293 research outputs found
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
Systematic investigation of the effects of disorder at the lowest order throughout the BCS-BEC crossover
A systematic investigation of the effects of disorder on the BCS-BEC
crossover at the lowest order in the impurity potential is presented for the
normal phase above the critical temperature Tc. Starting with the t-matrix
approach for the clean system, by which pairing correlations between
opposite-spin fermions evolve from the weak-coupling (BCS) to the
strong-coupling (BEC) limits by increasing the strength of the attractive
inter-particle interaction, all possible diagrammatic processes are considered
where the effects of a disordered potential are retained in the self-energy at
the lowest order. An accurate numerical investigation is carried out for all
these diagrammatic terms, to determine which of them are mostly important
throughout the BCS-BEC crossover. Explicit calculations for the values of Tc,
the chemical potential, and the Tan's contact are carried out. In addition, the
effect of disorder on the single-particle spectral function is analyzed, and a
correlation is found between an increase of Tc and a widening of the pseudo-gap
energy at Tc on the BCS side of unitarity in the presence of disorder, while on
the BEC side of unitarity the presence of disorder favors the collapse of the
underlying Fermi surface. The present investigation is meant to orient future
studies when the effects of disorder will be considered at higher orders, with
the purpose of limiting the proliferation of diagrammatic terms in which
interaction and disorder are considered simultaneously.Comment: 20 pages, 21 figure
Non-local equation for the superconducting gap parameter
The properties are considered in detail of a non-local (integral) equation
for the superconducting gap parameter, which is obtained by a coarse-graining
procedure applied to the Bogoliubov-deGennes (BdG) equations over the whole
coupling-vs-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-vs-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
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
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
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
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