, Essin, Moore, Vanderbilt ‘09 • Superconducting state Fu, Kane ‘08 • Excitonic Insulator Seradjeh, Moore, Franz ‘09

Surface Quantum Hall Effect Orbital QHE : E=0 Landau Level for Dirac fermions. “Fractional” IQHE 2 e σ = 2 xy B 2 h 1 � � 2 e 1 σ = + � � 0 n xy � � h 2 -1 2 e -2 σ = ν =1 chiral edge state xy 2 h Anomalous QHE : Induce a surface gap by depositing magnetic material � �� = ψ − σ ∇ − µ + ∆ σ ψ † H ( i v ) 2 2 e e 0 M z + − 2 h 2 h Mass due to Zeeman field M � M � 2 e σ = ∆ sgn( ) 2 xy M h TI E F E gap = 2| ∆ M | Chiral Edge State at Domain Wall : ∆ M � −∆ M

Superconducting Proximity Effect � �� = ψ − σ ∇ − µ ψ † ( H i v ) s wave superconductor + ∆ ψ ψ + ∆ ψ ψ † † * ↑ ↓ ↓ ↑ S S Topological insulator proximity induced superconductivity at surface • Half an ordinary superconductor • Similar to 2D spinless p x +ip y topological superconductor, except : - Does not violate time reversal symmetry � -k Dirac point - s-wave singlet superconductivity - Required minus sign is provided by � � π Berry’s phase due to Dirac Point • Nontrivial ground state supports Majorana � k fermion bound states at vortices

Majorana Bound States on Topological Insulators 1. h/2e vortex in 2D superconducting state E γ † ∆ E h/2e γ = γ † 0 0 0 SC −∆ γ − = γ † TI E E Quasiparticle Bound state at E=0 Majorana Fermion γ 0 “Half a State” 2. Superconductor-magnet interface at edge of 2D QSHI m = ∆ − ∆ | | | | S M M S.C. m>0 E gap =2|m| QSHI m<0 Domain wall bound state γ 0

Majorana Fermions Topological Quantum Computing Kitaev, 2003 Measure ( ) + 0 0 1 1 / 2 12 34 12 34 • 2 Majorana bound states = 1 fermion Ψ = γ + γ i 1 2 - 2 degenerate states (full/empty) = 1 qubit • 2N separated Majoranas = N qubits • Quantum Information is stored non locally t Braid - Immune from local decoherence • Adiabatic Braiding performs unitary operations - Non Abelian Statistics 0 0 Create 12 34 Potential Condensed Matter Hosts : • Quasiparticles in fractional Quantum Hall effect at ν =5/2 Moore, Read ’91 • s-wave superconductor / Topological Insulator structure Fu, Kane ‘08 • semiconductor - magnet - superconductor structures Sau, Lutchyn, Tewari, Das Sarma ‘09 • .... among others

1D Majorana Fermions on Topological Insulators 1. 1D Chiral Majorana mode at superconductor-magnet interface E M SC k x TI = − γ ∂ γ � γ = γ − † v H i : “Half” a 1D chiral Dirac fermion k k F x 2. S-TI-S Josephson Junction φ = π φ 0 φ ≠ π SC SC TI Gapless non-chiral Majorana fermion for phase difference φ = π ( ) = − γ ∂ γ − γ ∂ γ + ∆ φ γ γ � v cos( / 2) H i i F L x L R x R L R

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

A Z 2 Interferometer for Majorana Fermions A signature for neutral Majorana fermions probed with charge transport N even = γ − γ † c i γ 2 1 2 = γ + γ e e c i 1 2 γ 1 h Φ = N 2 e N odd • Chiral electrons on magnetic domain wall split into a pair of chiral Majorana fermions γ 2 −γ 2 • “Z 2 Aharonov Bohm phase” converts an e h electron into a hole γ 1 G N Fu and Kane, PRL ‘09 Akhmerov, Nilsson, Beenakker, PRL ‘09

Majorana Fermions in Three Dimensions Majorana bound states arise as solutions to three dimensional BdG theories � �� ( ) ( ) � � = τ − µ σ ∇ + µ + ∆ τ + ∆ τ H i m ( ) r Re ( ) r Im ( ) r � � z x z x y Qi, Hughes, Zhang Model for m > 0 Superconducting pairing edge of 3D topological insulator Trivial Insulator at surface m < 0 Topological Insulator Majorana bound state Minimal O(3) n-vector model : “hedgehog” configuration � � � � � = − γ ⋅∇ + Γ⋅ n r ( ) H i n ( ) r (γ 1 , γ 2 , γ 3 ), (Γ 1 , Γ 2 , Γ 3 ) : 8x8 Dirac matrices � ( ) ( ) = = ∆ ∆ n ( ) r n n n , , Re( ),Im( ), m 1 2 3

Topological Classification of Defects • Adiabatic approximation: r away from defect H varies slowly ∈ 3 k T crystal momentum = ( , ) k r H H ∈ 2 r S enclosing surface 2 S in real space − • Particle-Hole symmetry (Class D) : − = −Ξ Ξ 1 H ( k r , ) H ( , ) k r • � 2 Topological Invariant : signals an enclosed zero mode = � Q 5 = Chern Simons 5 form µ 3 2 S d k d r Q ( , ) mod 2 A F 5 Analogous to Qi,Hughes Zhang formula 3 2 × T • O(3) n-vector model : µ = Hedgehog number mod 2

Non-Abelian Exchange Statistics in 3D Exchange a pair of hedgehogs: � � = n ( ) r n 1 � � = n ( ) r n 2 2 π rotation : Wavefunction of Majorana bound ( ) [ ] π = � O (3) state changes sign 1 2 Interchange rule: Ising Anyons π γ γ γ → γ i 1 2 = 1 2 Nayak, Wilczek ’96 4 T e γ → − γ Ivanov ‘01 12 2 1

Braidless Operation In 3D, braids can be contracted to zero. There therefore must exist non trivial operations on stationary Majorana states. Gedanken expt: Spherical TI’s coated with superconductor, 4 vortices 0 12 1 2 + - - + 3 4 0 34

Braidless Operation In 3D, braids can be contracted to zero. There therefore must exist non trivial operations on stationary Majorana states. Gedanken expt: Spherical TI’s coated with superconductor, 4 vortices 2 1 ϕ 3 4 Braid 3 around 1 = γ γ 2 1 3 : T → 0 0 1 1 or 13 12 34 12 34 ϕ → ϕ + π 2 Advance phase

Braidless Operation In 3D, braids can be contracted to zero. There therefore must exist non trivial operations on stationary Majorana states. Gedanken expt: Spherical TI’s coated with superconductor, 4 vortices 2 1 3 4 ϕ + π 2 Braid 3 around 1 = γ γ 2 1 3 : T → 0 0 1 1 or 13 12 34 12 34 ϕ → ϕ + π 2 Advance phase

Fractional Josephson Effect Fu, Kane ’08 Kitaev ’01 Kwon, Sengupta, Yakovenko ‘04 γ 1 γ 2 1 1 o 12 34 e γ 3 γ 4 0 0 12 34 • 4 π perioidicity of E( φ ) protected by local conservation of fermion parity. • AC Josephson effect with half the usual frequency: f = eV/h