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

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.Comment: 12 pages, 8 figure

Bose-Einstein condensation of photons with nonlocal nonlinearity in a dye-doped graded-index microcavity

We consider a microcavity made by a graded-index (GRIN) glass, doped by dye molecules, placed within two planar mirrors and study Bose-Einstein condensation (BEC) of photons. The presence of the mirrors leads to an effective photon mass, and the index grading provides an effective trapping frequency; the photon gas becomes formally equivalent to a two dimensional Bose gas trapped in an isotropic harmonic potential. The inclusion of nonlinear effects provides an effective interaction between photons. We discuss, in particular, thermal lensing effects and nonlocal nonlinearity, and quantitatively compare our results with the reported experimental data.Comment: 13 pages, 6 figure

Spin-wave spectrum of a two-dimensional itinerant electron system: Analytic results for the incommensurate spiral phase in the strong-coupling limit

We study the zero-temperature spin fluctuations of a two-dimensional itinerant-electron system with an incommensurate magnetic ground state described by a single-band Hubbard Hamiltonian. We introduce the (broken-symmetry) magnetic phase at the mean-field (Hartree-Fock) level through a \emph{spiral spin configuration} with characteristic wave vector \gmathbf{Q} different in general from the antiferromagnetic wave vector \gmathbf{Q_{AF}}, and consider spin fluctuations over and above it within the electronic random-phase (RPA) approximation. We obtain a \emph{closed} system of equations for the generalized wave vector and frequency dependent susceptibilities, which are equivalent to the ones reported recently by Brenig. We obtain, in addition, analytic results for the spin-wave dispersion relation in the strong-coupling limit of the Hubbard Hamiltonian and find that at finite doping the spin-wave dispersion relation has a \emph{hybrid form} between that associated with the (localized) Heisenberg model and that associated with the (long-range) RKKY exchange interaction. We also find an instability of the spin-wave spectrum in a finite region about the center of the Brillouin zone, which signals a physical instability toward a different spin- or, possibly, charge-ordered phase, as, for example, the stripe structures observed in the high-Tc materials. We expect, however, on physical grounds that for wave vectors external to this region the spin-wave spectrum that we have determined should survive consideration of more sophisticated mean-field solutions.Comment: 30 pages, 4 eps figure

Gap equation with pairing correlations beyond mean field and its equivalence to a Hugenholtz-Pines condition for fermion pairs

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 ultra-cold Fermi gas in the intermediate regime between BCS and BEC, as well as with the available quantum Monte Carlo data in the same regime.Comment: 24 pages, 13 figure

From superconducting fluctuations to the bosonic limit in the response functions above the critical temperature

We investigate the density, current, and spin response functions above the critical temperature for a system of three-dimensional fermions interacting via an attractive short-range potential. In the strong-coupling (bosonic) limit of this interaction, we identify the dominant diagrammatic contributions for a ``dilute'' system of composite bosons which form as bound-fermion pairs, and compare them with the usual (Aslamazov-Larkin, Maki-Thompson, and density-of-states) terms occurring in the theory of superconducting fluctuations above the critical temperature for a clean system in the weak-coupling limit. We show that, at the zeroth order in the diluteness parameter for the composite bosons, the Aslamazov-Larkin term still represents formally the dominant contribution to the density and current response functions, while the Maki-Thompson and density-of-states terms are strongly suppressed. Corrections to the Aslamazov-Larkin term are then considered at the next order in the diluteness parameter for the composite bosons. The spin response function is also examined, and it is found to be exponentially suppressed in the bosonic limit only when appropriate sets of diagrams are considered simultaneously.Comment: 10 pages, 6 figure

Density and spin response of a strongly-interacting Fermi gas in the attractive and quasi-repulsive regime

Recent experimental advances in ultra-cold Fermi gases allow for exploring response functions under different dynamical conditions. In particular, the issue of obtaining a "quasi-repulsive" regime starting from a Fermi gas with an attractive inter-particle interaction while avoiding the formation of the two-body bound state is currently debated. Here, we provide a calculation of the density and spin response for a wide range of temperature and coupling both in the attractive and quasi-repulsive regime, whereby the system is assumed to evolve non-adiabatically toward the "upper branch" of the Fermi gas. A comparison is made with the available experimental data for these two quantities.Comment: 8 pages, 7 figures, to appear on Phys. Rev. Let

Size shrinking of composite bosons for increasing density in the BCS to Bose-Einstein crossover

We consider a system of fermions in the continuum case at zero temperature, in the strong-coupling limit of a short-range attraction when composite bosons form as bound-fermion pairs. We examine the density dependence of the size of the composite bosons at leading order in the density ("dilute limit"), and show on general physical grounds that this size should decrease with increasing density, both in three and two dimensions. We then compare with the analytic zero-temperature mean-field solution, which indeed exhibits the size shrinking of the composite bosons both in three and two dimensions. We argue, nonetheless, that the two-dimensional mean-field solution is not consistent with our general result in the "dilute limit", to the extent that mean field treats the scattering between composite bosons in the Born approximation which is known to break down at low energy in two dimensions.Comment: Revised version to be published on Eur. Phys. Jour. B, 7 pages, 1 figur