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Topological Insulators = Topological Insulators Akira Furusaki (Condensed Matter Theory Lab.) = topological insulators (3d and 2d) Outline • Introduction: band theory • Example of topological insulators: integer quantum Hall effect • New members: Z2 topological insulators • Table of topological insulators Insulator • A material which resists the flow of electric current. insulating materials Band theory of electrons in solids Ion (+ charge) electron Electrons moving in the lattice of ions • Schroedinger equation Periodic electrostatic potential from 2 ions and other electrons (mean-field) ' h 2 $ %) ( +V (r)"! (r)= E! (r) V (r + a)= V (r) & 2m # Bloch’s theorem: ikr " " # (r)= e un,k (r), ! < k < , un,k (r + a)= un,k (r) a a En (k) Energy band dispersion n : band index Metal and insulator in the band theoryE Electrons are fermions (spin=1/2). Each state (n,k) can accommodate up to two electrons (up, down spins). k ! 0 ! Pauli principle " a a Metal empty states E E Apply electric field k k ! 0 ! ! 0 ! " a a " a a Flow of electric current occupied states k y kx Apply electric field Band Insulator E ~ Band gap V (k = ! / a) k ! 0 ! " a a All the states in the lower band are completely filled. (2 electrons per unit cell) Electric current does not flow under (weak) electric field. Digression: other (named) insulators • Peierls insulator (lattice deformation) E k ! 0 ! " a a • Mott insulator (Coulomb repulsion) Large Coulomb energy! Electrons cannot move. • Anderson insulator (impurity scattering) electron Random scattering causes interference of electron’s wave function. standing wave Anderson localization Is that all? No ! Yet another type of insulators: Topological insulators ! A topological insulator is a band insulator which is characterized by a topological number and which has gapless excitations at its boundaries. Prominent example: quantum Hall effect • Classical Hall effect + r ! n : electron density B + ! + ! Electric current I = !nevW + r r ! v + v E ! Electric field E = B e + ! ! c B Hall voltage VH = EW = I W ! ne B Hall resistance R = Lorentz force H ! ne r r F = "evr ! B 1 Hall conductance ! xy = RH Integer quantum Hall effect (von Klitzing 1980) ! xy = !H h = 25812.807! e2 Quantization of Hall conductance ! xx e2 ! = i xy h exact, robust against disorder etc. Integer quantum Hall effect • Electrons are confined in a two-dimensional plane. (ex. AlGaAs/GaAs interface) • Strong magnetic field is applied (perpendicular to the plane) r AlGaAs GaAs B E cyclotron motion Landau levels: 1 eB En = h!c (n + 2 ), !c = , n = 0,1,2,... mc k "! ! TKNN number (Thouless-Kohmoto-Nightingale- den Nijs) TKNN (1982); Kohmoto (1985) e2 " = ! C xy h r ik !r r r first Chern number (topological invariant) " = e uk (r) * * 1 2 2 ' (u (u (u (u $ C = d k d r% ) " integer valued 2 i ! ! % k k k k " * filled band & ( y ( x ( x ( y # 1 2 r r r r = d k "k ! A(kx ,k y ) A(kx ,k y )= u r !k u r 2$i # k k Topological distinction of ground states projection operator n empty bands m filled bands k y map from BZ to Grassmannian kx # 2 [U (m + n) U (m)"U (n)]= ! IQHE homotopy class Edge states • There is a gapless chiral edge mode along the sample boundary. r B E "! ! kx !" xy Number of edge modes = = C e2 / h Robust against disorder (chiral fermions cannot be backscattered) m(y) Effective field theory y H = #iv(! x" x +! y" y )+ m(y)! z domain wall fermion Topological Insulators (definition ??) • (band) insulator with a nonzero gap to excitated states • topological number stable against any (weak) perturbation • gapless edge mode • When the gapless mode appears/disappears, the bulk (band) gap closes. Quantum Phase Transition • Low-energy effective theory = topological field theory (Chern-Simons) 5 Fractional quantum Hall effect !at= 2 • 2nd Landau level • Even denominator (cf. Laughlin states: odd denominator) • Moore-Read (Pfaffian) state z j = x j + iy j ) 1 & z 2 ' $ 2 "! i * MR = Pf #(zi " z j ) e Pf (Aij )= det Aij ' z z $ i< j ( i " j % Pf( ) is equal to the BCS wave function of px+ipy pairing state. Bound state of two spinless ferions: P-wave & angular momentum=1 Excitations above the Moore-Read state obey non-Abelian statistics. Spinless px+ipy superconductor in 2 dim. r • Order parameter !