Superconductivity, generalized random phase approximation and linear scaling methods

The superfluid weight is an important observable of superconducting materials since it is related to the London penetration depth of the Meissner effect. It can be computed from the change in the grand potential (or free energy) in response to twisted boundary conditions in a torus geometry. Here we review the Bardeen-Cooper-Schrieffer mean-field theory emphasizing its origin as a variational approximation for the grand potential. The variational parameters are the effective fields that enter in the mean-field Hamiltonian, namely the Hartree-Fock potential and the pairing potential. The superfluid weight is usually computed by ignoring the dependence of the effective fields on the twisted boundary conditions. However, it has been pointed out in recent works that this can lead to unphysical results, particularly in the case of lattice models with flat bands. As a first result, we show that taking into account the dependence of the effective fields on the twisted boundary conditions leads in fact to the generalized random phase approximation. Our second result is providing the mean-field grand potential as an explicit function of the one-particle density matrix. This allows us to derive the expression for the superfluid weight within the generalized random phase approximation in a transparent manner. Moreover, reformulating mean-field theory as a well-posed minimization problem in terms of the one-particle density matrix is a first step towards the application to superconducting systems of the linear scaling methods developed in the context of electronic structure theory.