Topological Insulators andSuperconductors

Akira Furusaki

2012/2/8 1YIPQS 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

– 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 Bi1-xSbx

Bi2Se3 , Bi2Te3 , Bi2Tl2Se, …..

Topological insulators

Examples: integer quantum Hall effect, polyacetylen,

quantum spin Hall effect, 3D topological insulator, ….

•　 band insulators

•　 characterized by a topological number (Z or

Z2) Chern #, winding #, …

•　 gapless excitations at boundariesstable

free fermions (ignore e-e int.)

6

in broader sense

Topologicalinsulator

non-topological (vacuum)

Band insulators• An electron in a periodic potential (crystal)

• Bloch’s theorem Brillouin zone

2 2

2

2

dx V x x E x V x a V x

m dx

ikxk k k kx e u x u x a u x k

a a

E

ka

a

empty

occupiedband gap Band insulator

0

8

Energy band structure:

kukEkHk nn

or , ,a mapping

Topological equivalence (adiabatic continuity)

Band structures are equivalent if they can be continuously deformedinto one another without closing the energy gap

Topological distinction of ground states

m filledbands

n emptybands

xk

yk

map from BZ to Grassmannian

nUmUnmU2 IQHE (2 dim.)

homotopy class

deformed “Hamiltonian”

†1 0

0 1n

m

Q k U U U U m n

9

2 †1, , trQ Q Q Q m n

: Brillouin zone ( -space) Q k U m n U m U n

Berry phase of Bloch wave function

10

Berry connection kukuikAk

Berry curvature kAkFk

Berry phase C SC kFdkdkA 2

Example: 2-level Hamiltonian (spin ½ in magnetic field)

kukdkukH

zyx

yxz

didd

idddkdkH

Integer QHE

11

Integer quantum Hall effect (von Klitzing 1980)

Hxy

xx Quantization of Hall conductance2

: integerxy

ei ih

807.258122e

h

exact, robust against disorder etc.

12

Integer Quantum Hall Effect

Ch

exy

2

Chern number

21 ,

2 k x yC d k A k ki

filled band

,

, x y

x y k

k k k

A k k k k

integer valued

13

(TKNN: Thouless, Kohmoto, Nightingale & den Nijs 1982)

B

Hxy

xx

= number of edge modes crossing EF

Berry connection

bulk-edge correspondence

x

eBk y

x

y

Lattice model for IQHE (Haldane 1988)

• Graphene: a single layer of graphite– Relativistic electrons in a pencil

Geim & Novoselov: Nobel prize 2010

pypx

E

K

K’

K’

K’

K

K

0 0 0

0 0 0

0 0 0

0 0 0

x y

x y

Fx y

x y

p ip

p ipH v

p ip

p ip

ABAB

K K’

B A

Matrix element for hoppingbetween nearest-neighbor sites: t

3

2 300F

ta cv

Dirac masses• Staggered site energy (G. Semenoff 1984)

• Complex 2nd-nearest-neighbor hopping (Haldane 1988) No net magnetic flux through a unit cell

, on A sites

, on B sites

M

M

2it e

point: K F x x y y zK H v p p m

'' point: 'K F x x y y zK H v p p m

23 3 sinm M t

2' 3 3 sinm M t 2

xy

eC

h Hall conductivity

1sgn sgn '

2C m m

1C 1C

0C

0C

Chern insulator

0C

Breaks inversion symmetry

Breaks time-reversal symmetry

xm

x

Massive Dirac fermion: a minimal model for IQHE

zyyxx xmivH

zyyxx mivH

parity anomaly mxy sgn2

1

Domain wall fermion

i

dxxmv

ikyyxx

y

1''

1exp,

0

vkE 0m 0m

16

(2+1)d Chern-Simons theory for EM

Quantum spin Hall effect

(2D Z2 topological insulator)

17

2D Quantum spin Hall effect

• time-reversal invariant band insulator• spin-orbit interaction• gapless helical edge mode (Kramers’ pair) 　　

Kane & Mele (2005, 2006); Bernevig & Zhang (2006)

B

B

up-spin electrons

down-spin electrons

L S

E

xk

Sz is not conserved in general. Topological index: Z Z2

18

valence band

conduction band

19

Quantum spin Hall insulator

Bulk energy gap & gapless edge states

Helical edge states:(i) Half an ordinary 1D electron gas(ii) 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

Kane-Mele Haldane Haldane spin-flip2 2H H H H

up-spin electrons down-spin electrons

†spin-flip

. .

( )R i ij z jn n

H i c s r c

i i s s 2K 1yis

(1) (1)U U

0R 1C C

: electron spins

0R 0C 0, 1

Effective Hamiltonian• • •

: 1 A sublattice, 1 B sublatticez z

: 1 K point, 1 K' pointz z

: s 1 up spin, s 1 down spinz zs

0 F x z x y yH i v

SO SO z z zH s

R R x z y y xH s s

M zH M

complex 2nd nearest-neighbor hopping (Haldane)

spin-flip hopping

staggered site potential (Semenoff)

1total totalH H

Time-reversal symmetry2K 1x yi s Chern # = 0

complex conjugation

Z2 index Kane & Mele (2005); Fu & Kane (2006)

nu k

Bloch wave of occupied bands mn m nw k u k u k

Time-reversal invariant momenta: a a a G a aw w

antisymmetric

Z2 index

4

1

Pf1 1

det

a

aa

w

w

Quantum spin Hall insulator

E

kvalence band

conduction band

0

1 0

an odd numberof crossing

Trivial insulator

E

kvalence band

conduction band

0

an even numberof crossing

FEFE

Time reversal symmetry

23

Time reversal operator

*

*

k k

k k

2 1

Kramers’ theorem

All states are doubly degenerate.

time-reversal pair

Z2: stability of gapless edge states

(1) A single Kramers doublet

(2) Two Kramers doublets

24

E

k

E

k

Kramers’ theorem

E

k

E

k

stable

Two pairs of edge states are unstable against perturbations that respect TRS.

