1. Topological insulators
Pavel Buividovich
(Regensburg)

2. Hall effect Classical treatment
Dissipative motion for point-like particles (Drude theory)
Steady motion

3. Classical Hall effect Cyclotron frequency Drude conductivity Current
Resistivity tensor
Hall resistivity (off-diag component of resistivity tensor)
- Does not depend on disorder
Measures charge/density
of electric current carriers
- Valuable experimental tool

4. Classical Hall effect: boundaries
Clean system limit:
INSULATOR!!!
Importance of
matrix structure Naïve look at longitudinal components:
INSULATOR AND CONDUCTOR SIMULTANEOUSLY!!!
Conductance happens exclusively due to boundary states!
Otherwise an insulating state

5. Quantum Hall Effect Non-relativistic Landau levels
Model the boundary by a confining potential V(y) = mw2y2/2

6. Quantum Hall Effect Number of conducting states =
no of LLs below Fermi level
Hall conductivity σ ~ n
Pairs of right- and left- movers
on the “Boundary”
NOW THE QUESTION:
Hall state without magnetic
Field???

7. Chern insulator [Haldane’88]
Originally, hexagonal lattice, but we consider square
Two-band model, similar to Wilson-Dirac [Qi, Wu, Zhang]
Phase diagram
m=2 Dirac point at kx,ky=±π
m=0 Dirac points at (0, ±π), (±π,0)
m=-2 Dirac point at kx,ky=0

8. Chern insulator [Haldane’88]
Open B.C. in y direction, numerical diagonalization

9. Quantum Hall effect: general formula
Response to a weak electric field, V = -e E y
(Single-particle states)
Electric Current (system of multiple fermions)
Velocity operator
vx,y from
Heisenberg
equations

10. Quantum Hall effect and Berry flux TKNN invariant
Berry connection
Berry curvature
Integral of Berry curvature = multiple of 2π
(wave function is single-valued on the BZ)
Berry curvature in terms of projectors
TKNN = Thouless, Kohmoto, Nightingale, den Nijs

11. Digression: Berry connection
Adiabatically time-dependent Hamiltonian H(t) = H[R(t)] with parameters R(t). For every t, define an eigenstate
However, does not solve the Schroedinger equation
Substitute
Adiabatic evolution along the loop yields a nontrivial phase
Bloch momentum: also adiabatic parameter

12. Example: two-band model
General two-band Hamiltonian Projectors
Berry curvature in terms of projectors
Two-band Hamiltonian: mapping of sphere on the torus,
VOLUME ELEMENT
For the Haldane model
m>2: n=0
2>m>0: n=-1
0>m>-2: n=1
-2>m : n = 0
CS number change =
Massless fermions =
Pinch at the surface

13. Electromagnetic response and effective action
Along with current, also charge density is generated
Response in covariant form
Effective action for this response
Electromagnetic Chern-Simons
= Magnetic Helicity
Winding of
magnetic flux
lines

14. Topological inequivalence of insulators

15. QHE and adiabatic pumping
Consider the Quantum Hall state
in cylindrical geometry
ky is still a good quantum number
Collection of 1D Hamiltonians
Switch on electric field Ey, Ay = - Ey t “Phase variable”
2 π rotation of Φ , time Δt = 2 π/ Ly Ey
Charge flow in this time ΔQ = σH Δt Ey Ly = CS/(2 π) 2 π = CS
Every cycle of Φ moves CS unit charges to the boundaries

16. QHE and adiabatic pumping
More generally, consider a parameter-dependent Hamiltonian
Define the current response
Similarly to QHE derivation
Polarization
EM response

17. Model: electrons in 1D periodic potentials
Quantum theory of electric polarization [King-Smith,Vanderbilt’93 (!!!)]
Classical dipole moment
But what is X for PBC???
Mathematically,
X is not a good operator
Resta formula:
Model: electrons in 1D periodic potentials
Bloch Hamiltonians a
Discrete levels at finite interval!!

18. Quantum theory of electric polarization
Many-body fermionic theory Slater determinant

19. Quantum theory of electric polarization
King-Smith and Vanderbilt formula
Polarization =
Berry phase of 1D theory
(despite no curvature)
Formally, in tight-binding models X is always integer-valued
BUT: band structure implicitly remembers about continuous
space and microscopic dipole moment
We can have e.g. Electric Dipole Moment
for effective lattice Dirac fermions
In QFT, intrinsic property
In condmat, emergent phenomenon
C.F. lattice studies of CME

20. From (2+1)D Chern Insulators to (1+1)D Z2 TIs
1D Hamiltonian Particle-hole symmetry
Consider two PH-symmetric hamiltonians h1(k) and h2(k)
Define continuous interpolation
For
Now h(k,θ) can be assigned
the CS number
= charge flow in a cycle of θ

21. From (2+1)D Chern Insulators to (1+1)D Z2 TIs
Particle-hole symmetry implies P(θ) = -P(2π - θ)
On periodic 1D lattice of unit spacing,
P(θ) is only defined modulo P(θ) +P(2π - θ) = 0 mod 1
P(0) or P(π) = 0 or ½ Z2 classification
Relative parity of CS numbers
Generally, different h(k,θ) = different CS numbers
Consider two interpolations h(k,θ) and h’(k,θ)
C[h(k, θ)]-C[h’(k,θ)] = 2 n

22. Relative Chern parity and level crossing
Now consider 1D Hamiltonians with open boundary conditions
CS = numer of left/right zero level crossings in [0, 2 π]
Particle-hole symmetry: zero level at θ also at 2 π – θ
Odd CS zero level at π (assume θ=0 is a trivial insul.)

