Organizing Principles for Understanding Matter
Symmetry
Topology
Interplay between symmetry and topology has led to a new understanding of electronic phases of matter.
• Conceptual simplification
• Conservation laws
• Distinguish phases of matter by pattern of broken symmetries
• Properties insensitive to smooth deformation
• Quantized topological numbers
• Distinguish topological phases of matter
genus = 0 genus = 1
symmetry group p31msymmetry group p4
Topology and Quantum Phases
Topological Equivalence : Principle of Adiabatic Continuity
Quantum phases with an energy gap are topologically equivalent if they can be smoothly deformed into one another without closing the gap.
Topologically distinct phases are separated by quantum phase transition.
Topological Band Theory
Describe states that are adiabatically connected to non interacting fermions
Classify single particle Bloch band structuresEg ~ 1 eV
Band Theory of Solidse.g. Silicon
E
adiabatic deformation
excited states
topological quantumcritical point
GapEG
Ground state E0
( ) : Bloch Hamiltonans Brillouin zone (torwith ener
usgy
) gap
H k
E
k
Topological Electronic Phases
Many examples of topological band phenomena
States adiabatically connected to independent electrons:
- Quantum Hall (Chern) insulators - Topological insulators - Weak topological insulators - Topological crystalline insulators - Topological (Fermi, Weyl and Dirac) semimetals …..
Beyond Band Theory: Strongly correlated statesState with intrinsic topological order
- fractional quantum numbers - topological ground state degeneracy - quantum information
- Symmetry protected topological states - Surface topological order ……
Many real materialsand experiments
Topological Superconductivity
Much recent conceptual progress, but theory is
still far from the real electrons
Proximity induced topological superconductivity
Majorana bound states, quantum information
Tantalizing recent experimental progress
Topological Band Theory
Lecture #1: Topology and Band Theory
Lecture #2: Topological Insulators in 2 and 3 dimensions
Topological Semimetals
Lecture #3: Topological Superconductivity
Majorana Fermions
Topological quantum computation
General References :
“Colloquium: Topological Insulators” M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)
“Topological Band Theory and the Z2 Invariant,” C. L. Kane in “Topological insulators” edited by M. Franz and L. Molenkamp, Elsevier, 2013.
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
- Weyl semimetal
- Higher dimensions
- Topological Defects
The Insulating State
The Integer Quantum Hall State
g cE
2D Cyclotron Motion, sxy = e2/h
E
Insulator vs Quantum Hall state
What’s the difference? Distinguished by Topological Invariant
0 p/a-p/a
E
0 p/a-p/a/h e
a
atomic insulator
atomic energylevels
Landaulevels
3p
4s
3s
2
3
1
gE
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 g 1 2
1
r r
Gauss Bonnet Theorem :
2
10
r 0
0
A good math book : Nakahara, ‘Geometry, Topology and Physics’
Band Theory of SolidsBloch 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 p/a-p/a
Egap
=kx
ky
p/a
p/a
-p/a
-p/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 kA k k
( ) kA A k
Berry phase : change in phase on a closed loop C C Cd A k
Berry curvature : kF A 2C S
d k FFamous example : eigenstates of 2 level Hamiltonian
( ) ( ) z x y
x y z
d d idH
d id d
k d k
( ) ( ) ( ) ( )H u uk k d k k 1 ˆSolid Angle swept out by ( )2C d k
C
d̂
S
Topology in one dimension : Berry phase and electric polarization
Classical electric polarization :
+Q-Q1D insulator
Proposition: The quantum polarization is a Berry phase
( )2 BZ
eP A k dk
see, e.g. Resta, RMP 66, 899 (1994)
k
-p/a
p/a0
dipole moment
lengthP
bound P Bound charge density
End charge ˆendQ P n
( ) ( )i u u kA k k
BZ = 1D Brillouin Zone = S1
Circumstantial evidence #1 :
• The end charge is not completely determined by the bulk polarization P because integer charges can be added or removed from the ends :
• The Berry phase is gauge invariant under continuous gauge transformations, but is not gauge invariant under “large” gauge transformations.
( )( ) ( )i ku k e u kP P en ( / ) ( / ) 2a a n
Changes in P, due to adiabatic variation are well defined and gauge invariant
( ) ( , ( ))u k u k t
1 0 2 2C S
e eP P P dk dkd
A F
when with
k
-p/a p/a
l1
0
Q P eend mod
C
S
gauge invariant Berry curvature
The polarization and the Berry phase share the same ambiguity:
They are both only defined modulo an integer.
