Problem: Frequency-Dependent Conductivity

      Consider a two-dimensional metal with a square lattice and a half-filled tight-binding band, . Assume that the mean free path, l, is independent of the wavenumber of the electrons.

   (a) Reduce the solution to the Boltzmann equation (Eq. ) to a two-dimensional electron system. Express the real and imaginary parts of the frequency-dependent conductivity and , respectively) in terms of , and the lattice spacing a.

(b) Calculate the area under the curve. Compare the result to the free-electron case. Discuss (qualitatively) why the oscillator sum rule is violated for this single-band model and how it can be restored.  

(c) Show that the real and imaginary parts of the conductivity satisfy the Kramers-Kronig relation.