Topological insulators

Pavel Buividovich(Regensburg)

Hall effect

Dissipative motion for point-like particles (Drude theory)

Classical treatment

Steady motion

Classical Hall effectCyclotron frequencyDrude conductivity Current

Resistivity tensor

Hall resistivity (off-diag component of resistivity tensor)

- Does not depend on disorder- Measures charge/densityof electric current carriers

- Valuable experimental tool

Classical Hall effect: boundariesClean 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

Quantum Hall EffectNon-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”

NOW THE QUESTION:Hall state without magneticField???

Chern insulator [Haldane’88]Originally, hexagonal lattice, but we consider square

Two-band model, similar to Wilson-Dirac [Qi, Wu, Zhang]

Phase diagramm=2 Dirac point at kx,ky=±πm=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

Quantum Hall effect: general formulaResponse 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

Integral of Berry curvature = multiple of 2π(wave function is single-valued on the BZ)

Berry curvature in terms of projectors

Quantum Hall effect and Berry flux

TKNN invariant

Berry curvature Berry connection

TKNN = Thouless, Kohmoto, Nightingale, den Nijs

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 phaseBloch momentum: also adiabatic parameter

Example: two-band model

Berry curvature in terms of projectors

General two-band Hamiltonian 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

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 HelicityWinding of magnetic flux lines

Topological inequivalence of insulators

QHE and adiabatic pumpingConsider the Quantum Hall statein cylindrical geometryky 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 = CSEvery cycle of Φ moves CS unit charges to the boundaries

QHE and adiabatic pumpingMore generally, consider a parameter-dependent Hamiltonian

Define the current responseSimilarly to QHE derivation

Polarization EM

response

Quantum theory of electric polarization

[King-Smith,Vanderbilt’93 (!!!)]Classical dipole momentBut what is X for PBC???

Mathematically, X is not a good operator

Resta formula:Model: electrons in 1D periodic potentials

Bloch Hamiltonians

aDiscrete levels at finite interval!!

Quantum theory of electric polarizationMany-body fermionic theory Slater

determinant

Quantum theory of electric polarizationKing-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

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

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

numbersConsider two interpolations h(k,θ) and

h’(k,θ)C[h(k, θ)]-C[h’(k,θ)] = 2 n

Relative Chern parity and level crossingNow 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 – θOdd CS zero level at π (assume θ=0 is a trivial insul.)

Relative Chern parity and θ-termOnce 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

(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

(4+1)D Chern insulators: Dirac modelsIn 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 modelsCritical 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

Charge flows into the bulk= (3+1)D anomaly

2 Weyl fermions = 1 Domain-wall fermion (Dirac)

Z2 classification of time-reversal invariant topological insulators in

(3+1)D and in (2+1)Dfrom (4+1)D Chern insulatorsConsider 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

Effective EM action of 3D TRI topinsulatorsDimensional reduction from (4+1)D effective

action

In the bulk, P3=1/2 theta-angle = πElectric 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

Real 3D topological insulator: Bi1-xSbx

Band inversion at intermediate concentration

(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 invariantThis 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

Periodic table of Topological Insulators

Chern invariants are only defined in odd dimensions

Kramers theoremTime-reversal operator for Pauli electrons

Anti-unitary symmetry

Single-particle Hamiltonian in momentum space (Bloch Hamiltonian)

If [h,θ]=0

Consider some eigenstate

Kramers theoremEvery 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 kFor 1D lattice, unit spacing

TRIM: k = {±π, 0}Assume

States at TRIM are always doubly degenerate Kramers degeneracy

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)

Kane-Mele model: role of SO couplingSimple 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 implementationHeavy adatom in the centre of hexagonal lattice(SO is big for heavy atomswith high orbitals occupied)

Spin-momentum lockingTwo edge states with opposite spins: left/up, right/down

Insensitive to disorder as long asT is not violated

Magnetic disorderis dangerous

Topological Mott insulatorsGraphene tight-binding model with nearest- and next-nearest-neighbour interactions

By tuning U, V1 and V2 we can generate an effective SOcoupling. Not in real graphene,But what about artificial?

Also, spin transport on the surface of 3D Mott TI[Pesin,Balents’10]

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:1304.5693

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