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

Extracting the condensate density from projection experiments with Fermi gases

A debated issue in the physics of the BCS-BEC crossover with trapped Fermi atoms is to identify characteristic properties of the superfluid phase. Recently, a condensate fraction was measured on the BCS side of the crossover by sweeping the system in a fast (nonadiabatic) way from the BCS to the BEC sides, thus ``projecting'' the initial many-body state onto a molecular condensate. We analyze here the theoretical implications of these projection experiments, by identifying the appropriate quantum-mechanical operator associated with the measured quantities and relating them to the many-body correlations occurring in the BCS-BEC crossover. Calculations are presented over wide temperature and coupling ranges, by including pairing fluctuations on top of mean field.Comment: 4 pages, 4 figure

The influence of long-range hopping on ferromagnetism in the Hubbard model

The phase diagram of the Hubbard model in an external magnetic field is examined by extrapolation of small-cluster exact-diagonalization calculations. Using a general expression for the hopping matrix elements ($t_{ij}\sim q^{|i-j|}$) the influence of long-range hopping (band asymmetry) on ferromagnetism in this model is studied. It is found that the long-range hopping (nonzero $q$) stabilizes ferromagnetism in an external magnetic field for $n > 1$. In the opposite limit $n \leq 1$ the fully polarized ferromagnetic state is generally suppressed with increasing $q$. The critical value of magnetic field $h$ below which the ferromagnetic state becomes unstable is calculated numerically.Comment: 8 pages, 3 Postscript figures, Late

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

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

Temperature dependence of a vortex in a superfluid Fermi gas

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 novel 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 inter-particle interaction is of the contact type, and permits 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.Comment: 18 pgaes, 16 figure

Low density ferromagnetism in the Hubbard model

A single-band Hubbard model with nearest and next-nearest neighbour hopping is studied for $d=1$, 2, 3, using both analytical and numerical techniques. In one dimension, saturated ferromagnetism is found above a critical value of $U$ for a band structure with two minima and for small and intermediate densities. This is an extension of a scenario recently proposed by M\"uller--Hartmann. For three dimensions and non-pathological band structures, it is proven that such a scenario does not work.Comment: 4 pages, 3 postscript figure