Topological insulators
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Topological insulators. Pavel Buividovich (Regensburg). Hall effect. Classical treatment. Dissipative motion for point-like particles ( Drude theory). Steady motion. Cyclotron frequency. Drude conductivity. Current. Resistivity tensor. Classical Hall effect.

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Topological insulators

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Topological insulators



Hall effect

Classical treatment

Dissipative motion for point-like particles (Drude theory)

Steady motion

Cyclotron frequency

Drude conductivity


Resistivity tensor

Classical Hall effect

  • Hall resistivity (off-diag component of resistivity tensor)

  • - Does not depend on disorder

  • Measures charge/density

  • of electric current carriers

  • - Valuable experimental tool

Clean system limit:


Importance of

matrix structure Naïve look at longitudinal components:


Classical Hall effect: boundaries

Conductance happens exclusively due to boundary states!

Otherwise an insulating state

Non-relativistic Landau levels

Model the boundary by a confining potential V(y) = mw2y2/2

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”


  • Hall state without magnetic

  • Field???

Quantum Hall Effect

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

Chern insulator [Haldane’88]

Open B.C. in y direction, numerical diagonalization

Chern insulator [Haldane’88]

Response to a weak electric field, V = -e E y

(Single-particle states)

Electric Current (system of multiple fermions)

Velocity operator




Quantum Hall effect: general formula

Quantum Hall effect and Berry fluxTKNN 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

Adiabatically time-dependent Hamiltonian H(t) = H[R(t)] with parameters R(t). For every t, define an eigenstate

However, does not solve the Schroedingerequation


Digression: Berry connection

Adiabatic evolution along the loop yields a nontrivial phase

Bloch momentum: also adiabatic parameter

General two-band Hamiltonian Projectors

Berry curvature in terms of projectors

  • Two-band Hamiltonian: mapping of sphere on the torus,


  • For the Haldane model

    • m>2: n=0

    • 2>m>0: n=-1

    • 0>m>-2: n=1

    • -2>m : n = 0

Example: two-band model

CS number change =

Massless fermions =

Pinch at the surface

Along with current, also charge density is generated

Response in covariant form

Electromagnetic response and effective action

Effective action for this response

Electromagnetic Chern-Simons

= Magnetic Helicity

Winding of

magnetic flux


Topological inequivalence of insulators

Consider the Quantum Hall state

in cylindrical geometry

ky is still a good quantum number

Collection of 1D Hamiltonians

QHE and adiabatic pumping

Switch on electric field Ey, Ay = - Ey t “Phase variable”

2 πrotation ofΦ , timeΔt = 2 π/ LyEy

Charge flow in this timeΔQ = σHΔt Ey Ly = CS/(2 π) 2 π = CS

Every cycle of Φ moves CS unit charges to the boundaries

More generally, consider a parameter-dependent Hamiltonian

Define the current response

Similarly to QHE derivation

QHE and adiabatic pumping


EM response

Classical dipole moment

But what is X for PBC???


X is not a good operator

Resta formula:

Quantum theory of electric polarization[King-Smith,Vanderbilt’93 (!!!)]

Model: electrons in 1D periodic potentials

Bloch Hamiltonians


Discrete levels at finite interval!!

Many-body fermionic theorySlater determinant

Quantum theory of electric polarization

King-Smith and Vanderbilt formula

Polarization =

Berry phase of 1D theory

(despite no curvature)

Quantum theory of electric polarization

  • 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

1D Hamiltonian Particle-hole symmetry

Consider two PH-symmetric hamiltoniansh1(k) and h2(k)

Define continuous interpolation


Now h(k,θ) can be assigned

the CS number

= charge flow in a cycle of θ

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 1 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

From (2+1)D Chern Insulators to (1+1)D Z2 TIs

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.)

Relative Chern parity and level crossing

Once again, EM response for electrically polarized system

Corresponding effective action

For bulk Z2 TI with periodic BC P(x) = 1/2

Relative Chern parity and θ-term

  • TI = Topological field theory in the bulk:

  • no local variation can changeΦ

  • 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 statesat the left/right boundary

Consider the 4D single-particle hamiltonianh(k)

Similarly to (2+1)D Chern insulator, electromagnetic response

C2 is the “Second Chern Number”

(4+1)D Chern insulators (aka domain wall fermions)

Effective EM action

Parallel E and B in 3D generate current along 5th dimension

In continuum space

Five (4 x 4) Dirac matrices:{Γµ , Γν} = 2 δµν

Lattice model = (4+1)D Wilson-Dirac fermions

In momentum space

(4+1)D Chern insulators: Dirac models

Critical values of mass CS numbers

(where massless modes exist)

(4+1)D Chern insulators: Dirac models

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

Consider two 3D hamiltonians

h1(k) and h2(k), Define extrapolation

“Magnetoelectric polarization”

Z2 classification of time-reversal invariant topological insulators in (3+1)D and in (2+1)Dfrom (4+1)D Chern insulators

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

Dimensional reduction from (4+1)D effective action

In the bulk, P3=1/2 theta-angle = π

Electric current responds to the gradient of P3

At the boundary,

Effective EM action of 3D TRI topinsulators

  • 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

Band inversion at intermediate concentration

Real 3D topological insulator: Bi1-xSbx

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 parametrizationsh(k,θ) and h’(k,θ)


between them

(4+1)D CSI Z2TRI in (3+1)D Z2TRI in (2+1D)

This is also interpolation between h1 and h2

Berry curvature of φ vanishes on the boundary

Periodic table of Topological Insulators

Chern invariants are only defined in odd dimensions

Time-reversal operator for Pauli electrons

Anti-unitary symmetry

Single-particle Hamiltonian in momentum space

(Bloch Hamiltonian)

If [h,θ]=0

Consider some eigenstate

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}


Kramers theorem

States at TRIM are always doubly degenerate

Kramers degeneracy

  • Contact || x between two (2+1)D Tis

  • kxis 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

Z2 classification of (2+1)D TI

  • Odd number of crossings = odd number of massless modes

  • Topologically protected (no smooth deformations remove)

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)

Kane-Mele model: role of SO coupling

Two edge states with opposite spins: left/up, right/down

Spin-momentum locking

Insensitive to disorder as long as

T is not violated

Magnetic disorder

is dangerous

Graphene tight-binding model with nearest- and

next-nearest-neighbour interactions

Topological Mott insulators

By tuning U, V1 and V2 we

can generate an effective SO


Not in real graphene,

But what about artificial?

Also, spin transport on the surface of 3D Mott TI


- “Primer on topological insulators”, A. Altland and L. Fritz

- “Topological insulator materials”, Y. Ando, ArXiv:1304.5693

- “Topological field theory of time-reversal invariant insulators”, X.-L. Qi, T. L. Hughes, S.-C. Zhang, ArXiv:0802.3537

Some useful references (and sources of pictures/formulas for this lecture :-)

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