Superconductivity, generalized random phase approximation and linear scaling methods

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