(k ) " #k #$k " !0 (kx + ik y ) • Chiral (Majorana) edge state E !0 k • Majorana bound state in a quantum vortex hc vortex ! = e Bogoliubov-de Gennes equation h ( u u 1 r 2 & ( # &u# & 0 #& # & # h = pr + eA ! E $ ! $ ! $ !$ ! = )$ ! 0 ( ) F $ + ! ' $ ! $ * * !$ ! $ ! 2m %( " % v " %( ' h0 "% v " % v " &u# & v* # ) : $ ! ( ') : $ ! particle-hole symmetry $ ! $ * ! % v " %u " 2 $ n = n#0 , #0 " !0 / EF + zero mode ! 0 = 0 " = " (= ! ) Majorana (real) fermion! interchanging vortices braid groups, non-Abelian statistics " i ! " i+1 i i+1 ! i+1 # "! i Quantum spin Hall effect (Z2 top. Insulator) Kane & Mele (2005, 2006); Bernevig & Zhang (2006) • Time-reversal invariant band insulator r • Strong spin-orbit interaction "L #!r • Gapless helical edge mode (Kramers pair) r B up-spin electrons down-spin electrons r ! B No spin rotation symmetry • Quantum spin Hall insulator is characterized by Z2 topological index ! ! =1 an odd number of helical edge modes; Z2 topological insulator ! = 0 an even (0) number of helical edge modes 1 0 # of a pair of zeros of & r r * # Pf ui (k )isy u j (k ) %$ "! Kane-Mele model graphene + SOI [PRL 95, 146802 (2005)] s e Quantum spin Hall effect " = xy 2! Experiment HgTe/(Hg,Cd)Te quantum wells CdTe HgCdTe CdTe Konig et al. [Science 318, 766 (2007)] 3-dimensional Z2 topological insulator Moore & Balents; Roy; Fu, Kane & Mele (2006, 2007) (strong) topological insulator surface Dirac fermion bulk: band insulator surface: an odd number of surface Dirac modes bulk insulator characterized by Z2 topological numbers Ex: tight-binding model with SO int. on the diamond lattice [Fu, Kane, & Mele; PRL 98, 106803 (2007)] trivial band insulator: trivial insulator 0 or an even number of surface Dirac modes Z2 topological insulator Surface Dirac fermions topological insulator • A “half” of graphene K K’ E K’ K ky K K’ kx • An odd number of Dirac fermions in 2 dimensions cf. Nielsen-Ninomiya’s no-go theorem photon Experiments p, E • Angle-resolved photoemission spectroscopy (ARPES) Hsieh et al., Nature 452, 970 (2008) Bi1-xSbx An odd (5) number of surface Dirac modes were observed. Experiments II Bi2Se3 “hydrogen atom” of top. ins. Xia et al., arXiv:0812.2078 band calculation Q: Are there other 3D topological insulators? Yes! Let’s make a table of all possible topological insulators. Classification of topological insulators Topological insulators are stable against (weak) perturbations. Random deformation of Hamiltonian Natural framework: random matrix theory (Wigner, Dyson, Altland & Zirnbauer) Assume only basic discrete symmetries: (1) time-reversal symmetry 0 no TRS * !1 TRS = +1 TRS withT ! = +T (integer spin) TH T = H " -1 TRS withT = !T (hal-odd integer spin) (2) particle-hole symmetry 0 no PHS " !1 ! CH C = !H PHS = +1 PHS withC = +C (odd parity: p-wave) -1 PHS withC " = !C (even parity: s-wave) (3) TRS ! PHS = chiral symmetry [sublattice symmetry (SLS)] !1 TCH (TC) = !H 3!3+1 =10 10 random matrix ensembles IQHE Z2 TPI MR Pfaffian Examples of topological insulators in 2 spatial dimensions Integer quantum Hall Effect Z2 topological insulator (quantum spin Hall effect) also in 3D Moore-Read Pfaffian state (spinless p+ip superconductor) Table of topologocal insulators in 1, 2, 3 dim. Schnyder, Ryu, Furusaki & Ludwig, PRB (2008) Examples: (a)Integer Quantum Hall Insulator, (b) Quantum Spin Hall Insulator, (c) 3d Z2 Topological Insulator, (d) Spinless chiral p-wave (p+ip) superconductor (Moore-Read), (e)Chiral d-wave superconductor, (f) superconductor, (g) 3He B phase. Reordered Table Kitaev, arXiv:0901.2686 Periodic table for topological insulators Classification in any dimension Summary • Many topological insulators of non-interacting fermions have been found. interacting fermions?? • Gapless boundary modes (Dirac or Majorana) stable against any (weak) perturbation disorder • Majorana fermions to be found experimentally in solid-state devices.
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