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Time-dependent Bogolubov--de-Gennes equations and non-validity of Ginzburg--Landau theory

Rupert L. Frank, Christian Hainzl, Benjamin Schlein, Robert Seiringer

Published Apr 22, 2015• 1 citation
OpenAlex

Abstract

We study the time-dependent Bogolubov--de-Gennes equations for generic translation-invariant fermionic many-body systems, and show that even slightly above the critical temperature the order parameter does not decay in time, in contrast to what was predicted previously in the literature on the basis of Ginzburg--Landau theory. The full non-linear structure of the equations is necessary to understand this behavior.

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We study the time-dependent Bogolubov--de-Gennes equations for generic translation-invariant fermionic many-body systems, and show that even slightly above the critical temperature the order parameter does not decay in time, in contrast to what was predicted previously in the literature on the basis of Ginzburg--Landau theory. The full non-linear structure of the equations is necessary to understand this behavior.
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.
Using a functional approach to the Keldysh formalism we develop a time-dependent Ginzburg-Landau equation for the long-wavelength and low-frequency dynamics of a weak-coupling superconductor, which is also valid in situations that deviate significantly from thermal equilibrium. The theory takes the interactions between quasiparticles and Cooper pairs into account exactly and is explicitly Galilean-invariant at zero temperature. However, the temporal and spatial dependence of the order parameter is assumed to be such that the effects of Landau damping can be neglected. We also consider the screening properties of the superconductor and show that the inclusion of the polarizability of the electron gas is necessary for a complete description.
We derive a nonlinear differential equation for the gap parameter of a superfluid Fermi system by performing a suitable coarse graining of the Bogoliubov--de Gennes (BdG) equations throughout the BCS-BEC crossover, with the aim of replacing the time-consuming solution of the original BdG equations by the simpler solution of this novel equation. We perform a favorable numerical test on the validity of this new equation over most of the temperature-coupling phase diagram, by an explicit comparison with the full solution of the original BdG equations for an isolated vortex. We also show that the new equation reduces both to the Ginzburg-Landau equation for Cooper pairs in weak coupling close to the critical temperature and to the Gross-Pitaevskii equation for composite bosons in strong coupling at low temperature.
In this paper we report on the results of a numerical study of the nonlinear time-dependent Bardeen--Cooper--Schrieffer (BCS) equation, often also denoted as Bogoliubov--de--Gennes (BdG) equation, for a one-dimensional system of fermions with contact interaction. We show that the full equation and its linear approximation give rise to completely different evolutions. This strongly contradicts the existence of a diffusive nonlinear time-dependent Ginzburg--Landau equation, in accordance with a recent theoretical result by Frank, Hainzl, Schlein, and Seiringer. We include a full description on the numerical implementation of the partial differential equations.
We construct a time-dependent Ginzburg-Landau (TDGL) theory for the superfluid atomic Fermi-gases near the Feshbach resonance from the fermion-boson model on the basis of the functional integral formalism. We show that the GL coefficients in the TDGL equation are complex numbers except in the BEC limit. The complex TDGL equation describes both damping and propagating dynamics, which leads to very rich superfluid dynamics near the Feshbach resonance. We predict multiple plane-wave modes corresponding to the in-phase and out-of-phase like oscillations between fermionic and bosonic superfluids and a galaxylike spiral pattern for a vortex.
The Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover is derived by the path-integral method. In addition to the standard Ginzburg-Landau equation, a second equation describing the total atom density is obtained. These two coupled equations are necessary to describe both homogeneous and inhomogeneous systems. The Ginzburg-Landau theory is valid near the transition temperature Tc{T}_{c}Tc​ on both sides of the crossover. In the weakly interacting BEC region, it is also accurate at zero temperature where the Ginzburg-Landau equation can be mapped onto the Gross-Pitaevskii (GP) equation. The applicability of GP equation at finite temperature is discussed. On the BEC side, the fluctuation of the order parameter is studied and the renormalization to the molecule coupling constant is obtained.