8 M ar 1 99 7 An Explicit Solution of the BCS Model and Demonstration of Symmetry Breaking

An explicit solution of the BCS model, whose partition function is given by Tr e−βH(λ) where H(λ) = 1 Ld ∑ k,σ e(k)a + kσakσ + λ Ld 1 L2d ∑ k,p a + k↑a + −k↓a−p↓ap↑ is presented. The generating functional for the connected amputated Greens functions can be computed explicitely in arbitrary dimension d and for arbitrary energy momentum relation e(k). Symmetry breaking can be seen explicitely. That is, for attractive coupling limL→∞ limr→0〈ak,k0,↑a−k,−k0,↓〉β,r,L = 0 but limr→0 limL→∞〈ak,k0,↑a−k,−k0,↓〉β,r,L = ∆ k2 0+e(k) 2+∆2 where the expectation value 〈·〉β,r,L is taken with respect to Hr(λ) = H(λ)+ ∑ k[rak↑a−k↓+ r̄a + −k↓a + k↑] and ∆ is determined by the minimum of an effective potential. The more general interaction 1 L3d ∑