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arXiv:1707.07972 · Physical review. B./Physical review. B

Nonlocal equation for the superconducting gap parameter

Stefano Simonucci, Giancarlo Calvanese Strinati

Published Aug 3, 2017• 5 citations
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Abstract

The properties are considered in detail of a nonlocal (integral) equation for the superconducting gap parameter, which is obtained by a coarse-graining procedure applied to the Bogoliubov-de Gennes (BdG) equations over the whole coupling-versus-temperature phase diagram associated with the superfluid phase. It is found that the limiting size of the coarse-graining procedure, which is dictated by the range of the kernel of this integral equation, corresponds to the size of the Cooper pairs over the whole coupling-versus-temperature phase diagram up to the critical temperature, even when Cooper pairs turn into composite bosons on the BEC side of the BCS-BEC crossover. A practical method is further implemented to solve numerically this integral equation in an efficient way, which is based on a novel algorithm for calculating the Fourier transforms. Application of this method to the case of an isolated vortex, throughout the BCS-BEC crossover and for all temperatures in the superfluid phase, helps clarifying the nature of the length scales associated with a single vortex and the kinds of details that are in practice disposed off by the coarse-graining procedure on the BdG equations.

DOI
10.1103/physrevb.96.054502

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The properties are considered in detail of a nonlocal (integral) equation for the superconducting gap parameter, which is obtained by a coarse-graining procedure applied to the Bogoliubov-de Gennes (BdG) equations over the whole coupling-versus-temperature phase diagram associated with the superfluid phase. It is found that the limiting size of the coarse-graining procedure, which is dictated by the range of the kernel of this integral equation, corresponds to the size of the Cooper pairs over the whole coupling-versus-temperature phase diagram up to the critical temperature, even when Cooper pairs turn into composite bosons on the BEC side of the BCS-BEC crossover. A practical method is further implemented to solve numerically this integral equation in an efficient way, which is based on a novel algorithm for calculating the Fourier transforms. Application of this method to the case of an isolated vortex, throughout the BCS-BEC crossover and for all temperatures in the superfluid phase, helps clarifying the nature of the length scales associated with a single vortex and the kinds of details that are in practice disposed off by the coarse-graining procedure on the BdG equations.
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.
The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.
Most theoretical treatments of inhomogeneous superconductivity/fermionic superfluidity have been based on the Bogoliubov-deGennes equations (or, else, on their various simplified forms), which implement a standard mean-field decoupling in the presence of spatial inhomogeneities. This approach is reliable even at finite temperature for weak interparticle attraction, when the Cooper pair size is much larger than the average interparticle distance (corresponding to the BCS limit of the BCS-BEC crossover). However, it loses accuracy for increasing attraction when the Cooper pair size becomes comparable or even smaller than the average interparticle distance (corresponding to the BEC limit of the BCS-BEC crossover), in particular when finite-temperature effects are considered. In these cases, inclusion of pairing fluctuations beyond mean field is required, a task that turns out to be especially difficult in the presence of inhomogeneities. Here, we implement the inclusion of pairing fluctuations directly on a coarse-graining version of the Bogoliubov-deGennes equations, which makes it simpler and faster to obtain a solution over the whole sector of the temperature-coupling phase diagram of the BCS-BEC crossover in the broken-symmetry phase. We apply this method in the presence of a supercurrent flow, such that problems related to the Josephson effect throughout the BCS-BEC crossover can be addressed under a variety of circumstances. This is relevant in the view of recent experimental data with ultracold Fermi atoms, to which the results of the present approach are shown to favorably compare in the companion paper.
The proximity effect, which arises at the interface between two fermionic superfluids with different critical temperatures, is examined with a nonlocal (integral) equation whose kernel contains information about the size of Cooper pairs that leak across the interface. This integral approach avoids reference to the boundary conditions at the interface that would be required with a differential approach. The temperature dependence of the pair penetration depth on the normal side of the interface is determined over a wide temperature range also varying the interparticle coupling along the BCS side of the BCS-BEC crossover independently on both sides of the interface. In this way, the size of Cooper pairs evolves from being much larger than (BCS limit) the interparticle distance to being comparable with (unitarity limit, halfway between the BCS and BEC limits) the interparticle distance. Conditions are then found for which the proximity effect is optimized in terms of the extension of the pair penetration depth.
The effects of quantum confinement on the superconducting ground state are studied within the Bogoliubov–de Gennes (BdG) formalism and an attractive Hubbard model. We consider a periodic arrangement of two-dimensional superconducting grains composed by N × N atoms surrounded by insulating, metallic or superconducting stripes with a thickness of s atoms, leading to 2(N + s)2 coupled self-consistent BdG equations for a supercell of (N + s) × (N + s) atoms. These equations determine the spatial variation of superconducting gap as functions of temperature, electron–electron interaction, and hopping integrals analyzing three types of boundary stripes. The results show a clear enhancement of the superconducting gap and critical temperature induced by the electron confinement in the grain, being larger for the insulating boundary case. Finally, the numerical solutions of BdG equations are compared with those obtained by applying the BCS theory to each grain site.
The equation for the gap parameter represents the main equation of the pairing theory of superconductivity. Although it is formally defined through a single-particle property, physically, it reflects the pairing correlations between opposite-spin fermions. Here, we exploit this physical connection and cast the gap equation in an alternative form which explicitly highlights these two-particle correlations by showing that it is equivalent to a Hugenholtz-Pines condition for fermion pairs. At a formal level, a direct connection is established in this way between the treatment of the condensate fraction in condensate systems of fermions and bosons. At a practical level, the use of this alternative form of the gap equation is expected to make easier the inclusion of pairing fluctuations beyond mean field. As a proof-of-concept of the new method, we apply the modified form of the gap equation to the long-pending problem about the inclusion of the Gorkov-Melik-Barkhudarov correction across the whole BCS-BEC crossover, from the BCS limit of strongly overlapping Cooper pairs to the BEC limit of dilute composite bosons, and for all temperatures in the superfluid phase. Our numerical calculations yield excellent agreement with the recently determined experimental values of the gap parameter for an ultracold Fermi gas in the intermediate regime between BCS and BEC, as well as with the available quantum Monte Carlo data in the same regime.