Topological Insulators and Superconductors Akira Furusaki 2012/2/8 YIPQS Symposium

Condensed matter physics Diversity of materials Understand their properties Find new states of matter Emergent behavior of electron systems at low energy Spontaneous symmetry breaking crystal, magnetism, superconductivity, …. Fermi liquids (non-Fermi liquids) (high-Tc) superconductivity, quantum criticality, … Insulators Mott insulators, quantum Hall effect, topological insulators, … “More is different” (P.W. Anderson)

Outline Topological insulators: introduction Examples: Integer quantum Hall effect Quantum spin Hall effect 3D Z2 topological insulator Topological superconductor Classification Summary and outlook

Introduction Topological insulator an insulator with nontrivial topological structure massless excitations live at boundaries bulk: insulating, surface: metallic Many ideas from field theory are realized in condensed matter systems anomaly domain wall fermions … Recent reviews: Z. Hasan & C.L. Kane, RMP 82, 3045 (2010) X.L. Qi & S.C. Zhang, RMP 83, 1057 (2011)

Recent developments 2005- Insulators which are invariant under time reversal can have topologically nontrivial electronic structure 2D: Quantum Spin Hall Effect theory C.L. Kane & E.J. Mele 2005; A. Bernevig, T. Hughes & S.C. Zhang 2006 experiment L. Molenkamp’s group (Wurzburg) 2007 HgTe 3D: Topological Insulators in the narrow sense L. Fu, C.L. Kane & E.J. Mele 2007; J. Moore & L. Balents 2007; R. Roy 2007 Z. Hasan’s group (Princeton) 2008 Bi1-xSbx Bi2Se3 , Bi2Te3 , Bi2Tl2Se, …..

Topological insulators in broader sense 　band insulators 　characterized by a topological number (Z or Z2) Chern #, winding #, … 　gapless excitations at boundaries free fermions (ignore e-e int.) Topological insulator non-topological (vacuum) stable Examples: integer quantum Hall effect, polyacetylen, quantum spin Hall effect, 3D topological insulator, ….

Band insulators An electron in a periodic potential (crystal) Bloch’s theorem Brillouin zone empty occupied Band insulator band gap

Energy band structure: a mapping Topological equivalence (adiabatic continuity) Band structures are equivalent if they can be continuously deformed into one another without closing the energy gap

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

Berry phase of Bloch wave function Berry connection Berry curvature Berry phase Example: 2-level Hamiltonian (spin ½ in magnetic field)

Integer QHE

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

Integer Quantum Hall Effect (TKNN: Thouless, Kohmoto, Nightingale & den Nijs 1982) Chern number integer valued = number of edge modes crossing EF bulk-edge correspondence filled band Berry connection

Lattice model for IQHE (Haldane 1988) Graphene: a single layer of graphite Relativistic electrons in a pencil Geim & Novoselov: Nobel prize 2010 B A py px E K K’ K K’ A B Matrix element for hopping between nearest-neighbor sites: t

Dirac masses Staggered site energy (G. Semenoff 1984) Complex 2nd-nearest-neighbor hopping (Haldane 1988) No net magnetic flux through a unit cell Breaks inversion symmetry Breaks time-reversal symmetry Hall conductivity Chern insulator

Massive Dirac fermion: a minimal model for IQHE parity anomaly (2+1)d Chern-Simons theory for EM Domain wall fermion

Quantum spin Hall effect (2D Z2 topological insulator)

2D Quantum spin Hall effect Kane & Mele (2005, 2006); Bernevig & Zhang (2006) time-reversal invariant band insulator spin-orbit interaction gapless helical edge mode (Kramers’ pair) up-spin electrons down-spin electrons conduction band valence band Sz is not conserved in general. Topological index: Z Z2

Quantum spin Hall insulator Bulk energy gap & gapless edge states Helical edge states: Half an ordinary 1D electron gas Protected by time reversal symmetry

Kane-Mele model Two copies of Haldane’s model (spin up & down) + spin-flip term Invariant under time-reversal transformation Spin-flip term breaks symmetry two copies of Chern insulators a new topological number: Z2 index up-spin electrons down-spin electrons

Effective Hamiltonian complex 2nd nearest-neighbor hopping (Haldane) spin-flip hopping staggered site potential (Semenoff) Time-reversal symmetry Chern # = 0 complex conjugation

Z2 index Kane & Mele (2005); Fu & Kane (2006) Quantum spin Hall insulator Trivial insulator valence band conduction band valence band conduction band an odd number of crossing an even number of crossing Bloch wave of occupied bands Time-reversal invariant momenta: antisymmetric Z2 index

Time reversal symmetry Time reversal operator Kramers’ theorem time-reversal pair All states are doubly degenerate.

