Introduction to topological insulators and superconductors September 16

Introduction to topological insulators and superconductors 古崎 昭 （理化学研究所） September 16, 2009 topological insulators (3 d and 2 d)

Band theory of electrons in solids ion (+ charge) electron Electrons moving in the lattice of ions • Schroedinger equation Periodic electrostatic potential from ions and other electrons (mean-field) Bloch’s theorem: Energy band dispersion band index

Metal and insulator in the band theory Interactions between electrons are ignored. (free fermion) Each state (n, k) can accommodate up to two electrons (up, down spins). Pauli principle

Band Insulator Band gap All the states in the lower band are completely filled. (2 electrons per unit cell) Electric current does not flow under (weak) electric field.

Topological (band) insulators • バンド絶縁体 free fermions • トポロジカル数をもつ • 端にgapless励起(Dirac fermion)をもつ stable Condensed-matter realization of domain wall fermions Examples: integer quantum Hall effect, quantum spin Hall effect, Z 2 topological insulator, ….

Topological (band) insulators superconductors • BCS超伝導体 バンド絶縁体 超伝導gap • トポロジカル数をもつ (Dirac fermion)をもつ or Majorana) をもつ • 端にgapless励起(Dirac stable Condensed-matter realization of domain wall fermions Examples: p+ip superconductor, 3 He

Example 1: Integer QHE

Prominent example: quantum Hall effect • Classical Hall effect electron density Electric current Electric field Hall voltage Lorentz force Hall resistance Hall conductance

Integer quantum Hall effect (von Klitzing 1980) Quantization of Hall conductance exact, robust against disorder etc.

Integer quantum Hall effect • Electrons are confined in a two-dimensional plane. (ex. Al. Ga. As/Ga. As interface) • Strong magnetic field is applied (perpendicular to the plane) cyclotron motion Landau levels: Al. Ga. As

TKNN number (Thouless-Kohmoto-Nightingale-den Nijs) TKNN (1982); Kohmoto (1985) Chern number (topological invariant) integer valued filled band

Edge states • There is a gapless edge mode along the sample boundary. Number of edge modes Robust against disorder (chiral fermions cannot be backscattered) Bulk: (2+1)d Chern-Simons theory Edge: (1+1)d CFT

Effective field theory parity anomaly Domain wall fermion

Example 2: chiral p-wave superconductor

BCS理論 （平均場理論） Integrating out Ginzburg-Landau 超伝導秩序変数 S-wave (singlet) 高温超伝導体 P-wave (triplet) +

Spinless px+ipy superconductor in 2 dim. • Order parameter • Chiral (Majorana) edge state Hamiltonian density windingspectrum (wrapping) Gapped fermion number=1

Hamiltonian density Bogoliubov-de Gennes equation Particle-hole symmetry (charge conjugation) zeromode: Majorana fermion

Majorana edge state px+ipy superconductor:

• Majorana bound state in a quantum vortex Bogoliubov-de Gennes equation energy spectrum of vortex bound states zero mode Majorana (real) fermion! 2 N vortices GS degeneracy = 2 N

interchanging vortices i braid groups, non-Abelian statistics i+1 D. A. Ivanov, PRL (2001)

Fractional quantum Hall effect at • 2 nd Landau level • Even denominator (cf. Laughlin states: odd denominator) • Moore-Read (Pfaffian) state Pf( ) is equal to the BCS wave function of px+ipy pairing state. Excitations above the Moore-Read state obey non-Abelian statistics. Effective field theory: level-2 SU(2) Chern-Simons theory G. Moore & N. Read (1991); C. Nayak & F. Wilczek (1996)

Example 3: Z 2 topological insulator Quantum spin Hall effect

Quantum spin Hall effect (Z 2 top. Insulator) Kane & Mele (2005, 2006); Bernevig & Zhang (2006) • Time-reversal invariant band insulator • Strong spin-orbit interaction • Gapless helical edge mode (Kramers pair) up-spin electrons down-spin electrons If Sz is conserved, If Sz is NOT conserved, Chern # (Z) Z 2 Quantized spin Hall conductivity

(trivial) Band insulator Quantum Hall state Quantum Spin Hall state

Kane-Mele model (PRL, 2005) K K’ K K A B time reversal symmetry E K’ K’ kx ky

• Quantum spin Hall insulator is characterized by Z 2 topological index an odd number of helical edge modes; Z 2 topological insulator an even (0) number of helical edge modes 1 0 Kane-Mele model graphene + SOI [PRL 95, 146802 (2005)] Quantum spin Hall effect (if Sz is conserved) Edge states stable against disorder (and interactions)

Z 2 topological number

Z 2: stability of gapless edge states (1) A single Kramers doublet Kramers’ theorem (2) Two Kramers doublets opens a gap Odd number of Kramers doublet (1) Even number of Kramers doublet (2)

Experiment Hg. Te/(Hg, Cd)Te quantum wells Cd. Te Hg. Cd. Te Konig et al. [Science 318, 766 (2007)]

