Problem 1
(a) How does the electron density of states change in the free electron model when a magnetic field is applied?

(a) How does the specific heat in the free electron model change when an magnetic field is applied?

(b) Why does the longitudinal resistivity $\rho_{xx}$ go to zero in the Quantum Hall effect?

Problem 2
(a) How is an impulse response function related to the generalized susceptibility?

(b) Explain how you can calculate the dielectric function of some material theoretically, how you could measure it experimentally, and how you could check it with the Kramers-Kronig relations.

(c) Draw the dielectric function for a semiconductor with a bandgap of 1 eV.

Problem 3
A metal-insulator transition can be induced by electron-electron interactions.

(a) Explain why electron-electron interactions are difficult to describe.

(b) A simple model for electron-electron interactions is electron screening. Explain what screening is and how it can be used to explain a Mott transition.

(c) Single electron effects are another simple way to include electron-electron interactions. Explain the single-electron charging effect.

Problem 4
The pyroelectric coefficient describes how the polarization of a crystal changes as the temperature changes.

(a) Assume Landau's theory of a second order phase transition can be used to describe polarization of the crystal. Sketch the pyroelectric coefficient as a function of temperature.

(b) If you measured the pryoelectric coefficient as a function of temperature and the electric susceptibilty as a function of temperature you would be able to predict the specific heat as a function of temperature. Explain how this is possible.