Microscopic Derivation of the Ginzburg-Landau Model

Abstract

Abstract. We present a summary of our recent rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic BardeenCooper-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

“…The validity of this approximation was discussed by Gor'kov [9] and, later, by de Gennes [7] and Eilenberger [2]. In our previous work [4] (see also [5,6]) we identified a precise parameter regime where this approximation is valid and we gave the first mathematical derivation of GL theory from BCS theory with quantitative error bounds. In this paper we continue our investigation and discuss the critical temperature in the BCS model.…”

“…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. …”

“…as h → 0. In [4] this was shown for D > 0, but in [6] it was remarked that the same proof works even for D ≤ 0, provided the normalization of α * is changed accordingly. The result (2.15) neither implies nor is implied by our Theorem 2.4 here.…”

“…Therefore, to complete the proof of Theorem 2.4 it remains to prove the following bound. 6 Lemma 6.2. Assume that an admissible Γ satisfies F T (Γ) ≤ F (0) T and define σ and ∆ by (6.14) and (6.19).…”

Differential Equations of Quantum Mechanics

“…In a remarkable work, R. Frank, Ch. Hainzl, R. Seiringer, J.P. Solovej ( [132]) have shown that, for non-dynamical magnetic fields, the nanoscopic approximation of the BdG system is given by the Ginzburg-Landau one (see also [133,134,72]).…”