Example 4: 3 -dimensional Z 2 topological insulator

3 -dimensional Z 2 topological insulator Moore & Balents; Roy; Fu, Kane & Mele (2006, 2007) Z 2 topological insulator surface Dirac fermion bulk: band insulator surface: an odd number of surface Dirac modes characterized by Z 2 topological numbers bulk 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 E K’ K’ K K K’ kx • An odd number of Dirac fermions in 2 dimensions cf. Nielsen-Ninomiya’s no-go theorem ky

Experiments photon • Angle-resolved photoemission spectroscopy (ARPES) Bi 1 -x. Sbx Hsieh et al. , Nature 452, 970 (2008) An odd (5) number of surface Dirac modes were observed. p, E

Experiments II Bi 2 Se 3 “hydrogen atom” of top. ins. a single Dirac cone Xia et al. , Nature Physics 5, 398 (2009) ARPES experiment Band calculations (theory)

トポロジカル絶縁体・超伝導体の分類 Schnyder, Ryu, AF, and Ludwig, PRB 78, 195125 (2008) ar. Xiv: 0905. 2029 (Landau 100)

Classification of topological insulators/superconductors Schnyder, Ryu, AF, and Ludwig, PRB (2008) Kitaev, ar. Xiv: 0901. 2686

Zoo of topological insulators/superconductors

Classification of topological insulators/SCs 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 TRS = (2) particle-hole symmetry PHS = (3) TRS PHS 0 no TRS +1 TRS with -1 TRS with 0 no PHS +1 PHS with -1 PHS with (integer spin) (half-odd integer spin) (odd parity: p-wave) (even parity: s-wave) chiral symmetry [sublattice symmetry (SLS)]

(2) particle-hole symmetry px+ipy dx 2 -y 2+idxy Bogoliubov-de Gennes

10 random matrix ensembles IQHE Z 2 TPI px+ipy dx 2 -y 2+idxy Examples of topological insulators in 2 spatial dimensions Integer quantum Hall Effect Z 2 topological insulator (quantum spin Hall effect) also in 3 D Moore-Read Pfaffian state (spinless p+ip superconductor)

Table of topological insulators in 1, 2, 3 dim. Schnyder, Ryu, Furusaki & Ludwig, PRB (2008) Examples: (a) Integer Quantum Hall Insulator, (b) Quantum Spin Hall Insulator, (c) 3 d Z 2 Topological Insulator, (d) Spinless chiral p-wave (p+ip) superconductor (Moore-Read), (e) Chiral d-wave superconductor, (f) superconductor, (g) 3 He B phase.

Classification of 3 d topological insulators/SCs strategy (bulk boundary) • Bulk topological invariants integer topological numbers： 3 random matrix ensembles (AIII, CI, DIII) BZ: Brillouin zone • Classification of 2 d Dirac fermions 13 classes (13=10+3) AIII, CI, DIII Bernard & Le. Clair (‘ 02) AII, CII • Anderson delocalization in 2 d nonlinear sigma models Z 2 topological term (2) or WZW term (3)

Topological distinction of ground states deformed “Hamiltonian” n empty bands m filled bands map from BZ to Grassmannian IQHE (2 dim. ) homotopy class

In classes AIII, BDI, CI, DIII, Hamiltonian can be made off-diagonal. Projection operator is also off-diagonal. topological insulators labeled by an integer Discrete symmetries limit possible values of Z 2 insulators in CII (chiral symplectic)

The integer number # of surface Dirac (Majorana) fermions bulk insulator surface Dirac fermion

(3+1)D 4 -component Dirac Hamiltonian AII: TRS DIII: AIII: PHS ch. S

(3+1)D 8 -component Dirac Hamiltonian CI: CII:

Classification of 3 d topological insulators strategy (bulk boundary) • Bulk topological invariants integer topological numbers： 3 random matrix ensembles (AIII, CI, DIII) • Classification of 2 d Dirac fermions 13 classes (13=10+3) AIII, CI, DIII Bernard & Le. Clair (‘ 02) AII, CII • Anderson delocalization in 2 d nonlinear sigma models Z 2 topological term (2) or WZW term (3)

Nonlinear sigma approach to Anderson localization • • (fermionic) replica Matrix field Q describing diffusion Localization massive Extended or critical massless Wegner, Efetov, Larkin, Hikami, …. topological Z 2 term or WZW term

Table of topological insulators in 1, 2, 3 dim. Schnyder, Ryu, Furusaki & Ludwig, PRB (2008) ar. Xiv: 0905. 2029 Examples: (a) Integer Quantum Hall Insulator, (b) Quantum Spin Hall Insulator, (c) 3 d Z 2 Topological Insulator, (d) Spinless chiral p-wave (p+ip) superconductor (Moore-Read), (e) Chiral d-wave superconductor, (f) superconductor, (g) 3 He B phase.

Reordered Table Periodic table of topological insulators Classification in any dimension Kitaev, ar. Xiv: 0901. 2686

Classification of topological insulators/superconductors Schnyder, Ryu, AF, and Ludwig, PRB (2008) Kitaev, ar. Xiv: 0901. 2686

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 Andreev bound states in p-wave superfluids Z 2 T. I. + s-wave SC Majorana bound state 3 He-B (Fu & Kane)