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The superfluid weight is an important observable of superconducting materialssince it is related to the London penetration depth of the Meissner effect. Itcan be computed from the change in the grand potential (or free energy) inresponse to twisted boundary conditions in a torus geometry. Here we review theBardeen-Cooper-Schrieffer mean-field theory emphasizing its origin as avariational approximation for the grand potential. The variational parametersare the effective fields that enter in the mean-field Hamiltonian, namely theHartree-Fock potential and the pairing potential. The superfluid weight isusually computed by ignoring the dependence of the effective fields on thetwisted boundary conditions. However, it has been pointed out in recent worksthat this can lead to unphysical results, particularly in the case of latticemodels with flat bands. As a first result, we show that taking into account thedependence of the effective fields on the twisted boundary conditions leads infact to the generalized random phase approximation. Our second result isproviding the mean-field grand potential as an explicit function of theone-particle density matrix. This allows us to derive the expression for thesuperfluid weight within the generalized random phase approximation in atransparent manner. Moreover, reformulating mean-field theory as a well-posedminimization problem in terms of the one-particle density matrix is a firststep towards the application to superconducting systems of the linear scalingmethods developed in the context of electronic structure theory.

Superconductivity, generalized random phase approximation and linear scaling methods