Incompatibility of time-dependent Bogoliubov--de-Gennes and Ginzburg--Landau equations

TL;DR

This paper demonstrates that the time-dependent Bogoliubov--de-Gennes equations do not align with the Ginzburg--Landau equations for certain initial states, highlighting the importance of their full non-linear structure.

Contribution

It shows that the time-dependent Bogoliubov--de-Gennes equations do not follow the Ginzburg--Landau dynamics near critical temperature, emphasizing the need to consider their full non-linear form.

Findings

Order parameter remains nearly constant over time.

Ginzburg--Landau equations predict decay of the order parameter.

Full non-linear structure is essential for accurate dynamics.

Abstract

We study the time-dependent Bogoliubov--de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg--Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time. The full non-linear structure of the equations is necessary to understand this behavior.