München 1997 – scientific programme

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MP: Theoretische und Mathematische Grundlagen der Physik

MP 9: Statistische Mechanik

MP 9.7: Fachvortrag

Thursday, March 20, 1997, 16:20–16:40, HS 117

An Explicit Solution of the BCS Model and Demonstration of Symmetry Breaking — •Detlef Lehmann — Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

An explicit solution of the BCS model, whose partition function is given by Tre^{−β H(λ)} with Hamilton operator H(λ)=1/L^d ∑ _k,σ e( k) a _kσ⁺ a _kσ +λ/L^d1/L^2d∑ _{k, p} a _k↑⁺ a_{− k↓}⁺ a_{− p↓}a _p↑ is presented. The generating functional for the connected amputated Greens functions can be computed explicitely in arbitrary dimension and for arbitrary energy momentum relation e( k). For e(k)=k²/2m−µ, the results coincide with those obtained by mean field theory. In particular, for attractive coupling the model shows symmetry breaking. That is, lim_L→∞lim_{r→ 0} ⟨ a_k,k0,↑ a_{−k,−k0,↓}⟩_β,r,L=0 but lim_{r→ 0}lim_L→∞⟨ a_k,k0,↑ a_{−k,−k0,↓}⟩_β,r,L=△/k₀²+e(k)²+|△|² where the expectation value ⟨·⟩_β,r,L is taken with respect to H_r(λ)=H(λ)+ra_k↑a_−k↓+r a_−k↓⁺a_k↑⁺ and |△| is determined by the minimum of an effective potential. For repulsive coupling, all connected Greens functions vanish in the infinite volume limit. The main feature which makes this model explicitely solvable is the fact that the interaction is given by a product of two quadratic factors (∑_k a_k↑⁺ a_−k↓⁺)(∑_p a_−p↓a_p↑). Possible generalizations to more complicated models are shortly discussed.