Z2: stability of gapless edge states (1) A single Kramers doublet E k E k Kramers’ theorem stable (2) Two Kramers doublets E k E k Two pairs of edge states are unstable against perturbations that respect TRS.

Experiment HgTe/(Hg,Cd)Te quantum wells QSHI CdTe HgCdTe CdTe Konig et al. [Science 318, 766 (2007)] Trivial Ins. QSHI

Z2 topological insulator in 3 spatial dimensions

3 dimensional Topological insulator Band insulator Metallic surface: massless Dirac fermions (Weyl fermions) Z2 topologically nontrivial Theoretical Predictions made by: Fu, Kane, & Mele (2007) Moore & Balents (2007) Roy (2007) an odd number of Dirac cones/surface

Surface Dirac fermions topological insulator “1/4” of graphene An odd number of Dirac fermions in 2 dimensions cf. Nielsen-Ninomiya’s no-go theorem ky kx E K K’

Experimental confirmation Bi1-xSbx 0.09<x<0.18 theory: Fu & Kane (PRL 2007) exp: Angle Resolved Photo Emission Spectroscopy Princeton group (Hsieh et al., Nature 2008) 5 surface bands cross Fermi energy Bi2Se3 ARPES exp.: Xia et al., Nature Phys. 2009 a single Dirac cone p, E photon Other topological insulators: Bi2Te3, Bi2Te2Se, …

Response to external EM field Qi, Hughes & Zhang, 2008 Essin, Moore & Vanderbilt 2009 Integrate out electron fields to obtain effective action for the external EM field axion electrodynamics (Wilczek, …) time reversal trivial insulators topological insulators (2+1)d Chern-Simons theory topological insulator

Topological magnetoelectric effect Magnetization induced by electric field Polarization induced by magnetic field

Topological superconductors

Topological superconductors BCS superconductors Quasiparticles are massive inside the superconductor Topological numbers Majorana (Weyl) fermions at the boundaries stable topological superconductor vacuum (topologically trivial) Examples: p+ip superconductor, fractional QHE at , 3He

Majorana fermion Particle that is its own anti-particle Neutrino ? Ettore Majorana mysteriously disappeared in 1938 Particle that is its own anti-particle Neutrino ? In superconductors: condensation of Cooper pairs nothing (vacuum) particle hole Quasiparticle operator This happens at E=0.

2D p+ip superconductor (similar to IQHE) (px+ipy)-wave Cooper pairing Hamiltonian for Nambu spinor (spinless case) Majorana Weyl fermion along the edge angular momentum = wrapping # = 1 px+ipy px-ipy

(p+ip) superconductor Majorana zeromode in a quantum vortex Zero-energy Majorana bound state (p+ip) superconductor vortex zero mode energy spectrum near a vortex Majorana fermion If there are 2N vortices, then the ground-state degeneracy = 2N.

(p+ip) superconductor interchanging vortices braid groups, non-Abelian statistics (p+ip) superconductor i i+1 D.A. Ivanov, PRL (2001) topological quantum computing ? Majorana zeromode is insensitive to external disturbance (long coherence time).

