Microscopic derivation of Ginzburg-Landau theory

Abstract

We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

The External Field Dependence of the BCS Critical Temperature

“…Let us discuss the similarities and differences between this paper and our previous paper [6]. In [6] it was shown that for…”

“…In [6] this was shown for D > 0, but in [8] it was remarked that the same proof works even for D ≤ 0, provided the normalization of α * is changed accordingly. …”

“…Usually, the dependence on h is understood and we suppress it in the notation, abbreviating F T ( ) = F T,h ( ) and h = h h . We refer to [6] for an explanation why the ratio of scales parameter h enters into the free energy in the way it does and for a discussion of the fact that direct and exchange term of the particle interaction have been neglected. In this paper we are concerned with the minimization problem inf F T,h ( ) : admissible and, in particular, whether this infimum is realized for with α ≡ 0 (normal state) or with α ≡ 0 (superconducting state).…”

“…The particles, however, interact on the microscopic scale. The ratio between the microscopic and the macroscopic scales will be denoted by the small parameter h. Our main result in [6] about the connection between BCS and GL theory says that in the limit of small h the BCS free energy functional separates into two parts, namely, a translation invariant BCS functional describing the microscopic structure and a GL functional involving the macroscopic objects. In particular, if we normalize scales so that the macroscopic scale is of order one (and therefore the microscopic scale is of order h), the BCS-minimizing Cooper-pair wave function α is to leading order of the form 1) provided the temperature T is such that (T c − T )/T c is of order h 2 .…”

Multi-Vortex Non-radial Solutions to the Magnetic Ginzburg-Landau Equations

“…It can be derived from the microscopic theory due to Bardeen, Cooper and Schrieffer [8,10]. Stationary states of superconductors occupying (for simplicity) the plane R 2 , are described by pairs (ψ, A), where ψ : R 2 → C is the order parameter and A : R 2 → R 2 is the magnetic potential.…”