Download Solid State 3, Problem Set 2 Lecturer: Eytan Grosfeld

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.
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