Variational extensions of BCS theory

Abstract

A variational principle is devised which optimizes the characteristic function at thermodynamical equilibrium. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are an unnormalized 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 thermodynamic relations. The connection with the standard minimum principle for thermodynamic potentials is discussed. We apply our variational principle to the derivation of equations which are tailored for (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. When the trial spaces are chosen to be of the independent-quasi-particle type, we obtain an extension of the Hartree–Fock–Bogoliubov theory which optimizes the characteristic function. The expansion of the latter in powers of its sources yields for the fluctuations and correlations compact formulae in which the RPA kernel emerges variationally. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature T, the level density and the strength of the pairing force, are investigated analytically and numerically. When the single-particle level spacing is small compared to the BCS gap Δ at zero temperature, pairing correlations are effective, for both even and odd projected cases, at all temperatures below the BCS critical temperature T c. There exists a crossover temperature T × such that odd–even effects disappear for T such that T ×< T< T c. Below T ×, the free-energy difference between odd and even systems decreases quasi-linearly with T. The low temperature entropy for odd systems has the Sackur–Tetrode form. When the level spacing is comparable with Δ, pairing in odd systems is predicted to take place only between two critical temperatures, thus exhibiting a reentrance effect.