Variational extensions of BCS theory

A variational principle is devised which optimizes the characteristic funct ion at thermodynamical equilibrium. The Bloch equation is used as a constra int to define the equilibrium state, and the trial quantities are an unnorm alized density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity yield for the latter a Bloch -like equation with an imaginary time running backwards. General conditions for the trial spaces are given that warrant the preservation of thermodyna mic relations. The connection with the standard minimum principle for therm odynamic potentials is discussed. We apply our variational principle to the derivation of equations which are tailored for (i) the consistent evaluati on of fluctuations and correlations and (ii) the restoration through projec tion of broken symmetries. When the trial spaces are chosen to be of the in dependent-quasi-particle type, we obtain an extension of the Hartree-Fock-B ogoliubov theory which optimizes the characteristic function. The expansion of the latter in powers of its sources yields for the fluctuations and cor relations compact formulae in which the RPA kernel emerges variationally. V ariational expressions for thermodynamic quantities or characteristic funct ions are also obtained with projected trial states, whether an invariance s ymmetry is broken or not. In particular, the projection on even or odd part icle number is worked out for a pairing Hamiltonian, which leads to new equ ations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature T, the level density and the strengt h of the pairing force, are investigated analytically and numerically. When the single-particle level spacing is small compared to the BCS gap Delta a t zero temperature, pairing correlations are effective, for both even and o dd projected cases, at all temperatures below the BCS critical temperature T-x. There exists a crossover temperature T, such that odd-even effects dis appear for T such that T-x < T < T-c. Below T-x, the free-energy difference between odd and even systems decreases quasi-linearly with T. The low temp erature entropy for odd systems has the Sackur-Tetrode form. When the level spacing is comparable with Delta, pairing in odd systems is predicted to t ake place only between two critical temperatures, thus exhibiting a reentra nce effect. (C) 1999 Elsevier Science B.V. All rights reserved.