The BCSBEC crossover: From ultracold Fermi gases to nuclear systems
Abstract
This report aims at covering the overlap between ultracold Fermi gases and nuclear matter in the context of the BCSBEC crossover. By this crossover, the phenomena of BardeenCooperSchrieffer (BCS) superfluidity and BoseEinstein condensation (BEC), which share the same kind of spontaneous symmetry breaking, are smoothly connected through the progressive reduction of the size of the fermion pairs involved as the fundamental entities in both phenomena. This size ranges, from large values when Cooper pairs are strongly overlapping in the BCS limit of a weak interparticle attraction, to small values when composite bosons are nonoverlapping in the BEC limit of a strong interparticle attraction, across the intermediate unitarity limit where the size of the pairs is comparable with the average interparticle distance.
The BCSBEC crossover has recently been realized experimentally, and essentially in all of its aspects, with ultracold Fermi gases. This realization, in turn, has raised the interest of the nuclear physics community in the crossover problem, since it represents an unprecedented tool to test fundamental and unanswered questions of nuclear manybody theory. Here, we focus on the several aspects of the BCSBEC crossover, which are of broad joint interest to both ultracold Fermi gases and nuclear matter, and which will likely help to solve in the future some open problems in nuclear physics (concerning, for instance, neutron stars). Similarities and differences occurring in ultracold Fermi gases and nuclear matter will then be emphasized, not only about the relative phenomenologies but also about the theoretical approaches to be used in the two contexts. Common to both contexts is the fact that at zero temperature the BCSBEC crossover can be described at the meanfield level with reasonable accuracy. At finite temperature, on the other hand, inclusion of pairing fluctuations beyond mean field represents an essential ingredient of the theory, especially in the normal phase where they account for precursor pairing effects.
After an introduction to present the key concepts of the BCSBEC crossover, this report discusses the meanfield treatment of the superfluid phase, both for homogeneous and inhomogeneous systems, as well as for symmetric (spin or isospinbalanced) and asymmetric (spin or isospinimbalanced) matter. Pairing fluctuations in the normal phase are then considered, with their manifestations in thermodynamic and dynamic quantities. The last two Sections provide a more specialized discussion of the BCSBEC crossover in ultracold Fermi gases and nuclear matter, respectively. The separate discussion in the two contexts aims at cross communicating to both communities topics and aspects which, albeit arising in one of the two fields, share a strong common interest.
Contents
 1 Introduction

2 BCS mean field
 2.1 The homogeneous infinite matter case for a contact interaction
 2.2 Solution at for a contact potential
 2.3 Extension to finite temperature
 2.4 Bogoliubovde Gennes and HartreeFockBogoliubov equations for inhomogeneous Fermi systems
 2.5 GinzburgLandau and GrossPitaevskii equations
 2.6 Spinimbalanced (polarized) systems

3 Pairing fluctuations
 3.1 NozièresSchmittRink approach and its extensions
 3.2 Intra and interpair correlations
 3.3 Singleparticle spectral function and pseudogap
 3.4 Gor’kovMelikBarkhudarov (screening) corrections
 3.5 NSR theory in nuclear physics and the approach by Zimmermann and Stolz
 3.6 Residual interaction among composite bosons
 3.7 Pairing fluctuations in polarized Fermi gases

4 BCSBEC crossover in ultracold Fermi gases
 4.1 FanoFeshbach resonances and the interactioninduced crossover
 4.2 Main experimental results
 4.3 The unitary limit
 4.4 Theoretical approaches to the unitary limit and comparison with experiments
 4.5 Tan contact
 4.6 Josephson effect
 4.7 Collective modes and anisotropic expansion
 4.8 Quantum vortices and moment of inertia

