Time Reversal Invariant Ζ Topological Insulator D Bloch Hamiltonians subject to the T constraint 1 ( ) ΘH Θ = H( ) with Θ = 1 are classified by a Ζ topological invariant (ν =,1) Understand via Bul-Boundary correspondence : dge States for <<π/a ν= : Conventional Insulator ν=1 : Topological Insulator Kramers degenerate at time reversal invariant momenta * = * + G *= *=π/a *= *=π/a ven number of bands crossing Fermi energy Odd number of bands crossing Fermi energy
3D Topological Insulators There are 4 surface Dirac Points due to Kramers degeneracy y Λ 4 Λ 3 x Λ 1 Λ D Dirac Point Surface Brillouin Zone ν = : Wea Topological Insulator =Λ a =Λ b OR =Λ a =Λ b How do the Dirac points connect? Determined by 4 bul Z topological invariants ν ; (ν 1 ν ν 3 ) y Related to layered D QSHI ; (ν 1 ν ν 3 ) ~ Miller indices Fermi surface encloses even number of Dirac points x ν = 1 : Strong Topological Insulator Fermi circle encloses odd number of Dirac points Topological Metal : 1/4 graphene Berry s phase π Robust to disorder: impossible to localize y x F
Bi 1-x Sb x Theory: Predict Bi 1-x Sb x is a topological insulator by exploiting inversion symmetry of pure Bi, Sb (Fu,Kane PRL 7) xperiment: ARPS (Hsieh et al. Nature 8) Bi 1-x Sb x is a Strong Topological Insulator ν ;(ν 1,ν,ν 3 ) = 1;(111) 5 surface state bands cross F between Γ and M Bi Se 3 ARPS xperiment : Y. Xia et al., Nature Phys. (9). Band Theory : H. Zhang et. al, Nature Phys. (9). ν;(ν1,ν,ν3) = 1;() : Band inversion at Γ nergy gap: ~.3 ev : A room temperature topological insulator Control F on surface by exposing to NO Simple surface state structure : Similar to graphene, except only a single Dirac point F
Unique Properties of Topological Insulator Surface States Half an ordinary DG ; ¼ Graphene Spin polarized Fermi surface Charge Current ~ Spin Density Spin Current ~ Charge Density F π Berry s phase Robust to disorder Wea Antilocalization Impossible to localize, Klein paradox xotic States when broen symmetry leads to surface energy gap: Quantum Hall state, topological magnetoelectric effect Fu, Kane 7; Qi, Hughes, Zhang 8, ssin, Moore, Vanderbilt 9 Superconducting state Fu, Kane 8
Orbital QH : Surface Quantum Hall ffect = Landau Level for Dirac fermions. Fractional IQH 1-1 - σ xy e = n + h 1 e σ xy = h e σ xy = h B ν=1 chiral edge state Anomalous QH : Induce a surface gap by depositing magnetic material H = ψ ( iv σ µ + Mσz ) ψ e e F Mass due to xchange field e σ xy = sgn( M ) h + h M M TI h gap = M Chiral dge State at Domain Wall : M M
Topological Superconductors, Majorana Fermions and Topological Quantum Computation 1. Bogoliubov de Gennes Theory. Majorana bound states, Kitaev model 3. Topological superconductor 4. Periodic Table of topological insulators and superconductors 5. Topological quantum computation 6. Proximity effect devices
BCS Theory of Superconductivity mean field theory : * Ψ ΨΨ Ψ Ψ Ψ ΨΨ = ΨΨ ( ) H H BdG Ψ Ψ = Ψ Ψ Bogoliubov de Gennes Hamiltonian H BdG H = * H Intrinsic anti-unitary particle hole symmetry 1 ΞHBdGΞ = Ξ =+1 Particle hole redundancy H BdG ϕ =Ξϕ = Ξ ϕ= τϕ* x = τ x same state 1 = 1 ΞH Ξ = 1 ( ) H ( ) Bloch - BdG Hamiltonians satisfy BdG BdG Topological classification problem similar to time reversal symmetry
1D Ζ Topological Superconductor : ν =,1 Bul-Boundary correspondence : Discrete end state spectrum ND (Kitaev, ) ν= trivial ν=1 topological - Γ Γ =Γ = Zero mode Γ =Γ = = Majorana fermion bound state Majorana Fermion : Particle = Antiparticle = Real part of a Dirac fermion : =Ψ+Ψ ; Ψ= + i 1 1 i( ) ; i = Ψ Ψ Ψ = { } 1 = 1 i, = δ i j ij Half a state Two Majorana fermions define a single two level system occupied H iε ε = 1 = ΨΨ ε empty
Kitaev Model for 1D p wave superconductor H µ N= tcc ( + c c) µ cc+ ( cc + c c) i i i+ 1 i+ 1 i i i i i+ 1 i+ 1 i c = ( c c ) HBdG ( ) c H ( ) = τ (tcos µ ) + τ sin = d( ) τ BdG z x µ >t : Strong pairing phase trivial superconductor d x t d() d z + µ <t : Wea pairing phase topological superconductor d x d() d z Similar to SSH model, except different symmetry : ( d, d, d x y z) = ( d x, d y, d z)
Majorana Chain c + i i 1i i µ cc iµ i i 1i i ( i i+ 1+ i+ 1 i) ( 1 ii+ 1 i1i+ 1) ( cc i i+ 1 ci+ 1ci) i ( 1 ii+ 1 i1 i+ 1) t c c c c it + + H = i t + t 1 1i i i 1i + 1 i For =t : nearest neighbor Majorana chain t = µ, t = t 1 1i t 1 t i t 1 >t trivial SC t 1 <t topological SC Unpaired Majorana Fermion at end
