Superconductivity, generalized random phase approximation and linear scaling methods

Abstract

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.

24 pages, 1 figure. Final published version with modified abstract and introduction to provide a better presentation of the results

References in corpus (7)

• Classification of topological insulators and superconductors in three spatial dimensions
• Superfluidity and Quantum Geometry in Twisted Multilayer Systems
• Band geometry, Berry curvature and superfluid weight
• Quantum breakdown of superconductivity in low-dimensional materials
• Pairing fluctuations and the superfluid density through the BCS-BEC crossover
• Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
• Pairing and superconductivity in quasi one-dimensional flat band systems: Creutz and sawtooth lattices

Cited by in corpus (5)

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• Geometric Stiffness in Interlayer Exciton Condensates
• Superfluid weight in the isolated band limit within the generalized random phase approximation
• Supersolid phase in two-dimensional soft-core bosons at finite temperature
• Superfluid weight in disordered flat-band superconductors as a competition between localization functionals