Eur. Phys. J. B (2016) 89: 133
https://doi.org/10.1140/epjb/e2016-60975-y

Regular Article

Comparing the full time-dependent Bogoliubov-de-Gennes equations to their linear approximation: a numerical investigation

Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle, 72076 Tübingen, Germany

^a e-mail: joni@seyrichs.de

Received: 22 December 2015
Received in final form: 6 April 2016
Published online: 25 May 2016

Abstract

In this paper we report on the results of a numerical study of the nonlinear time-dependent Bardeen-Cooper-Schrieffer (BCS) equations, often also denoted as Bogoliubov-de-Gennes (BdG) equations, for a one-dimensional system of fermions with contact interaction. We show that, even above the critical temperature, the full equations and their linear approximation give rise to completely different evolutions. In contrast to its linearization, the full nonlinear equation does not show any diffusive behavior in the order parameter. This means that the order parameter does not follow a Ginzburg-Landau-type of equation, in accordance with a recent theoretical result in [R.L. Frank, C. Hainzl, B. Schlein, R. Seiringer, to appear in Lett. Math. Phys., arXiv:1504.05885 (2016)]. We include a full description on the numerical implementation of the partial differential BCS/BdG equations.