akira furusaki- topological insulators
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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 +=
( ) !"#$
$
%
&
'
'
(
)
"
="
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