Circumstantial evidence #2 :
( ) ( ) ( ) ( )2 2 kBZ BZ
dk ieP e u k r u k u k u k
kr i
A slightly more rigorous argument:
Construct Localized Wannier Orbitals :
( )( ) ( )2
ik R r
BZ
dkR e u k
( ) ( )
( ) ( )2 kBZ
P e R r R R
ieu k u k
Wannier states are gauge dependent, but for a sufficiently smooth gauge, they are localized states associated with a Bravais Lattice point R
( )R r
R
“ ”
R
( )R r
r
Su Schrieffer Heeger Model model for polyacetalenesimplest “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
p/a-p/a
Provided symmetry requires dz(k)=0, the states with dt>0 and dt<0 are distinguished byan integer winding number. Without extra symmetry, all 1D band structures are topologically equivalent.
A,iB,i
dt>0 : Berry phase 0
P = 0
dt<0 : Berry phase pP = e/2
Gap 4|dt|
Peierl’s instability → dt
A,i+1
“Chiral” Symmetry :
Reflection Symmetry :
Symmetries of the SSH model
• Artificial symmetry of polyacetylene. Consequence of bipartite lattice with only A-B hopping:
• Requires dz(k)=0 : integer winding number
• Leads to particle-hole symmetric spectrum:
( ), 0 ( ) ( ) (or )z z zH k H k H k
iA iA
iB iB
c c
c c
z E z E z E EH E
( ) ( )x xH k H k
• Real symmetry of polyacetylene.
• Allows dz(k)≠0, but constrains dx(-k)= dx(k), dy,z(-k)= -dy,z(k)
• No p-h symmetry, but polarization is quantized: Z2 invariant
P = 0 or e/2 mod e
Domain Wall StatesAn interface between different topological states has topologically protected midgap states
Low energy continuum theory :For small dt focus on low energy states with k~p/a xk q q i
a
;
( )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 y0
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
-p/a p/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 bywhich 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 InvariantThouless, Kohmoto, Nightingale and den Nijs 82
View cylinder as 1D system with subbands labeled by0
1( )m
yk mR
0
0
, ( )2
mx x y
m
eQ d dk k k ne
F
kx
kyTKNN number = Chern number
21 1( )
2 2 CBZ
n d k d
F k A k
2
xy
en
h
kym(0)
kym(f0)
Distinguishes topologically distinct 2D band structures. Analogous to Gauss-Bonnet thm.
Alternative calculation: compute sxy via Kubo formula
C
kx
( ) , ( )mm x x yE k E k k
TKNN Invariant Thouless, Kohmoto, Nightingale and den Nijs 82
Physical meaning: Hall conductivity
Laughlin Argument: Thread magnetic flux f0 = h/e through a 1D cylinder Polarization changes by sxy f0
21( )
2 BZ
d k
F k
2
xy
en
h
1 2
1 1
2 2C Cn d d
A k A k
( ) ( ) ( )i u u kΑ k k k
kx
ky
C1
C2
p/a
p/a
-p/a
-p/a
For a 2D band structure, define
E Iyk P ne
ne-ne
Graphene E
k
Two band model
Novoselov et al. ‘05www.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 vortices in ( , )x yd d
( ) vH K q q
Massless Dirac Hamiltonian
k K-K +K
Berry’s phase p around Dirac point
A
B
†Ai Bj
ij
H t c c
3
1
ˆ ˆ( ) cos sinj jj
t x y
d k k r k r
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 StatesGapless states at the interface between topologically distinct phases
IQHE staten=1
Egap
Domain wallbound state y0
Fv ( ) ( )x x y y zH i k m x
F
0
( ') '/ v
0 ( ) ~
x
y
m x dxik yx e e
Vacuumn=0
Edge states ~ skipping orbitsLead 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=1m-= -m+
n=0m-= +m+
yin
|t|=1disorder
Bulk - Boundary Correspondence
Bulk – Boundary Correspondence :
NR (NL) = # Right (Left) moving chiral fermion branches intersecting EF
DN = NR - NL is a topological invariant characterizing the boundary.
DN = 1 – 0 = 1
DN = 2 – 1 = 1
E
kyKK’
EF
Haldane Model
E
kyKK’
EF
The boundary topological invariant DN characterizing the gapless modes
Difference in the topological invariantsDn characterizing the bulk on either side=
Weyl Semimetal
Gapless “Weyl points” in momentum space are topologically protected in 3D
A sphere in momentum space can have a Chern number:
2 ( )S Sn d k F k
S
nS=+1: S must enclose a degenerate Weyl point: Magnetic monopole for Berry flux
0( ) )v( x x y y z zH k q q q q [ ] 0( or v with det v )ia i a iaq
k
E
kx,y,z
Total magnetic charge in Brillouin zone must be zero: Weyl pointsmust come in +/- pairs.
Surface Fermi Arc
n1=n2=0
n0=1
k0
k1
k2kz
kx
ky
E
ky
EF
kz=k1, k2
E
ky
EF
kz=k0
Surface BZ
kz
ky
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 DefectsConsider 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
2 cn
G B
Burgers’ vector
3D Chern number
(vector ┴ layers)
Are there other ways to engineer1D chiral dirac fermions?
Teo, Kane ‘10