Topological Insulators Aira Furusai (Condensed Matter Theory Lab.) = topological insulators (3d and 2d)
Outline Introduction: band theory Example of topological insulators: integer quantum Hall effect New members: Z 2 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 ' %) & 2 h ( 2m 2 + V $ " = # ( r) ( r) E ( r) Periodic electrostatic potential from ions and other electrons (mean-fiel ( r + a) V( r) V = Bloch s theorem: # " a ir ( r) = e u ( r), < <, u ( r + a) = u ( r) n, " a n, E n ( ) Energy band dispersion n : n, band index
Metal and insulator in the band theory E Electrons are fermions (spin=1/2). " 0 a a Each state (n,) can accommodate up to two electrons (up, down spins). Pauli principle
Metal E empty states E Apply electric field " 0 a a " 0 a a occupied states y Flow of electric current x Apply electric field
Band Insulator E " 0 a a ~ Band gap V ( = / a) All the states in the lower band are completely filled. (2 electrons per unit cell) Electric current does not flow under (wea) electric field.
Digression: other (named) insulators Peierls insulator (lattice deformation) E Mott insulator (Coulomb repulsion) " 0 a a 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 + + + + + + v r e W E r B r Lorentz force r F r r = " ev B n : electron density Electric current Electric field Hall voltage Hall resistance Hall conductance I = nevw E = v c B B V H = EW = ne B R H = ne 1 xy = R H I
Integer quantum Hall effect (von Klitzing 1980) xy = H h 2 e = 25812. 807 xx Quantization of Hall conductance xy 2 e = i 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) B r AlGaAs GaAs cyclotron motion E Landau levels: E eb = n mc ( n + 1 ), =, = 0,1,2,... n h c 2 c "
TKNN number (Thouless-Kohmoto-Nightingaleden Nijs) TKNN (1982); Kohmoto (1985) C h e xy 2 = " first Chern number (topological invariant) " " # $ % % & ' ( ( ( ( ) ( ( ( ( = y x x y u u u u r d d i C * * 2 2 2 1 * ( ) y x A d i, 2 1 2 r r " = # $ filled band ( ) y x u u A r r r r =, integer valued (r ) u e r i r r r r = "
Topological distinction of ground states projection operator n empty bands m filled bands y map from BZ to Grassmannian x 2 [ U( m + n) U( m) " U( n) ] = # IQHE homotopy class
Edge states There is a gapless chiral edge mode along the sample boundary. B r E Number of edge modes = " e 2 / xy = h C " x Robust against disorder (chiral fermions cannot be bacscattered) Effective field theory ( ) x " x + y " y + m( y) z H = # iv domain wall fermion m ( y) y
Topological Insulators (definition??) (band) insulator with a nonzero gap to excitated states topological number stable against any (wea) perturbation gapless edge mode When the gapless mode appears/disappears, the bul (band) gap closes. Quantum Phase Transition Low-energy effective theory = topological field theory (Chern-Simons)
Fractional quantum Hall effect at 2 nd Landau level Even denominator (cf. Laughlin states: odd denominator) Moore-Read (Pfaffian) state z = x + iy * j MR j ) = Pf ' ( z i j 1 " z j & $ # i< j % 2 " ( z " z ) e i j z i 2 = 5 2 ( Aij) det Aij Pf = Pf( ) is equal to the BCS wave function of p x +ip y pairing state. Bound state of two spinless ferions: P-wave & angular momentum=1 Excitations above the Moore-Read state obey non-abelian statistics.
Spinless p x +ip y superconductor in 2 dim. Order parameter r ( ) " # # " 0( x + i y ) $ Chiral (Majorana) edge state E 0
Majorana bound state in a quantum vortex vortex = hc e Bogoliubov-de Gennes equation & h $ %( $ * ( ' h * 0 #& u $ "% v # " & u# = ) $ % v " 0 0 ( ) F 2 n = n # 0, # 0 " 0 / E F zero mode h = 1 r p + ea r 2 E 2m * & u# & v particle-hole symmetry # ) : $ ( ') : $ * % v " % u " 0 = 0 " = " + ( = ) & ( $ %( + # " ' & $ % Majorana (real) fermion u v # " interchanging vortices braid groups, non-abelian statistics i i+1 " i " i+1 # " i +1 i
Quantum spin Hall effect (Z 2 top. Insulator) Time-reversal invariant band insulator Strong spin-orbit interaction Kane & Mele (2005, 2006); Bernevig & Zhang (2006) Gapless helical edge mode (Kramers pair) B r r r " L # up-spin electrons B r down-spin electrons No spin rotation symmetry
Quantum spin Hall insulator is characterized by Z 2 topological index =1 an odd number of helical edge modes; Z 2 topological insulator = 0 an even (0) number of helical edge modes 1 0 # of a pair of zeros of r r Pf ui isy u j $% ( ) ( ) # " & * Kane-Mele model graphene + SOI [PRL 95, 146802 (2005)] Quantum spin Hall effect " s xy = e 2
Experiment HgTe/(Hg,Cd)Te quantum wells CdTe HgCdTe CdTe Konig et al. [Science 318, 766 (2007)]
3-dimensional Z 2 topological insulator Moore & Balents; Roy; Fu, Kane & Mele (2006, 2007) (strong) topological insulator bul: band insulator surface: an odd number of surface Dirac modes characterized by Z 2 topological numbers surface Dirac fermion bul insulator Ex: tight-binding model with SO int. on the diamond lattice [Fu, Kane, & Mele; PRL 98, 106803 (2007)] trivial insulator Z 2 topological insulator trivial band insulator: 0 or an even number of surface Dirac modes
Surface Dirac fermions topological insulator A half of graphene K K E K K K K x y An odd number of Dirac fermions in 2 dimensions cf. Nielsen-Ninomiya s no-go theorem
Experiments photon p, E Angle-resolved photoemission spectroscopy (ARPES) Bi 1-x Sb x Hsieh et al., Nature 452, 970 (2008) An odd (5) number of surface Dirac modes were observed.
Experiments II Bi 2 Se 3 hydrogen atom of top. ins. Xia et al., arxiv:0812.2078 band calculation
Q: Are there other 3D topological insulators? Yes Let s mae a table of all possible topological insulators.
Classification of topological insulators Topological insulators are stable against (wea) perturbations. Random deformation of Hamiltonian Natural framewor: random matrix theory (Wigner, Dyson, Altland & Zirnbauer) Assume only basic discrete symmetries: (1) time-reversal symmetry * 1 TH T = H TRS = 0 no TRS +1 TRS with T = +T " -1 TRS with T = T (integer spin) (hal-odd integer spin (2) particle-hole symmetry CH " C 1 = H ( )1 TC = H TCH PHS = 0 no PHS +1 PHS with C = +C " -1 PHS with C = C (odd parity: p-wave) (even parity: s-wave (3) TRS PHS = chiral symmetry [sublattice symmetry (SLS)] 3 3+ 1 = 10
10 random matrix ensembles IQHE Z 2 TPI MR Pfaffian Examples of topological insulators in 2 spatial dimensions Integer quantum Hall Effect Z 2 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, Furusai & Ludwig, PRB (200 Examples: (a)integer Quantum Hall Insulator, (b) Quantum Spin Hall Insulator, (c) 3d Z 2 Topological Insulator, (d) Spinless chiral p-wave (p+ip) superconductor (Moore (e)chiral d-wave superconductor, (f) superc (g) 3 He 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 (wea) perturbation disorder Majorana fermions to be found experimentally in solid-state devices