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Final exam problems

Problem 1

Consider the tight binding Hamiltonian

H=-t*Sum_ijsigma[c_i,sigma⁺ c_j,sigma + h.c.] +U*Sum_in_i,upn_i,down+ V*Sum_i,jn_in_j

on a two-dimensional square lattice.The sum in the first and last terms is over nearest neighbor sites i,j only.

(a) Show that within BCS theory this H may give rise to superconductivity with (i) extended s-wave, (ii) p-wave, and (iii) d-wave symmetry. What is the form of the gap function in each case?

(b) Find equations determining T_c for each case.

(c) Discuss qualitatively for what parameter ranges you expect each solution to be favored.

Problem 2

Show explicitely that the order parameter approaches zero as (T_c-T)^1/2 as T approaches T_c for the following cases:
(a) Energy gap in BCS theory.
(b) Magnetization in Stoner theory.
(c) Magnetization in Weiss theory.

Problem 3

Consider a Hubbard model

H=-t*Sum_ijsigma[c_i,sigma⁺ c_j,sigma + h.c.] +U*Sum_in_i,upn_i,down

on a 3-dimensional simple cubic lattice, with a half-filled band (one electron per site) and U>0. The sum in the first term is over nearest neighbor sites i,j only.

(a) Show that mean field theory can describe the transition to an antiferromagnetic state in this model for any U>0. Find equations for the quasiparticle energy, quasiparticle operators and the self-consistency condition.
(b) Find an expression for the critical temperature in weak coupling assuming a constant density of states.

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