23. Relative Chern parity and θ-term
Once again, EM response for electrically polarized system
Corresponding effective action
For bulk Z2 TI with periodic BC P(x) = 1/2
TI = Topological field theory in the bulk:
no local variation can change Φ
Current can only flow at the boundary where P changes
Theta angle = π, Charge conjugation only allows
theta = 0 (Z2 trivial) or theta = π (Z2 nontrivial)
Odd number of localized states at the left/right boundary

24. (4+1)D Chern insulators (aka domain wall fermions)
Consider the 4D single-particle hamiltonian h(k)
Similarly to (2+1)D Chern insulator, electromagnetic response
C2 is the “Second Chern Number”
Effective EM action
Parallel E and B in 3D generate current along 5th dimension

25. (4+1)D Chern insulators: Dirac models
In continuum space
Five (4 x 4) Dirac matrices: {Γµ , Γν} = 2 δµν
Lattice model = (4+1)D Wilson-Dirac fermions
In momentum space

26. (4+1)D Chern insulators: Dirac models
Critical values of mass CS numbers
(where massless modes exist)
Open boundary conditions in the 5th dimension
|C2| boundary modes on the left/on the right boundaries
Effective boundary Weyl Hamiltonians
2 Weyl fermions =
1 Domain-wall fermion (Dirac)
Charge flows into the bulk
= (3+1)D anomaly

27. Z2 classification of time-reversal invariant topological insulators in (3+1)D and in (2+1)D from (4+1)D Chern insulators
Consider two 3D hamiltonians
h1(k) and h2(k), Define extrapolation
“Magnetoelectric polarization”
Time-reversal implies P(θ) = -P(2π - θ)
P(θ) is only defined modulo 1 => P(θ) +P(2π - θ) = 0 mod 1
P(0) or P(π) = 0 or ½ => C[h(k, θ)]-C[h’(k,θ)] = 2 n

28. Effective EM action of 3D TRI topinsulators
Dimensional reduction from (4+1)D effective action
In the bulk, P3=1/ theta-angle = π
Electric current responds to the gradient of P3
At the boundary,
Spatial gradient of P3: Hall current
Time variation of P3: current || B
P3 is like “axion” (TME/CME)
Response to electrostatic field near boundary
Electrostatic potential A0

29. Real 3D topological insulator: Bi1-xSbx
Band inversion at intermediate concentration

30. (4+1)D CSI Z2TRI in (3+1)D Z2TRI in (2+1D)
Consider two 2D hamiltonians
h1(k) and h2(k), Define extrapolation
h(k,θ) is like 3D Z2 TI Z2 invariant
This invariant does not depend on parametrization?
Consider two parametrizations h(k,θ) and h’(k,θ)
Interpolation
between them
This is also interpolation between h1 and h2
Berry curvature of φ vanishes on the boundary

31. Periodic table of Topological Insulators
Chern invariants are only defined in odd dimensions

32. Kramers theorem Time-reversal operator for Pauli electrons
Anti-unitary symmetry
Single-particle Hamiltonian in momentum space
(Bloch Hamiltonian)
If [h,θ]=0
Consider some eigenstate

33. Kramers theorem Every eigenstate has a partner at (-k)
With the same energy!!!
Since θ changes spins, it cannot be
Example: TRIM
(Time Reversal Invariant Momenta)
-k is equivalent to k
For 1D lattice, unit spacing
TRIM: k = {±π, 0}
Assume
States at TRIM are always doubly degenerate
Kramers degeneracy

34. Z2 classification of (2+1)D TI
Contact || x between two (2+1)D Tis
kx is still good quantum number
There will be some midgap states crossing zero
At kx = 0, π (TRIM) double degeneracy
Even or odd number of crossings Z2 invariant
Odd number of crossings = odd number of massless modes
Topologically protected (no smooth deformations remove)

35. Kane-Mele model: role of SO coupling
Simple theoretical model for (2+1)D TRI topological insulator
[Kane,Mele’05]: graphene with strong spin-orbital coupling
- Gap is opened
- Time reversal is not broken
- In graphene, SO coupling
is too small
Possible physical implementation
Heavy adatom in the
centre of hexagonal lattice
(SO is big for heavy atoms
with high orbitals occupied)

36. Spin-momentum locking
Two edge states with opposite spins: left/up, right/down
Insensitive to disorder as long as
T is not violated
Magnetic disorder
is dangerous

37. Topological Mott insulators
Graphene tight-binding model with nearest- and
next-nearest-neighbour interactions
By tuning U, V1 and V2 we
can generate an effective SO
coupling.
Not in real graphene,
But what about artificial?
Also, spin transport on the surface of 3D Mott TI
[Pesin,Balents’10]

38. Some useful references (and sources of pictures/formulas for this lecture :-)
- “Primer on topological insulators”, A. Altland and L. Fritz
- “Topological insulator materials”, Y. Ando, ArXiv:
- “Topological field theory of time-reversal invariant insulators”, X.-L. Qi, T. L. Hughes, S.-C. Zhang, ArXiv: