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Problem 15

Consider BCS theory for a model with constant density of states:
g(epsilon)=1/D for -D/2 < epsilon < D/2
and attractive interaction V throughout the band.
(a) Find T_c as function of n=1+mu/(D/2) in weak coupling (n=number of electrons per site, mu=chemical potential)
(b) Find the gap at T=0 in weak coupling, and the gap ratio 2*gap/kT_c
(c) Find T_c, the T=0 gap and the gap ratio in strong coupling (V/D >> 1) for mu=0
(d) Calculate the gap ratio versus V/D numerically for mu=0.

Problem 16

Consider the operator
O=Sum_k(c_kup⁺c_kup - c_kdown⁺c_kdown)
(a) Show that average(O)=0 within BCS.
(b) Assume a perturbation is added to the Hamiltonian of the form
H'=lambda*O
Find an expression for average(O) versus temperature in the presence of this perturbation. Make a schematic plot of the temperature dependence and find the analytic form at low temperatures.

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