D Ζ topological superconductor (broen T symmetry) Bul-Boundary correspondence: n = # Chiral Majorana Fermion edge states = xamples Spinless p x +ip y superconductor (n=1) Chiral triplet p wave superconductor (eg Sr RuO 4 ) (n=) T-SC m µ Read Green model : H= cc + ( ( ) cc + cc..) ( ) = ( x + iy) H ( ) = τ t cos + cos µ + τ sin + τ sin = d( ) τ Lattice BdG model : BdG ( z x y ) ( x x y y ) µ >4t : Strong pairing phase trivial superconductor d y d x d() d z Chern number d x µ <4t : Wea pairing phase topological superconductor d y d() d z Chern number 1
Majorana zero mode at a vortex Φ= h p e Hole in a topological superconductor threaded by flux Boundary condition on fermion wavefunction p 1 ( ) ( 1) + () ψ L = L A π m ψ ( x) e iq x ; q = ( m+ 1+ p) ψ m L p even p odd π v ε ~ L q q zero mode Without the hole : Caroli, de Gennes, Matricon theory ( 64) ε ~ F
Majorana Fermions and Topological Quantum Computing The degenerate states associated with Majorana zero modes define a topologically protected quantum memory (Kitaev 3) Majorana separated bound states = 1 fermion - degenerate states (full/empty) = 1 qubit N separated Majoranas = N qubits Quantum Information is stored non locally - Immune from local decoherence Braiding performs unitary operations Non-Abelian statistics Ψ= + i 1 Measure ( 1 34 + 1 34 ) 11 / Interchange rule (Ivanov 3) i j j i t Braid These operations, however, are not sufficient to mae a universal quantum computer Create 134
Potential condensed matter hosts for Majorana bound states Quasiparticles in fractional Quantum Hall effect at ν=5/ Moore Read 91 Unconventional superconductors - Sr RuO 4 Das Sarma, Naya, Tewari 6 - Fermionic atoms near feshbach resonance Gurarie 5 Proximity ffect Devices using ordinary s wave superconductors - Topological Insulator devices Fu, Kane 8 - Semiconductor/Magnet devices Sau, Lutchyn, Tewari, Das Sarma 9, Lee 9,...... among others Current Status : Not Observed
H Superconducting Proximity ffect = ψ ( iv σ µψ ) + S ψψ + ψψ * S proximity induced superconductivity at surface s wave superconductor Topological insulator Half an ordinary superconductor Similar to spinless p x +ip y superconductor, except : - Does not violate time reversal symmetry - s-wave singlet superconductivity - Required minus sign is provided by π Berry s phase due to Dirac Point Nontrivial ground state supports Majorana fermion bound states at vortices - Dirac point
Majorana Bound States on Topological Insulators 1. h/e vortex in D superconducting state SC TI h/e Quasiparticle Bound state at = Majorana Fermion Half a State = =. Superconductor-magnet interface at edge of D QSHI M S.C. QSHI m< m = S M m> gap = m Domain wall bound state
1D Majorana Fermions on Topological Insulators 1. 1D Chiral Majorana mode at superconductor-magnet interface M SC x TI = : Half a 1D chiral Dirac fermion. S-TI-S Josephson Junction H = i v F x SC φ SC φ = π φ π TI Gapless non-chiral Majorana fermion for phase difference φ = π H = i + i v ( ) F L xl R xr cos( φ/ ) L R
Manipulation of Majorana Fermions Control phases of S-TI-S Junctions Tri-Junction : A storage register for Majoranas φ φ 1 + Majorana present Create A pair of Majorana bound states can be created from the vacuum in a well defined state >. Braid A single Majorana can be moved between junctions. Allows braiding of multiple Majoranas Measure Fuse a pair of Majoranas. States,1> distinguished by presence of quasiparticle. supercurrent across line junction 1 φ π φ π φ π
Another route to the D p+ip superconductor Semiconductor - Magnet - Superconductor structure Superconductor Semiconductor Magnetic Insulator Single Fermi circle with Berry phase π ε Topological superconductor with Majorana edge states and Majorana bound states at vortices. Sau, Lutchyn, Tewari, Das Sarma 9 Rashba split DG bands F Zeeman splitting Variants : - use applied magnetic field to lift Kramers degeneracy (Alicea 1) - Use 1D quantum wire (eg InAs ). A route to 1D p wave superconductor with Majorana end states. (Oreg, von Oppen, Alicea, Fisher 1 Challenge : requres very low electron density high purity.
Periodic Table of Topological Insulators and Superconductors Anti-Unitary Symmetries : - Time Reversal : - Particle - Hole : Unitary (chiral) symmetry : ΘH Θ =+ Θ =± 1 ( ) H( ) ; 1 ΞH Ξ = Ξ =± 1 ( ) H( ) ; 1 1 ΠH( ) Π = H( ) ; Π ΘΞ 8 antiunitary symmetry classes Altland- Zirnbauer Random Matrix Classes Complex K-theory Real K-theory Kitaev, 8 Schnyder, Ryu, Furusai, Ludwig 8 Bott Periodicity d d+8