Topological Insulators and Superconductors

Transcription

1 Topological Insulators and Superconductors Lecture #1: Topology and Band Theory Lecture #: Topological Insulators in and 3 dimensions Lecture #3: Topological Superconductors, Majorana Fermions an Topological quantum compuation General References : M.Z. Hasan and C.L. Kane, RMP in press, arxiv: X.L. Qi and S.C. Zhang, Physics Today (1). J.E. Moore, Nature 464, 194 (1). My collaborators : Gene Mele, Liang Fu, Jeffrey Teo, Zahid Hasan

2 Topology and Band Theory I. Introduction - Insulating State, Topology and Band Theory II. Band Topology in One Dimension - Berry phase and electric polarization - Su Schrieffer Heeger model : domain wall states and Jackiw Rebbi problem - Thouless Charge Pump III. Band Topology in Two Dimensions - Integer quantum Hall effect - TKNN invariant - Edge States, chiral Dirac fermions IV. Generalizations - Bulk-Boundary correspondence - Higher dimensions - Topological Defects

3 The Insulating State Characterized by energy gap: absence of low energy electronic excitations Covalent Insulator e.g. intrinsic semiconductor Atomic Insulator e.g. solid Ar The vacuum electron E gap ~ 1 ev 4s E gap ~ 1 ev 3p Dirac Vacuum E gap = m e c ~ 1 6 ev Silicon positron ~ hole

4 The Integer Quantum Hall State D Cyclotron Motion, Landau Levels E E gap = ω c Energy gap, but NOT an insulator Quantized Hall conductivity : B J y E x J = σ E y xy x σ xy = n e h Integer accurate to 1-9

5 Topology The study of geometrical properties that are insensitive to smooth deformations Example: D surfaces in 3D A closed surface is characterized by its genus, g = # holes g= g=1 g is an integer topological invariant that can be expressed in terms of the gaussian curvature κ that characterizes the local radii of curvature κ = 1 rr 1 Gauss Bonnet Theorem : 1 κ = > κ = r S κ < κda = 4 π(1 g) A good math book : Nakahara, Geometry, Topology and Physics

6 Bloch Theorem : Lattice translation symmetry Band Theory of Solids T( R) ψ i = e kr ikr Bloch Hamiltonian H( k)η = e e ikr i ψ ψ = e kr u( k) H( k) u ( k) = E ( k) u ( k) n n n k = Brillouin Zone Torus, T Band Structure : A mapping d π/a π/a k H ( k) (or equivalently to E ( k) and u ( k) ) n n π/a k y E π/a π/a BZ k x k x = π/a E gap Topological Equivalence : adiabatic continuity Band structures are equivalent if they can be continuously deformed into one another without closing the energy gap

7 Berry Phase Phase ambiguity of quantum mechanical wave function u e u iφ ( ) ( k) k ( k) Berry connection : like a vector potential A= iu( k) k u( k) A A+ k φ( k) Berry phase : change in phase on a closed loop C Berry curvature : F= k Famous example : eigenstates of level Hamiltonian A dz dx id y H ( k) = dk ( ) σ = dx + id y dz H( ) u( ) = + ( ) u( ) γ C = A d k C γ C = Fdk S k k dk k 1 γ ( Solid Angle swept out by ˆ( ) ) C = dk C S ˆd

8 Topology in one dimension : Berry phase and electric polarization Electric Polarization P = dipole moment length P = ρ b -Q 1D insulator see, e.g. Resta, RMP 66, 899 (1994) +Q The end charge is not completely determined by the bulk polarization P because integer charges can be added or removed from the ends : Q= P mod e e Polarization as a Berry phase : P = A( k) dk π P is not gauge invariant under large gauge transformations. This reflects the end charge ambiguity π/a -π/a k P P en i ( k) + when uk ( ) e φ uk ( ) with ( / ) ( / ) = φπ a φ π a πn Changes in P, due to adiabatic variation are well defined and gauge invariant λ uk ( ) uk (, λ( t)) 1 e e P = P P = dk dkd π = π λ= 1 λ= A F λ C S gauge invariant Berry curvature S -π/a C π/a k

9 Su Schrieffer Heeger Model model for polyacetalene simplest two band model δ t > δ t < H= t+ tc c + t tc c + hc i ( δ ) Ai Bi ( δ ) Ai+ 1 Bi.. Hk ( ) = d( k) σ dx ( k) = ( t + δt) + ( t δt)cos ka d ( k) = ( t δt)sin ka d y z ( k) = A,i B,i A,i+1 a d y d(k) d y d(k) d x d x Gap 4 δt E(k) π/a π/a Peierl s instability δt δt> : Berry phase P = δt< : Berry phase π P = e/ k Provided symmetry requires d z (k)=, the states with δt> and δt< are topologically distinct. Without the extra symmetry, all 1D band structures are topologically equivalent.

