Topological Insulators and - PowerPoint Presentation

Slide1

Topological Insulators andSuperconductors

Akira Furusaki

2012/2/8

1

YIPQS SymposiumSlide2

Condensed matter physics

Diversity of materialsUnderstand their propertiesFind new states of matterEmergent 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)Slide3

Outline

Topological insulators: introductionExamples:Integer quantum Hall effectQuantum spin Hall effect3D Z

2 topological insulatorTopological superconductorClassification

Summary and outlookSlide4

Introduction

Topological insulatoran insulator with nontrivial topological structuremassless excitations live at boundaries

bulk: insulating, surface: metallicMany ideas from field theory are realized

in condensed matter systemsanomalydomain wall fermions

…Recent reviews:Z.

Hasan & C.L. Kane, RMP 82, 3045 (2010)X.L. Qi

& S.C. Zhang, RMP 83, 1057 (2011)Slide5

Recent developments 2005-

Insulators which are invariant under time reversal can have topologically nontrivial electronic structure2D: Quantum Spin Hall Effecttheory

C.L. Kane & E.J. Mele 2005; A. Bernevig

, T. Hughes & S.C. Zhang 2006experiment 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

Bi

1-x

Sb

x

Bi

2

Se

3

, Bi

2

Te

3

, Bi

2

Tl

2

Se, …..Slide6

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 boundaries

stable

f

ree

fermions (ignore e-e int.)

6

i

n broader sense

Topological

insulator

n

on-topological

(vacuum)Slide7

Band insulators

An electron in a periodic potential (crystal)

Bloch’s theorem Brillouin zone

empty

occupied

b

and gap

Band insulator

0Slide8

8

Energy band structure:

a m

apping

Topological equivalence (adiabatic continuity)

Band structures are equivalent if they can be continuously deformed

into one another

without closing the energy gapSlide9

Topological distinction of ground states

m

filled

bands

n

empty

bands

map from BZ to

Grassmannian

IQHE

(2 dim.)

homotopy class

deformed “Hamiltonian”

9Slide10

Berry phase of Bloch wave function

10Berry connection

Berry curvature

Berry phase

Example: 2-level Hamiltonian (spin ½ in magnetic field)Slide11

Integer QHE

11Slide12

I

nteger quantum Hall effect

(von Klitzing 1980)

Quantization of Hall conductance

e

xact, robust against disorder etc.

12Slide13

Integer Quantum Hall Effect

Chern

number

f

illed band

i

nteger valued

13

(TKNN:

Thouless

,

Kohmoto

, Nightingale & den

Nijs

1982)

= number of edge modes

crossing E

F

Berry connection

b

ulk-edge correspondenceSlide14

Lattice model for IQHE (Haldane 1988)

Graphene: a single layer of graphiteRelativistic electrons in a pencilGeim &

Novoselov: Nobel prize 2010

p

y

p

x

E

K

K’

K’

K’

K

K

A

B

A

B

K

K’

B

A

Matrix element for hopping

between nearest-neighbor sites:

tSlide15

Dirac masses

Staggered site energy (G. Semenoff 1984)

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

Hall conductivity

Chern

insulator

Breaks inversion symmetry

Breaks time-reversal symmetrySlide16

Massive Dirac

fermion

: a minimal model for IQHE

p

arity anomaly

D

omain wall

fermion

16

(2+1)d

Chern

-Simons theory for EMSlide17

Quantum spin Hall effect

(2D Z2 topological insulator)

17Slide18

2D Quantum spin Hall effect

time-reversal invariant band insulatorspin-orbit interaction

gapless helical edge mode (Kramers’ pair)

Kane &

Mele (2005, 2006);

Bernevig & Zhang (2006)

u

p-spin electrons

down

-spin electrons

S

z

is not conserved in general.

Topological index: Z Z

2

18

v

alence band

conduction

bandSlide19

19

Quantum spin Hall insulatorBulk energy gap & gapless edge states

Helical edge states:

Half an ordinary 1D electron gas

Protected by time reversal symmetrySlide20

Kane-Mele model

Two copies of Haldane’s model (spin up & down) + spin-flip termInvariant under time-reversal transformation

Spin-flip term breaks symmetry two copies of

Chern insulators

a new topological number: Z2

index

u

p-spin electrons

down-

spin electronsSlide21

Effective Hamiltonian

c

omplex 2

nd

nearest-neighbor hopping (Haldane)

s

pin-flip hopping

s

taggered site potential (

Semenoff

)

Time-reversal symmetry

Chern

# = 0

c

omplex conjugationSlide22

Z2 index

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

Bloch wave of occupied bands

Time-reversal invariant

momenta

:

antisymmetric

Z

2

index

Quantum spin Hall

insulator

v

alence band

c

onduction band

a

n

odd

number

of crossing

Trivi

al insulator

v

alence band

c

onduction band

a

n

even

number

of crossingSlide23

Time reversal symmetry

23Time reversal operator

Kramers

’ theorem

All states are doubly degenerate.

t

ime-reversal pairSlide24

Z

2: 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

.Slide25

Experiment

HgTe/(Hg,Cd)Te quantum wells

Konig

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

CdTe

HgCdTe

CdTe

25

QSHI

Trivial Ins.Slide26

Z2

topological insulatorin 3 spatial dimensions

26Slide27

3 dimensional Topological insulator

Band insulatorMetallic surface: massless Dirac fermions

(Weyl fermions)

