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

TL;DR

This study numerically compares the nonlinear time-dependent Bogoliubov--de-Gennes equations with their linear approximation in a 1D fermionic system, revealing significant differences in dynamics even above the critical temperature.

Contribution

It provides a detailed numerical analysis showing that the full nonlinear equations behave differently from linearized versions, especially regarding diffusive behavior of the order parameter.

Findings

Full equations do not exhibit diffusive behavior above critical temperature

Linear approximation predicts diffusive evolution of the order parameter

Full nonlinear equations do not follow Ginzburg--Landau-type dynamics

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 results. We include a full description on the numerical implementation of the partial differential BCS\ BdG equations.