Engineering topological superconductors 3D topological insulator + s-wave superconductor (Fu & Kane, 2008) Quantum wire with strong spin-orbit coupling + B field + s-SC Race is on for the search of elusive Majorana! Z2 TPI s-SC S-wave SC Dirac mass for the (2+1)d surface Dirac fermion Similar to a spinless p+ip superconductor Majorana zeromode in a vortex core (cf. Jakiw & Rossi 1981) (Das Sarma et al, Alicea, von Oppen, Oreg, … Sato-Fujimoto-Takahashi, ….) InAs, InSb wire B s-SC

Classification of topological insulators and superconductors

A: There are 5 classes of TPIs or TPSCs in each spatial dimension. Q: How many classes of topological insulators/superconductors exist in nature? A: There are 5 classes of TPIs or TPSCs in each spatial dimension. Generic Symmetries: time reversal charge conjugation (particle hole) SC

Classification of free-fermion Hamiltonian in terms of generic discrete symmetries Time-reversal symmetry (TRS) Particle-hole symmetry (PHS) BdG Hamiltonian TRS PHS = Chiral symmetry (CS) anti-unitary spin 0 spin 1/2 triplet singlet anti-unitary

10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997) TRS PHS Ch time evolution operator IQHE Wigner- Dyson Z2 TPI chiral px+ipy super- conductor Wigner-Dyson (1951-1963): “three-fold way” complex nuclei Verbaarschot & others (1992-1993) chiral phase transition in QCD Altland-Zirnbauer (1997): “ten-fold way” mesoscopic SC systems

10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997) TRS PHS Ch time evolution operator IQHE Wigner- Dyson Z2 TPI chiral px+ipy super- conductor “Complex” cases: A & AIII “Real” cases: the remaining 8 classes

How to classify topological insulators and SCs Gapless boundary modes are topologically protected. They are stable against any local perturbation. (respecting discrete symmetries) They should never be Anderson localized by disorder. Nonlinear sigma models for Anderson localization of gapless boundary modes + topological term (with no adjustable parameter) Z2 top. term WZW term bulk: d dimensions boundary: d -1 dimensions -term

NLSM topological terms Z2: Z2 topological term can exist in d dimensions d+1 dim. TI/TSC Z: WZW term can exist in d-1 dimensions d dim. TI/TSC

Classification of topological insulators/superconductors Standard (Wigner-Dyson) A (unitary) AI (orthogonal) AII (symplectic) TRS PHS CS d=1 d=2 d=3 0 0 0 +1 0 0 1 0 0 - Z -- - -- -- - Z2 Z2 AIII (chiral unitary) BDI (chiral orthogonal) CII (chiral symplectic) Chiral 0 0 1 +1 +1 1 1 1 1 Z -- Z Z -- -- Z -- Z2 D (p-wave SC) C (d-wave SC) DIII (p-wave TRS SC) CI (d-wave TRS SC) 0 +1 0 0 1 0 1 +1 1 +1 1 1 Z2 Z -- -- Z -- Z2 Z2 Z -- -- Z BdG IQHE QSHE Z2TPI polyacetylene (SSH) p+ip SC p SC d+id SC 3He-B (p+ip)x(p-ip) SC Schnyder, Ryu, AF, and Ludwig, PRB (2008)

Periodic table of topological insulators/superconductors A. Kitaev, AIP Conf. Proc. 1134, 22 (2009); arXiv:0901.2686 K-theory, Bott periodicity Ryu, Schnyder, AF, Ludwig, NJP 12, 065010 (2010) massive Dirac Hamiltonian Ryu, Takayanagi, PRD 82, 086914 (2010) Dp-brane & Dq-brane system

Summary and outlook Topological insulators/superconductors are new states of matter! There are many such states to be discovered. Junctions: TI + SC, TI + Ferromagnets, …. Search for Majorana fermions So far, free fermions. What about interactions?

Outlook Effects of interactions among electrons Topological insulators of strongly correlated electrons?? Fractional topological insulators ?? Topological order X.-G. Wen (no symmetry breaking) Fractional QH states Chern-Simons theory Low-energy physics described by topological field theory Fractionalization Symmetry protected topological states (e.g., Haldane spin chain in 1+1d)

Strongly correlated many-body systems have been (will remain to be) central problems High-Tc SC, heavy fermion SC, spin liquids, … but, very difficult to solve Theoretical approaches Analytical Application of new field theory techniques? AdS/CMT? …. Numerical Quantum Monte Carlo (fermion sign problem) Density Matrix RG (only in 1+1 d) New algorithms: tensor-network RG, …. Quantum information theory