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 v x,y from Heisenberg equations
Quantum Hall effect and Berry flux TKNN invariant Berry curvature Berry connection 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 CS number change = Massless fermions = Pinch at the surface m>2: n=0 2>m>0: n=-1 0>m>-2: n=1-2>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 k y is still a good quantum number Collection of 1D Hamiltonians Switch on electric field E y, A y = - E y t Phase variable 2 π rotation of Φ, time Δt = 2 π/ L y E y Charge flow in this time ΔQ = σ H Δt E y L y = 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 1D 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 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 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 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
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.)
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
(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 C 2 is the Second Chern Number Effective EM action Parallel E and B in 3D generate current along 5 th dimension
(4+1)D Chern insulators: Dirac models In continuum space Five (4 x 4) Dirac matrices: {Γ µ, Γ ν } = 2 δ µν 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) 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
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 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 3D 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 P3: Hall current Time variation of P3: current B P3 is like axion (TME/CME) Response to electrostatic field near boundary Electrostatic potential A 0
Real 3D topological insulator: Bi 1-x Sb x Band inversion at intermediate concentration
(4+1)D CSI Z2TRI in (3+1)D Z2TRI in (2+1D) Consider two 2D hamiltonians h 1 (k) and h 2 (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
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 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 k x is still good quantum number There will be some midgap states crossing zero At k x = 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 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, 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]
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