Boundary Superconductivity in the BCS Model

Abstract

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the onedimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg-Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.

Universality in low-dimensional BCS theory

“…Apart from the universality discussed here, also many other properties of BCS theory have been shown using this formulation: Most prominently, Ginzburg-Landau theory, as an effective theory describing superconductors close to the critical temperature, has been derived from BCS theory [DHM22a; DHM22b; FHSS12; FL16]. More recently, it has been shown that the effect of boundary superconductivity occurs in the BCS model [HRS22]. We refer to [HS16] for a more comprehensive review of developments in the mathematical formulation of BCS theory.…”

Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice

“…A closely related research direction is superconductivity: the existence of a boundary leads to boundary states in a superconductor with a higher critical temperature than the one of the bulk [SB20, SB21,HRS]. In this spirit, it seems very promising to also study the interplay of geometry and many-particle phenomena on Archimedean tilings.…”