Topological insulators Pavel Buividovich Regensburg Hall effect Classical

Topological insulators Pavel Buividovich (Regensburg)

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

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

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

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

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 magnetic Field? ? ?

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

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

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

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

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

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 CS number change = 2>m>0: n=-1 Massless fermions = 0>m>-2: n=1 Pinch at the surface -2>m : n = 0

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

Topological inequivalence of insulators

QHE and adiabatic pumping Consider the Quantum Hall state in cylindrical geometry ky is still a good quantum number Collection of 1 D 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

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

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 1 D periodic potentials Bloch Hamiltonians a Discrete levels at finite interval!!

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

Quantum theory of electric polarization King-Smith and Vanderbilt formula Polarization = Berry phase of 1 D 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 Z 2 TIs 1 D Hamiltonian Particle-hole symmetry Consider two PH-symmetric hamiltonians h 1(k) and h 2(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 Z 2 TIs • Particle-hole symmetry implies P(θ) = -P(2π - θ) • On periodic 1 D 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 Z 2 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

Relative Chern parity and level crossing Now consider 1 D 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 θ-term Once again, EM response for electrically polarized system Corresponding effective action For bulk Z 2 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 (Z 2 trivial) or theta = π (Z 2 nontrivial) • Odd number of localized states at the left/right boundary

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

(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

(4+1)D Chern insulators: Dirac models Critical values of mass (where massless modes exist) CS numbers Open boundary conditions in the 5 th dimension |C 2| 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

Z 2 classification of time-reversal invariant topological insulators in (3+1)D and in (2+1)D from (4+1)D Chern insulators Consider two 3 D hamiltonians h 1(k) and h 2(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 3 D TRI topinsulators Dimensional reduction from (4+1)D effective action In the bulk, P 3=1/2 theta-angle = π Electric current responds to the gradient of P 3 At the boundary, • Spatial gradient of P 3: Hall current • Time variation of P 3: current || B • P 3 is like “axion” (TME/CME) Response to electrostatic field near boundary Electrostatic potential A 0

Real 3 D topological insulator: Bi 1 -x. Sbx Band inversion at intermediate concentration

(4+1)D CSI Z 2 TRI in (3+1)D Z 2 TRI in (2+1 D) Consider two 2 D hamiltonians h 1(k) and h 2(k), Define extrapolation h(k, θ) is like 3 D Z 2 TI Z 2 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 h 1 and h 2 Berry curvature of φ vanishes on the boundary

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

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

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 1 D lattice, unit spacing TRIM: k = {±π, 0} Assume States at TRIM are always doubly degenerate Kramers degeneracy

Z 2 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 Z 2 invariant • Odd number of crossings = odd number of massless modes • Topologically protected (no smooth deformations remove)

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)

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

Topological Mott insulators Graphene tight-binding model with nearest- and next-nearest-neighbour interactions By tuning U, V 1 and V 2 we can generate an effective SO coupling. Not in real graphene, But what about artificial? Also, spin transport on the surface of 3 D Mott TI [Pesin, Balents’ 10]

Some useful references (and sources of pictures/formulas for this lecture : -) - “Primer on topological insulators”, A. Altland L. Fritz - “Topological insulator materials”, Y. Ando, Ar. Xiv: 1304. 5693 - “Topological field theory of time-reversal invariant insulators”, X. -L. Qi, T. L. Hughes, S. -C. Zhang, Ar. Xiv: 0802. 3537