Equation for the superfluid gap obtained by coarse graining the Bogoliubov–de Gennes equations throughout the BCS-BEC crossover

Abstract

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.

Finite-temperature effective field theory for dark solitons in superfluid Fermi gases

“…The comparison of the results of the local gradient expansion with the BdG data for a vortex performed in Ref. [23] explicitly confirmed the expectation that a local gradient expansion can fail in the BCS regime at low temperature, when the size of the Cooper pairs is quite large and comparable to that of the vortex. As a result, the gradient expansion can fail at low temperatures in the BCS regime.…”

“…[34] confirms the aforesaid reasoning regarding the range of validity of the local gradient expansion. First, a good agreement with the BdG theory [23] is found except in the BCS regime at low temperatures. Second, the agreement of EFT with BdG becomes gradually better when the temperature rises from 0 to T c .…”

“…The key approximation of the EFT is the local gradient expansion. This expansion is a common ingredient of the EFT and several other approaches, e.g., the GL formalism and the recently developed coarsegraining approach for the BdG equations [23]. Already in early works [38,39], it was pointed out that the range of validity of the local gradient expansion must be restricted by the condition that the pair field is slowly varying over distances characterizing the spatial extent of the electron pair correlation (the coherence length).…”

“…As a result, analysis of the BdG solutions for dark solitons, at least at present, has been performed only in the zero-temperature case [11,21,22]. Computationally less demanding extensions of the BdG approach based on coarsegraining have been developed recently [23] but not yet applied to solitons, because in the present form it is not time dependent. This inspires attempts to develop complementary approaches for ultracold Fermi gases exploiting only a macroscopic wave function.…”

Verification of an analytic fit for the vortex core profile in superfluid Fermi gases

“…Consequently the use of the BdG theory is mainly limited to the consideration of zero-temperature properties of single-vortex states [10][11][12] . Because of this big computational cost of the BdG theory, there is a recent interest in the development of effective field theories [20][21][22][23][24][25][26] . These effective field theories allow for a description of non-uniform excitations (e.g.…”