Topological Insulators and Superconductors
Lecture #1: Topology and Band Theory
Lecture #2: Topological Insulators in 2 and 3
dimensions
Lecture #3: Topological Superconductors, Majorana
Fermions an Topological quantum
compuation
General References :
M.Z. Hasan and C.L. Kane, RMP in press, arXiv:1002.3895
X.L. Qi and S.C. Zhang, Physics Today 63 33 (2010).
J.E. Moore, Nature 464, 194 (2010).
My collaborators :
Gene Mele, Liang Fu, Jeffrey Teo, Zahid Hasan
Topology and Band Theory
I. Introduction
- Insulating State, Topology and Band Theory
II. Band Topology in One Dimension
- Berry phase and electric polarization
- Su Schrieffer Heeger model :
domain wall states and Jackiw Rebbi problem
- Thouless Charge Pump
III. Band Topology in Two Dimensions
- Integer quantum Hall effect
- TKNN invariant
- Edge States, chiral Dirac fermions
IV. Generalizations
- Bulk-Boundary correspondence
- Higher dimensions
- Topological Defects
The Insulating State
Covalent Insulator
Characterized by energy gap: absence of low energy electronic excitations
The vacuum Atomic Insulator
e.g. solid Ar
Dirac
Vacuum Egap ~ 10 eV
e.g. intrinsic semiconductor
Egap ~ 1 eV
3p
4s
Silicon
Egap = 2 mec2
~ 106 eV
electron
positron ~ hole
The Integer Quantum Hall State
2D Cyclotron Motion, Landau Levels
Quantized Hall conductivity :
B
Jy
y xy xJ E
2
xyh
ne
Integer accurate to 10-9
Energy gap, but NOT an insulator
gap cE
E
gap cE
Ex
Topology The study of geometrical properties that are insensitive to smooth deformations
Example: 2D surfaces in 3D
A closed surface is characterized by its genus, g = # holes
g=0 g=1
g is an integer topological invariant that can be expressed in terms of the
gaussian curvature k that characterizes the local radii of curvature
4 (1 )S
dA gk
1 2
1
r rk
Gauss Bonnet Theorem :
2
10
rk 0k
0k
A good math book : Nakahara, ‘Geometry, Topology and Physics’
Band Theory of Solids
Bloch Theorem :
( ) ( ) ( ) ( )n n nH u E uk k k k
dT
k Brillouin Zone
Torus,
Topological Equivalence : adiabatic continuity
( ) ( )n nE uk k(or equivalently to and )
Band structures are equivalent if they can be continuously deformed
into one another without closing the energy gap
Band Structure :
( )Hk kE
kx /a /a
Egap
= kx
ky
/a
/a
/a
/a
BZ
( ) iT e k RRLattice translation symmetry ( )ie u k r
k
Bloch Hamiltonian ( ) Ηi iH e e k r k rk
A mapping
Berry Phase
Phase ambiguity of quantum mechanical wave function
( )( ) ( )iu e u kk k
Berry connection : like a vector potential ( ) ( )i u u k
A k k
( ) k
A A k
Berry phase : change in phase on a closed loop C CC
d A k
Berry curvature : k
F A 2
CS
d k F
Famous example : eigenstates of 2 level Hamiltonian
( ) ( )z x y
x y z
d d idH
d id d
k d k
( ) ( ) ( ) ( )H u u k k d k k 1 ˆSolid Angle swept out by ( )2
C d k
C
d̂
S
Topology in one dimension : Berry phase and electric polarization
Electric Polarization
+Q -Q 1D insulator
The end charge is not completely determined by the bulk polarization P
because integer charges can be added or removed from the ends :
Polarization as a Berry phase : ( )2
eP A k dk
P is not gauge invariant under “large” gauge transformations.
This reflects the end charge ambiguity
( )( ) ( )i ku k e u kP P en ( / ) ( / ) 2a a n
see, e.g. Resta, RMP 66, 899 (1994)
Changes in P, due to adiabatic variation are well defined and gauge invariant
( ) ( , ( ))u k u k t
1 02 2C S
e eP P P dk dkd
A F
when with
k
-/a
/a 0
k
-/a /a
1
0
dipole moment
lengthP
bP
mod Q P e
C
S
gauge invariant Berry curvature
Su Schrieffer Heeger Model model for polyacetalene
simplest “two band” model
† †
1( ) ( ) . .Ai Bi Ai Bi
i
H t t c c t t c c h c
( ) ( )H k k d
( ) ( ) ( )cos
( ) ( )sin
( ) 0
x
y
z
d k t t t t ka
d k t t ka
d k
0t
0t
a
d(k)
d(k)
dx
dy
dx
dy
E(k)
k
/a /a
Provided symmetry requires dz(k)=0, the states with t>0 and t<0 are topologically distinct.
Without the extra symmetry, all 1D band structures are topologically equivalent.
