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Topological Insulators

Akira Furusaki

(Condensed Matter Theory Lab.)

topological insulators (3d and 2d)

=

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Outline

Introduction: band theory

Example of topological insulators:

integer quantum Hall effect

New members: Z2 topological insulators

Table of topological insulators

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Insulator

A material which resists the flow ofelectric current.

insulating materials

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an t eory o e ectrons n

solids

Schroedinger equation

Ion (+ charge)

electron Electrons moving in the lattice of ions

( ) ( ) ( )rErrVm

!! ="#

$%&

'+() 2

2

2

h( ) ( )rVarV =+

Periodic electrostatic potential fro

ions and other electrons (mean-fie

Blochs theorem:

( ) ( ) ( ) ( )ruaruakaruer knknknikr

,,,

,, =+

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Metal and insulator in the band

theory

k

E

a

!

a

!" 0

Electrons are fermions (spin=1/2).

Each state (n,k) can accommodateup to two electrons (up, down spins).

Pauli principle

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Metal

k

E

a

!

a

!" 0

empty states

occupied states

Apply electric field

k

E

a

!

a

!" 0

Flow of electric current

xk

yk

Apply electric field

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Band Insulator

k

E

a

!

a

!" 0

Band gap ( )akV /~

!=

All the states in the lower band are completely filled.(2 electrons per unit cell)

Electric current does not flow under (weak) electric field.

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Digression: other (named)

insulators

Peierls insulator (lattice deformation)

k

E

a

!

a

!" 0

Mott insulator (Coulomb repulsion)

Large Coulomb energy! Electrons cannot move.

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Anderson insulator (impurity scattering)

electron

Random scattering causes interference of electrons wave function.standing wave

Anderson localization

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Is that all?

No !

Yet another type of insulators: Topological insulators !

A topological insulator is a band insulator

which is characterized by a topological number andwhich has gapless excitations at its boundaries.

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Prominent example: quantum Hall

effect

Classical Hall effect

B

r

e!!

!

!

!

!

!

+

+

+

+

+

+

E

r

v

r

W

:n electron density

Electric current nevWI !=

Bc

v

E=Electric field

Hall voltage Ine

BEWV

H

!

==

Hall resistancene

BR

H

!

=

Hall conductanceH

xyR

1=!

BveF

r

r

r

!"=

Lorentz force

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Integer quantum Hall effect (von Klitzing 1980)

Hxy !! =

xx!

Quantization of Hall conductance

heixy

2

=!

!= 807.258122

e

h

exact, robust against disorder etc.

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Integer quantum Hall effect

Electrons are confined in a two-dimensional plane.

(ex. AlGaAs/GaAs interface)

Strong magnetic field is applied

(perpendicular to the plane)

Landau levels:

( ) ,...2,1,0,,2

1 ==+= nmc

eBnE

ccn!!h

AlGaAs GaAsB

r

cyclotron motion

k

E

!!"

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TKNN number (Thouless-Kohmoto-Nightingale-den Nijs)

TKNN (1982); Kohmoto (1985)

C

h

exy

2

!="

first Chern number (topological invariant)

!! ""#

$

%%

&

'

(

(

(

()

(

(

(

(=

yxxy k

u

k

u

k

u

k

urdkd

iC

**

22

2

1

*

( )yxk kkAkd

i,

2

1 2rr

!"= #$

filled band

( )kkkyx

uukkA rrrr

!=,

integer valued

)(ruek

rki rr

r

r

!

="

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Topological distinction of ground states

projection operator

m filled

bands

n empty

bands

xk

yk

map from BZ to Grassmannian

( ) ( ) ( )[ ] !="+ nUmUnmU2# IQHE

homotopy class

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Edge states

There is a gapless chiral edge mode along the sample

boundary. Br

xk

E

!!"Number of edge modes C

he

xy=!=

/2

"

Effective field theory

( )zyyxx

ymivH !!! +"+"#=

( )ym

y

domain wall fermion

Robust against disorder (chiral fermions cannot be backscattered)

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Topological Insulators (definition ??)

(band) insulator with a nonzero gap to excitated states

topological numberstable against any (weak) perturbation

gapless edge mode

When the gapless mode appears/disappears, the bulk(band) gap closes. Quantum Phase Transition

Low-energy effective theory

= topological field theory (Chern-Simons)

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Fractional quantum Hall effect at

2nd Landau level

Even denominator (cf. Laughlin states: odd denominator) Moore-Read (Pfaffian) state

2

5=!

jjjiyxz +=

( ) !"#$

$

%

&

'

'

(

)

"

="