ExperimentHgTe/(Hg,Cd)Te quantum wells

Konig et al. [Science 318, 766 (2007)]

CdTeHgCdTeCdTe

25

QSHI

Trivial Ins.

Z2 topological insulator

in 3 spatial dimensions

26

3 dimensional Topological insulator

• Band insulator

• Metallic surface: massless Dirac fermions (Weyl fermions)

x

y

xk

yk

E

Theoretical Predictions made by:Fu, Kane, & Mele (2007)Moore & Balents (2007)Roy (2007)

Z2 topologically nontrivial

an odd number of Dirac cones/surface

Surface Dirac fermions

• “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’

K’

K’

K

K

topologicalinsulator

28

surface y x x yH i i

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, Ephoton

Bi2Te3, Bi2Te2Se, …Other topological insulators:

Response to external EM fieldieA

Integrate out electron fields to obtain effective action for the external EM field

2 23 3

eff 2 232 4

e eS dtdx F F dtdx E B

c c

axion electrodynamics (Wilczek, …)

Qi, Hughes & Zhang, 2008Essin, Moore & Vanderbilt 2009

0

trivial insulators

topological insulators

2

time reversal

topological insulator

vacuum 0 FF d AdA 2

surfacedtdx A A

(2+1)d Chern-Simons theory

2

2xy

e

h

Topological magnetoelectric effect2 2

3 3eff 2 232 4

e eS dtdx F F dtdx E B

c c

2

2

S eM E

hcB

Magnetization induced by electric field

Polarization induced by magnetic field2

2

S eP B

hcE

Topological superconductors

Topological superconductors

• BCS superconductors• Quasiparticles are massive inside the superconductor• Topological numbers• Majorana (Weyl) fermions at the boundaries

Examples: p+ip superconductor, fractional QHE at , 3He

topologicalsuperconductor

vacuum(topologically trivial)

stable

2

5

Majorana fermion

• Particle that is its own anti-particle• Neutrino ?• In superconductors: condensation of Cooper pairs

particle

hole

nothing (vacuum)

†uc vc † if u v Quasiparticle operator

Ettore Majoranamysteriously disappeared in 1938

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

2

2

2

2

x yF

p

x yF

pp ip

m pH d p

pp ip

p m

angular momentum =

px-ipypx+ipy

d̂ d d

2,x yp p S

2S

wrapping # = 1

k

E

2

†p

p

c

c

†k k

†

0

†

ikx ikxk k

k

x e e dk

x

*x p x pH H

Majorana zeromode in a quantum vortex

zero mode 00 00

Majorana fermionenergy spectrumnear a vortex

If there are 2N vortices, then the ground-state degeneracy = 2N.

(p+ip) superconductor

vortex

e

hc

Zero-energy Majorana bound state

E

0

interchanging vortices braid groups, non-Abelian statistics

1 ii

ii 1

D.A. Ivanov, PRL (2001)

(p+ip) superconductor

i i+1

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, ….)

s-SC

B

2

0 2x

x y x

pH gp s Bs

m InAs, InSb wire

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 Hamiltonianin terms of generic discrete symmetries

• Time-reversal symmetry (TRS)

• Particle-hole symmetry (PHS) BdG Hamiltonian

• TRS PHS = Chiral symmetry (CS)

HTTH 1*

HPPH 1*

2

2

0 no TRS

TRS 1 TRS with 1

1 TRS with 1

2

2

0 no PHS

PHS 1 PHS with 1

1 PHS with 1

spin 0

spin 1/2

HTPTPH 1

triplet

singlet

1Ch 0,PHSTRS 10133

jiji fHf

41

anti-unitary

anti-unitary

T

P

10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997)

42

Wigner-Dyson

chiral

super-conductor

IQHE

Z2 TPI

px+ipy

time evolution operatorTRS PHS Ch

• 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

exp iHt

10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997)

43

Wigner-Dyson

chiral

super-conductor

IQHE

Z2 TPI

px+ipy

time evolution operatorTRS PHS Ch

“Complex” cases: A & AIII“Real” cases: the remaining 8 classes HPPHHTTH 1*1* or

exp iHt

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 21 tr QrdS d

MQ

bulk: d dimensionsboundary: d -1 dimensions

21 ZMd Z2 top. term

WZW term ZMd

-term

(with no adjustable parameter)

44

NLSM topological terms HGd

Z2: Z2 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

46

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

QSHEZ2TPI

polyacetylene (SSH)

p+ip SC

d+id SC3He-B

Classification of topological insulators/superconductors

Schnyder, Ryu, AF, and Ludwig, PRB (2008)

p SC

(p+ip)x(p-ip) SC

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

period d = 2

period d = 8

Periodic table of topological insulators/superconductors

47Ryu, 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