A,i B,i
t>0 : Berry phase 0
P = 0
t<0 : Berry phase
P = e/2
Gap 4|t|
Peierl’s instability → t
A,i+1
Domain Wall States An interface between different topological states has topologically protected midgap states
Low energy continuum theory :
For small t focus on low energy states with k~/a xk q q ia
;
( )vF x x yH i m x
0t 0t
Massive 1+1 D Dirac Hamiltonian
“Chiral” Symmetry :
2v ; F ta m t
{ , } 0 z z E EH
0
( ') '/ v
0
1( )
0
x
Fm x dx
x e
Egap=2|m| Domain wall
bound state 0
m>0
m<0
2 2( ) ( )vFE q q m
Zero mode : topologically protected eigenstate at E=0
(Jackiw and Rebbi 76, Su Schrieffer, Heeger 79)
Any eigenstate at +E
has a partner at -E
Thouless Charge Pump
t=0
t=T
P=0
P=e
( , ) ( , )H k t T H k t
( , ) ( ,0)2
eP A k T dk A k dk ne
k
-/a /a
t=T
t=0 =
The integral of the Berry curvature defines the first Chern number, n, an integer
topological invariant characterizing the occupied Bloch states, ( , )u k t
In the 2 band model, the Chern number is related to the solid angle swept out by
which must wrap around the sphere an integer n times.
ˆ ( , ),k td
2
1
2 Tn dkdt
F
2
1 ˆ ˆ ˆ ( )4
k tT
n dkdt
d d dˆ ( , )k td
The integer charge pumped across a 1D insulator in one period of an adiabatic cycle
is a topological invariant that characterizes the cycle.
Integer Quantum Hall Effect : Laughlin Argument
Adiabatically thread a quantum of magnetic flux through cylinder.
2 xyI R E
0
T
xy xy
d hQ dt
dt e
1
2
dE
R dt
+Q -Q ( 0) 0
( ) /
t
t T h e
Just like a Thouless pump :
2
xy
eQ ne n
h
†( ) (0)H T U H U
TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 82
View cylinder as 1D system with subbands labeled by 0
1( )m
yk mR
0
0
, ( )2
m
x x y
m
eQ d dk k k ne
F
kx
ky TKNN number = Chern number
21 1( )
2 2 CBZ
n d k d
F k A k
2
xy
en
h
kym(0)
kym(0)
Distinguishes topologically distinct 2D band structures. Analogous to Gauss-Bonnet thm.
Alternative calculation: compute xy via Kubo formula
C
kx
( ) , ( )m
m x x yE k E k k
Graphene E
k
Two band model
Novoselov et al. ‘05 www.univie.ac.at
( ) ( )H k d k
( ) | ( ) |E k d k
ˆ ( , )x yk kd
Inversion and Time reversal symmetry require ( ) 0zd k
2D Dirac points at : point zeros in ( , )x yd d
( ) vH K q q Massless Dirac Hamiltonian
k K
-K +K
Berry’s phase around Dirac point
A
B
†
Ai Bj
ij
H t c c
Topological gapped phases in Graphene
1. Broken P : eg Boron Nitride
( ) v zH m K q q
ˆ# ( )n d ktimes wraps around sphere
m m
Break P or T symmetry :
2 2 2( ) | |vE m q q
Chern number n=0 : Trivial Insulator
2. Broken T : Haldane Model ’88
+K & -K
m m
Chern number n=1 : Quantum Hall state
+K
-K
2ˆ ( ) Sd k
2ˆ ( ) Sd k
Edge States Gapless states at the interface between topologically distinct phases
IQHE state
n=1
Egap
Domain wall
bound state 0
Fv ( ) ( )x x y y zH i m x
F
0
( ') '/ v
0 ( ) ~
x
y
m x dxik y
x e e
Vacuum
n=0
Edge states ~ skipping orbits
Lead to quantized transport
Chiral Dirac fermions are unique 1D states : “One way” ballistic transport, responsible for quantized
conductance. Insensitive to disorder, impossible to localize
Fermion Doubling Theorem :
Chiral Dirac Fermions can not exist in a purely 1D system.
0 F( ) vy yE k k
Band inversion transition : Dirac Equation
ky
E0
x
y
Chiral Dirac Fermions
m<0
m>0
n=1
m= m+
n=0
m= m+
in
|t|=1 disorder
Bulk - Boundary Correspondence
Bulk – Boundary Correspondence :
NR (NL) = # Right (Left) moving chiral fermion branches intersecting EF
N = NR - NL is a topological invariant characterizing the boundary.
N = 1 – 0 = 1
N = 2 – 1 = 1
E
ky K K’
EF
Haldane Model
E
ky K K’
EF
The boundary topological invariant
N characterizing the gapless modes Difference in the topological invariants
n characterizing the bulk on either side =
Generalizations
Higher Dimensions : “Bott periodicity” d → d+2
d=4 : 4 dimensional generalization of IQHE
( ) ( )ij i ju u d kA k k k
d F A A A
42
1[ ]
8Tr
Tn
F F
Boundary states : 3+1D Chiral Dirac fermions
Non-Abelian Berry connection 1-form
Non-Abelian Berry curvature 2-form
2nd Chern number = integral of 4-form over 4D BZ
no symmetry
chiral symmetry
Zhang, Hu ‘01
Topological Defects Consider insulating Bloch Hamiltonians that vary slowly in real space
defect line
s
( , )H H s k
2nd Chern number : 3 12
1[ ]
8Tr
T Sn
F F
Generalized bulk-boundary correspondence :
n specifies the number of chiral Dirac fermion modes bound to defect line
1 parameter family of 3D Bloch Hamiltonians
Example : dislocation in 3D layered IQHE
Gc 1
2cn
G B
Burgers’ vector
3D Chern number
(vector ┴ layers)
Are there other ways to engineer
1D chiral dirac fermions?
Teo, Kane ‘10