Theoretical Predictions made by:

Fu, Kane, &

Mele

(2007)

Moore &

Balents

(2007)

Roy (2007)

Z

2

topologically nontrivial

an

odd

number of Dirac cones/surfaceSlide28

Surface Dirac fermions

“1/4” of graphene

An odd number of Dirac fermions in 2 dimensions

cf. Nielsen-

Ninomiya’s no-go theorem

k

y

k

x

E

K

K’

K’

K’

K

K

topological

insulator

28Slide29

Experimental confirmation

Bi1-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

Bi2Se

3

ARPES exp.: Xia et al., Nature Phys. 2009 a single Dirac cone

p

, E

photon

Bi

2

Te

3

, Bi

2

Te

2

Se, …

Other topological insulators:Slide30

Response to external EM field

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

a

xion

electrodynamics (

Wilczek, …)

Qi

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

t

rivial insulators

topologic

al insulators

t

ime reversal

topological insulator

(2+1)d

Chern

-Simons theorySlide31

Topological magnetoelectric effect

Magnetization induced by electric field

Polar

ization induced by

magnet

ic fieldSlide32

Topological superconductorsSlide33

Topological superconductors

BCS superconductorsQuasiparticles are massive inside the superconductorTopological numbersMajorana

(Weyl) fermions at the boundaries

Examples:

p+ip superconductor, fractional QHE at ,

3He

t

opological

superconductor

v

acuum

(topologically trivial)

stableSlide34

Majorana fermion

Particle that is its own anti-particleNeutrino ?In superconductors: condensation of Cooper pairs

particle

hole

n

othing (vacuum)

Quasiparticle

operator

Ettore

Majorana

mysteriously disappeared in 1938

This happens at E=0.Slide35

2D

p+ip superconductor (similar to IQHE)

(

px+ipy

)-wave Cooper pairingHamiltonian for

Nambu spinor

(spinless

case)

Majorana

Weyl

fermion

along the edge

a

ngular momentum =

p

x

-ip

y

p

x

+ip

y

w

rapping # = 1Slide36

Majorana zeromode

in a quantum vortex

z

ero mode

Majorana

fermion

e

nergy spectrum

n

ear a vortex

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

N

.

(

p+ip

) superconductor

vortex

Zero-energy

Majorana

bound stateSlide37

interchanging vortices braid groups, non-

Abelian statistics

D.A.

Ivanov

, PRL (2001)

(

p+ip

) superconductor

i

i+1

t

opological quantum computing ?

Majorana

zeromode

is insensitive to external disturbance (long coherence time).Slide38

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!

Z

2

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

InAs

,

InSb

wireSlide39

Classification of topological insulators and superconductorsSlide40

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

40Slide41

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)

s

pin 0

s

pin 1/2

triplet

singlet

41

a

nti-unitary

a

nti-unitarySlide42

10 random matrix ensembles (symmetric spaces) Altland &

Zirnbauer (1997)42

Wigner-

Dyson

chiral

s

uper-

conductor

IQHE

Z

2

TPI

p

x

+ip

y

t

ime evolution operator

TRS 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 systemsSlide43

10 random matrix ensembles (symmetric spaces) Altland &

Zirnbauer (1997)43

Wigner-

Dyson

chiral

s

uper-

conductor

IQHE

Z

2

TPI

p

x

+ip

y

t

ime evolution operator

TRS PHS Ch

“Complex” cases: A & AIII

“Real”

cases: the remaining 8 classesSlide44

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

+

t

opological term

b

ulk:

d

dimensions

b

oundary:

d

-1

dimensions

Z

2

top. term

WZW

term

-

term

(w

ith no adjustable parameter)

44Slide45

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/TSCSlide46

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 --

- -- --

- Z

2

Z

2

AIII (

chiral

unitary)

BDI

(

chiral

orthogonal)

CII (

chiral

symplectic

)

Chiral

0 0 1

+1 +1 1

1 1 1

Z -- Z

Z

-- --

Z -- Z

2

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

Z

2

Z --

-- Z

--

Z

2

Z

2

Z

-- -- Z

BdG

IQHE

QSHE

Z

2

TPI

polyacetylene

(SSH)

p+ip

SC

d+id

SC

3

He-B

Classification

of topological insulators/superconductors

Schnyder

,

Ryu

, AF, and Ludwig, PRB (2008)

p SC

(

p+ip

)x(p-

ip

) SCSlide47

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

47

Ryu

,

Takayanagi

,

PRD 82, 086914 (2010)

Dp-brane

&

Dq-brane

systemSlide48

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?Slide49

Outlook

Effects of interactions among electronsTopological insulators of strongly correlated electrons??Fractional topological insulators ??

Topological order X.-G. Wen

(no symmetry breaking)Fractional QH states Chern

-Simons theoryLow-energy physics described by topological field theoryFractionalization

Symmetry protected topological states (e.g., Haldane spin chain in 1+1d)Slide50

Strongly correlated many-body systemshave been (will remain to be) central problems

High-Tc SC, heavy fermion SC, spin liquids, …but, very difficult to solveTheoretical approaches

AnalyticalApplication of new field theory techniques? AdS/CMT?

….NumericalQuantum Monte Carlo (

fermion sign problem)Density Matrix RG (only in 1+1 d)New algorithms: tensor-network RG, ….

Quantum information theory