Topological InsulatorsIon (+ charge) electron Electrons moving in the lattice of ions ... Bi 1-xSb x...
Transcript of Topological InsulatorsIon (+ charge) electron Electrons moving in the lattice of ions ... Bi 1-xSb x...
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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 of
electric current.
insulating materials
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Band theory of electrons insolids
• Schroedinger equation
Ion (+ charge)
electron Electrons moving in the lattice of ions
( ) ( ) ( )rErrVm
!! ="#
$%&
'+() 2
2
2
h( ) ( )rVarV =+
Periodic electrostatic potential from ions and other electrons (mean-field)
Bloch’s theorem:( ) ( ) ( ) ( )ruaru
ak
aruer
knknkn
ikr
,,, , , =+<<!=
""#
( )kEn Energy band dispersion :n band index
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Metal and insulator in the bandtheory
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 electron’s 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 and
which has gapless excitations at its boundaries.
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Prominent example: quantum Halleffect• Classical Hall effect
B
r
e!!
!
!
!
!
!
+
+
+
+
+
+
E
rvr
W
:n electron density
Electric current nevWI !=
Bc
vE =Electric field
Hall voltage Ine
BEWV
H
!==
Hall resistancene
BRH
!=
Hall conductanceH
xyR
1=!
BveF
rrr!"=
Lorentz force
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Integer quantum Hall effect (von Klitzing 1980)
Hxy !! =
xx! Quantization of Hall conductance
h
eixy
2
=!
!= 807.258122e
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 rr
rr!=,
integer valued
)(ruek
rki rr
rr!
="
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Topological distinction of ground statesprojection operator
m filledbands
n emptybands
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.B
r
xk
E
!!"
Number of edge modes Che
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 number stable 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 +=
( ) !"#$$
%
&
''
(
)
"=
"
<
2
2
MR
1Pf
iz
jiji
ji
ezzzz
* ( )ijij AA det Pf =
Pf( ) is equal to the BCS wave function of px+ipy pairing state.
Bound state of two spinless ferions:P-wave & angular momentum=1
Excitations above the Moore-Read state obey non-Abelian statistics.
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Spinless px+ipy superconductor in 2dim.• Order parameter
• Chiral (Majorana) edge state
)()( 0 yxkk ikkk +!"##"! $
r
k
E
0!
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• Majorana bound state in a quantum vortex
vortex e
hc=!
FnEn / ,
2
000 !"= ##$
Bogoliubov-de Gennes equation
!!"
#$$%
&=!!
"
#$$%
&!!"
#$$%
&
'(
(
v
u
v
u
h
h)
*
0
*
0 ( ) FEAepm
h !+=2
0
2
1 rr
zero mode
particle-hole symmetry !!"
#$$%
&'(!!
"
#$$%
&*
*
: :u
v
v
u))
!!"
#$$%
&'!!
"
#$$%
&
(
(+
v
u
00=! )( !="=" + Majorana (real) fermion!
interchanging vortices braid groups, non-Abelian statistics
i i+11+!
ii""
ii!! "#+1
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Quantum spin Hall effect (Z2 top.Insulator)• Time-reversal invariant band insulator• Strong spin-orbit interaction• Gapless helical edge mode (Kramers pair)
Kane & Mele (2005, 2006); Bernevig & Zhang (2006)
B
r
B
r!
up-spin electrons
down-spin electrons
!"rr#L
No spin rotation symmetry
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• Quantum spin Hall insulator is characterized byZ2 topological index !
1=! an odd number of helical edge modes; Z2 topological insulator
an even (0) number of helical edge modes0=!1 0
Kane-Mele model graphene + SOI [PRL 95, 146802 (2005)]
Quantum spin Hall effect!
"2
es
xy=
# of a pair of zeros of
( ) ( )!"#
$%& *
Pf kuisku jyi
rr
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ExperimentHgTe/(Hg,Cd)Te quantum wells
Konig et al. [Science 318, 766 (2007)]
CdTeHgCdTeCdTe
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3-dimensional Z2 topological insulatorMoore & Balents; Roy; Fu, Kane & Mele (2006, 2007)
bulkinsulator
surface Dirac fermion(strong) topological insulatorbulk: band insulatorsurface: an odd number of surface Dirac modescharacterized by Z2 topological numbers
Ex: tight-binding model with SO int. on the diamond lattice [Fu, Kane, & Mele; PRL 98, 106803 (2007)]
trivial insulator
Z2 topologicalinsulator
trivial band insulator: 0 or an evennumber of surface Dirac modes
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Surface Dirac fermions
• A “half” of graphene
• An odd number of Dirac fermions in 2 dimensions cf. Nielsen-Ninomiya’s no-go theorem
kykx
E
K
K’
K’
K’
K
K
topologicalinsulator
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Experiments• Angle-resolved photoemission spectroscopy (ARPES)
Bi1-xSbx
p, Ephoton
Hsieh et al., Nature 452, 970 (2008)
An odd (5) number of surface Dirac modes were observed.
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Experiments II
Bi2Se3
band calculation
Xia et al., arXiv:0812.2078
“hydrogen atom” of top. ins.
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Q: Are there other 3D topological insulators?
Yes!
Let’s make a table of all possible topological insulators.
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Topological insulators are stable against (weak) perturbations.
Classification of topological insulators
Random deformation of Hamiltonian
Natural framework: random matrix theory (Wigner, Dyson, Altland & Zirnbauer)
Assume only basic discrete symmetries:
(1) time-reversal symmetry
HTTH =!1*
0 no TRSTRS = +1 TRS with -1 TRS with
TT +=!
TT !="
(integer spin)(hal-odd integer spin)
(2) particle-hole symmetry
HCCH !=!" 1
0 no PHSPHS = +1 PHS with -1 PHS with
CC +=!
CC !="
(odd parity: p-wave)(even parity: s-wave)
( ) HTCTCH !=!1
10133 =+!
(3) TRS PHS chiral symmetry [sublattice symmetry (SLS)]! =
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10 random matrix ensembles
IQHE
Z2 TPI
MR Pfaffian
Examples of topological insulators in 2 spatial dimensionsInteger quantum Hall EffectZ2 topological insulator (quantum spin Hall effect) also in 3DMoore-Read Pfaffian state (spinless p+ip superconductor)
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Table of topologocal insulators in 1, 2, 3dim. Schnyder, Ryu, Furusaki & Ludwig, PRB (2008)
Examples:(a)Integer Quantum Hall Insulator, (b) Quantum Spin Hall Insulator,(c) 3d Z2 Topological Insulator, (d) Spinless chiral p-wave (p+ip) superconductor (Moore-Read),(e)Chiral d-wave superconductor, (f) superconductor,(g) 3He B phase.
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Reordered TableKitaev, arXiv:0901.2686
Periodic table for topological insulators
Classification in any dimension
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Summary• Many topological insulators of non-interacting
fermions have been found. interacting fermions??
• Gapless boundary modes (Dirac or Majorana) stable against any (weak) perturbation disorder
• Majorana fermions to be found experimentally in solid-state devices