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Solid State 3, Problem Set 2 Lecturer: Eytan Grosfeld Linear response theory 1. Density response function Calculate the imaginary part of the density reponse function χ(ω, q) for the clean 1D tight-binding chain (use the exact spectrum). 2. Transport on the surface of a topological insulator Electrons confined to the two-dimensional surface of a topological insulator tuned to the Dirac point are described by the continuum limit Hamiltonian H = vσ · p where σa are Pauli matrices (a = x, y) related to the electronic spin and v is a velocity. The momentum p is two-dimensional. Assume half filling and zero temperature. (a) Diagonalize the Hamiltonian and write the (two-component) wavefunctions associated with a given momentum k and either positive or negative Write an expression for the density operator ´ 2energies. iq·r ρ(q) = d r e ρ(r) [where ρ(r) = ψ↑† (r)ψ↑ (r) + ψ↓† (r)ψ↓ (r)] in the basis of the eigenstates of the Hamiltonian. (b) We now apply an electric field: calculate using the Golden rule the linear response to an electric field E(q → 0, ω) and extract the real (dissipative) part of the AC longitudinal conductivity σxx (ω) of the system in the limit ω → 0. 1