10 Domain Wall States An interface between different topological states has topologically protected midgap states δ t > δ t < Low energy continuum theory : For small δt focus on low energy states with k~π/a H= iv F σ x x + mx ( ) σ y π k + q ; q i a v F = ta ; m = δ t x Massive 1+1 D Dirac Hamiltonian Eq ( ) =± ( q) + m v F Chiral Symmetry : { σ, H} = σ ψ = ψ z z E E Any eigenstate at +E has a partner at -E Zero mode : topologically protected eigenstate at E= m> Domain wall bound state ψ (Jackiw and Rebbi 76, Su Schrieffer, Heeger 79) E gap = m m< ψ ( x) x m( x') dx'/ v 1 F = e

11 Thouless Charge Pump The integer charge pumped across a 1D insulator in one period of an adiabatic cycle is a topological invariant that characterizes the cycle. Hkt (, + T) = Hkt (, ) t= t=t P= P=e e P = ( A( k, T ) dk A( k,) dk ) = ne π 1 n = dkdt π F T The integral of the Berry curvature defines the first Chern number, n, an integer topological invariant characterizing the occupied Bloch states, ukt (,) -π/a t=t t= k π/a = In the band model, the Chern number is related to the solid angle swept out by which must wrap around the sphere an integer n times. 1 d ˆ( kt,) n = dkdt ˆ ( ˆ ˆ) k t 4π d d d T d ˆ( kt, ),

12 Integer Quantum Hall Effect : Laughlin Argument Adiabatically thread a quantum of magnetic flux through cylinder. -Q 1 dφ E = π R dt I = πr σ E xy +Q Φ ( t = ) = Φ ( t = T) = h/ e T dφ Q = σxy dt = dt Just like a Thouless pump : σ xy Q ne σ xy n h = = e h e HT ( ) = UH() U

13 TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 8 View cylinder as 1D system with subbands labeled by e m Q = dφ dkx ( kx, k y ( Φ )) = ne π F m φ m ky 1 Φ ( Φ ) = m+ R φ E ( k ) = E k, k ( Φ) m ( ) k x m x x y TKNN number = Chern number 1 1 ( ) π π C n dk d = = BZ Fk A k e σ xy = n h C k y k x k ym (φ ) k ym () Distinguishes topologically distinct D band structures. Analogous to Gauss-Bonnet thm. Alternative calculation: compute σ xy via Kubo formula

14 A B Graphene E Novoselov et al. 5 Two band model H ( k) = dk ( ) σ H = t caic < ij> Bj dˆ( k, k ) x y k E ( k) = ± dk ( ) Inversion and Time reversal symmetry require d ( k) = D Dirac points at k = ± K: point zeros in ( d, d ) H ( ± K+ q) = vσ q Berry s phase π around Dirac point x Massless Dirac Hamiltonian z y -K +K

15 Topological gapped phases in Graphene Break P or T symmetry : H( ± K+ q) = vq σ + m ± σz E( q) =± v q + m ± n = # times dk ˆ( ) wraps around sphere 1. Broken P : eg Boron Nitride +K & -K dk ˆ( ) S m = m + Chern number n= : Trivial Insulator. Broken T : Haldane Model 88 m = m Chern number n=1 : Quantum Hall state +K -K dk ˆ( ) S

16 Edge States Gapless states at the interface between topologically distinct phases IQHE state n=1 Vacuum n= Edge states ~ skipping orbits Lead to quantized transport Band inversion transition : Dirac Equation y Domain wall bound state ψ m< n=1 n= m = m + m = +m + x m> E gap H= iv( σ + σ ) + mx () σ F x x y y z E k y ik y ( )~ y ψ x e e x m( x') dx'/ v F E ( k ) = v k y F y Chiral Dirac Fermions Chiral Dirac fermions are unique 1D states : One way ballistic transport, responsible for quantized conductance. Insensitive to disorder, impossible to localize Fermion Doubling Theorem : Chiral Dirac Fermions can not exist in a purely 1D system. ψ in disorder t =1

17 Bulk - Boundary Correspondence N = N R - N L is a topological invariant characterizing the boundary. E N R (N L ) = # Right (Left) moving chiral fermion branches intersecting E F E F Haldane Model N = 1 = 1 E K K k y E F N = 1 = 1 K K k y Bulk Boundary Correspondence : The boundary topological invariant N characterizing the gapless modes = Difference in the topological invariants n characterizing the bulk on either side

18 Generalizations d=4 : 4 dimensional generalization of IQHE Zhang, Hu 1 A = u ( k) u ( k) dk ij i k j F= da+ A A 1 n = 8π T Tr[ F F] 4 Non-Abelian Berry connection 1-form Non-Abelian Berry curvature -form nd Chern number = integral of 4-form over 4D BZ Boundary states : 3+1D Chiral Dirac fermions Higher Dimensions : Bott periodicity d d+ no symmetry chiral symmetry

19 Topological Defects Consider insulating Bloch Hamiltonians that vary slowly in real space H = H( k, s) 1 parameter family of 3D Bloch Hamiltonians = 1 Tr[ F ] 8 F nd Chern number : 3 1 n π T S Generalized bulk-boundary correspondence : s defect line n specifies the number of chiral Dirac fermion modes bound to defect line Teo, Kane 1 Example : dislocation in 3D layered IQHE 1 n = Gc B π 3D Chern number (vector layers) G c Are there other ways to engineer 1D chiral dirac fermions? Burgers vector