Topological Insulators and Superconductors Akira Furusaki 201228 YIPQS

Topological Insulators and Superconductors Akira Furusaki 2012/2/8 YIPQS Symposium 1

Condensed matter physics • Diversity of materials – Understand their properties – Find new states of matter “More is different” (P. W. Anderson) • 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, …

Outline • Topological insulators: introduction • Examples: – – – Integer quantum Hall effect Quantum spin Hall effect 3 D Z 2 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 • 2 D: 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 Hg. Te • 3 D: Topological Insulators in the narrow sense – theory L. Fu, C. L. Kane & E. J. Mele 2007; J. Moore & L. Balents 2007; R. Roy 2007 – experiment Z. Hasan’s group (Princeton) 2008 Bi 1 -x. Sbx Bi 2 Se 3 , Bi 2 Tl 2 Se, …. .

Topological insulators • 　band insulators in broader sense free fermions (ignore e-e int. ) • 　characterized by a topological number (Z or Z 2) Chern #, winding #, … • 　gapless excitations at boundaries stable Topological non-topological (vacuum) insulator Examples: integer quantum Hall effect, polyacetylen, quantum spin Hall effect, 3 D topological insulator, …. 6

Band insulators • An electron in a periodic potential (crystal) • Bloch’s theorem Brillouin zone empty band gap Band insulator occupied 0

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 8

Topological distinction of ground states deformed “Hamiltonian” n empty bands m filled bands map from BZ to Grassmannian IQHE (2 dim. ) homotopy class 9

Berry phase of Bloch wave function Berry connection Berry curvature Berry phase Example: 2 -level Hamiltonian (spin ½ in magnetic field) 10

Integer QHE 11

Integer quantum Hall effect (von Klitzing 1980) Quantization of Hall conductance exact, robust against disorder etc. 12

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 13

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

Dirac masses • Staggered site energy (G. Semenoff 1984) Breaks inversion symmetry • Complex 2 nd-nearest-neighbor hopping (Haldane 1988) No net magnetic flux through a unit cell 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 16

Quantum spin Hall effect (2 D Z 2 topological insulator) 17

2 D 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 conduction band down-spin electrons valence band Sz is not conserved in general. Topological index: Z Z 2 18

Quantum spin Hall insulator Bulk energy gap & gapless edge states Helical edge states: (i) Half an ordinary 1 D electron gas (ii) Protected by time reversal symmetry 19

Kane-Mele model • Two copies of Haldane’s model (spin up & down) + spin-flip term down-spin electrons up-spin electrons • Invariant under time-reversal transformation • Spin-flip term breaks – – symmetry two copies of Chern insulators a new topological number: Z 2 index

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

Z 2 index Kane & Mele (2005); Fu & Kane (2006) Quantum spin Hall insulator conduction band Trivial insulator an odd number of crossing valence band conduction band an even number of crossing valence band Bloch wave of occupied bands Time-reversal invariant momenta: antisymmetric Z 2 index

Time reversal symmetry Time reversal operator Kramers’ theorem time-reversal pair All states are doubly degenerate. 23

Z 2: stability of gapless edge states (1) A single Kramers doublet E E Kramers’ theorem stable k k (2) Two Kramers doublets E E k k Two pairs of edge states are unstable against perturbations that respect TRS. 24

Experiment Hg. Te/(Hg, Cd)Te quantum wells Cd. Te Hg. Cd. Te Konig et al. [Science 318, 766 (2007)] Trivial Ins. QSHI 25

Z 2 topological insulator in 3 spatial dimensions 26

3 dimensional Topological insulator • Band insulator Z 2 topologically nontrivial • Metallic surface: massless Dirac fermions (Weyl fermions) 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 K E K’ K’ K K K’ kx ky • An odd number of Dirac fermions in 2 dimensions cf. Nielsen-Ninomiya’s no-go theorem 28

Experimental confirmation • Bi 1 -x. Sbx 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 • Bi 2 Se 3 ARPES exp. : Xia et al. , Nature Phys. 2009 photon p, E a single Dirac cone Other topological insulators: Bi 2 Te 3, Bi 2 Te 2 Se, …

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 , 3 He

Majorana fermion 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.

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

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

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

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

Classification of topological insulators and superconductors

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 40

Classification of free-fermion Hamiltonian in terms of generic discrete symmetries • Time-reversal symmetry (TRS) anti-unitary spin 0 spin 1/2 • Particle-hole symmetry (PHS) Bd. G Hamiltonian triplet anti-unitary singlet • TRS PHS = Chiral symmetry (CS) 41

10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997) TRS PHS Ch time evolution operator IQHE Wigner. Dyson Z 2 TPI chiral px+ipy superconductor • Wigner-Dyson (1951 -1963): “three-fold way” complex nuclei • Verbaarschot & others (1992 -1993) chiral phase transition in QCD 42 • 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 Z 2 TPI chiral px+ipy superconductor “Complex” cases: A & AIII “Real” cases: the remaining 8 classes 43

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) Z 2 top. term bulk: d dimensions boundary: d -1 dimensions WZW term -term 44

NLSM topological terms Z 2: Z 2 topological term can exist in d dimensions Z: WZW term can exist in d-1 dimensions d+1 dim. TI/TSC d dim. TI/TSC

Classification of topological insulators/superconductors Standard (Wigner-Dyson) Chiral Bd. G TRS PHS CS d=1 d=2 A (unitary) 0 0 0 -- Z AI (orthogonal) +1 0 0 -- -QSHE -- AII (symplectic) 1 0 0 -- Z 2 Z 2 TPI -- Z AIII (chiral unitary) 0 0 1 Z BDI (chiral orthogonal) +1 +1 1 Z CII (chiral symplectic) 1 1 1 Z IQHE -- polyacetylene (SSH) ---- 0 p SC Z 2 Z D (p-wave SC) 0 +1 C (d-wave SC) 0 1 0 -- Z DIII (p-wave TRS SC) 1 +1 1 Z 2 CI (d-wave TRS SC) +1 1 Schnyder, Ryu, AF, and Ludwig, PRB (2008) d=3 Z 2 p+ip SC -- d+id SC 1 (p+ip)x(p-ip) ---SC -Z 3 He-B Z 46

Periodic table of topological insulators/superconductors period d=2 period d=8 A. Kitaev, AIP Conf. Proc. 1134, 22 (2009); ar. Xiv: 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 47

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 – – (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+1 d) X. -G. Wen

• 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? • …. Ad. S/CMT? – Numerical • Quantum Monte Carlo (fermion sign problem) • Density Matrix RG (only in 1+1 d) • New algorithms: tensor-network RG, …. Quantum information theory