5 BCSBEC crossover in nuclear systems
 5.1 Deuteron in symmetric nuclear matter and protonneutron pairing
 5.2 Asymmetric nuclear matter and the BCSBEC crossover
 5.3 Protonneutron correlations at finite temperature
 5.4 Effect of pairing on the liquidgas transition in symmetric nuclear matter
 5.5 Neutronneutron pairing at zero temperature
 5.6 Neutron matter at finite temperature
 5.7 Shortrange correlations and generalized nuclear contact
 5.8 Quartet BEC with applications to nuclear systems: alpha condensation in infinite nuclear matter
 5.9 A glimpse at finite nuclei
 6 Concluding remarks
1 Introduction
1.1 Historical background
The idea behind the BCSBEC crossover dates back just after the birth of the BCS theory in 1957 Bardeen1957 (). The authors of this theory made a point to emphasize the differences between their theory for superconductors based on stronglyoverlapping Cooper pairs and the SchafrothButlerBlatt theory Schafroth1957 () resting on nonoverlapping composite bosons which undergo BoseEinstein condensation at low temperature. Subsequently, the interest in these two different situations has been kept disjoint for some time, until theoretical interest arose for unifying them as two limiting (BCS and BEC) situations of a single theory where they share the same kind of broken symmetry. In this way, the physical system passes through an intermediate situation, where the size of the pairs is comparable to the average interparticle spacing. (For a shortrange (contact) interaction, this intermediate situation is nowadays referred to as the “unitary” regime.) Pioneering work in this sense was done by the Russian school, motivated by the exciton condensation in semiconductors Keldysh1965 () or simply by intellectual curiosity Popov1966 (). The theory of the BCSBEC crossover took shape initially through the work by Eagles Eagles1969 () with possible applications to superconducting semiconductors, and later through the works by Leggett Leggett1980 () and Nozières and SchmittRink Nozieres1985 () where the formal aspects of the theory were developed at zero temperature and above the critical temperature, respectively.
The interest in the BCSBEC crossover grew up with the advent of hightemperature (cuprate) superconductors in 1987, in which the size of the pairs appears to be comparable to the interparticle spacing Randeria1989 (); Randeria1990 (); Micnas1990 (); Randeria1992 (); Drechsler1992 (); Haussmann1993 (); Pistolesi1994 (); Casas1994 (). Related interest in the BCSBEC crossover soon spread to some problems in nuclear physics Baldo1995 (), but a real explosion of this activity appeared starting from 2003 with the advent of the fully controlled experimental realization essentially of all aspects of the BCSBEC crossover in ultracold Fermi gases (see Refs. Inguscio2007 () and Zwerger2012 () for an experimental and theoretical overview, respectively). This realization, in turn, has raised the interest in the crossover problem especially of the nuclear physics community, as representing an unprecedented tool to test fundamental and unanswered questions of nuclear manybody theory. The Fermi gas at the unitary limit (UL), where fermions of opposite spins interact via a contact interaction with infinite scattering length, was actually introduced as a simplified model of dilute neutron matter Baker2000 (); Baker1999 (), and the possibility to realize this limit with ultracold atoms was hence regarded as extremely important for this field of nuclear physics.
In nuclear physics there is the paradigmatic example of a protonneutron bound state (the deuteron) for which in symmetric or asymmetric nuclear matter, as a function of density, one may find a transition from BEC to BCS Alm1993 (); Baldo1995 (); Stein1995 (); Lombardo2001b () (see also Ref.Andrenacci1999 () for the densityinduced BCSBEC crossover). Such a scenario may be realized in expanding nuclear matter generated from heavy ion collisions Baldo1995 () or in protoneutron stars Heckel2009 (). More recently, this density driven crossover has been studied together with the competing liquidgas phase transition Jin2010 (). Deuteron condensation also heavily competes with alphaparticle condensation, which is somehow related to pairing of pairs discussed also in condensedmatter physics. Nuclear physics is a precursor of the theory for quartet condensation Roepke1998 (), but theoretical studies of quartets came up later also in the area of ultracold atoms Capponi2007 (). Alphaparticle condensation is presently very much discussed in finite nuclei (Hoyle state) Yamada2012 (). It was also predicted that in the noncondensed phase, the deuterons give rise to a pseudogap formation Schnell1999 (). With today’s ultracold atoms experiments, it has become possible to test such theories Stewart2008 (); Gaebler2010 (); Feld2011 (). For neutrons, no bound state exists, but rather a virtual state at almost zero energy. As a consequence, a dilute gas of neutrons, as it exists in the inner crust of neutron stars, is almost in the unitary limit mentioned above. Recent studies of the dilute neutron gas, from (almost) unitarity at low density to the BCS limit at high density, were done within BCS theory Matsuo2006 (), QuantumMonteCarlo calculations Abe2009 (); Gezerlis2010 (), and the NozièresSchmittRink approach Ramanan2013 (). One can say that the equation of state and pairing properties of very dilute neutron matter, although inaccessible in experiments, are now known thanks to the analogy with ultracold atoms.
As these examples show, there are several aspects of the BCSBEC crossover which are of broad joint interest to both ultracold atoms and nuclear communities. This paper is thus meant to provide a comprehensive review which focuses mainly on these common aspects of ultracold atoms and nuclear physics. Along these views, this paper provides also a pedagogical review of the main essential aspects of the BCSBEC crossover. In the process of writing, we have mostly adopted the quantum manybody diagrammatic techniques in line with our own technical expertise, and we have mainly focused on the topics to which we have provided original contributions over the last several years. Accordingly, for readers interested in complementary theoretical approaches to the problem we refer to other reviews which cover various aspects of the BCSBEC crossover. In particular, we can refer to reviews on the application to stronglyinteracting Fermi gases (in particular, at unitarity) of Quantum Monte Carlo methods Bulgac2012 (); Carlson2012 (), functionalrenormalizationgroup techniques Diehl2010 (), epsilon Nishida2012 () and virial Liu2013 () expansions. For the application of functionalintegral approaches (in particular, to the superfluid phase) we refer instead to the original research works of Refs. Pistolesi1996 (); Hu2006 (); Diener2008 ().
In addition, it should be mentioned that further reviews cover a number of aspects on the BCSBEC crossover which share a partial overlap with the material discussed here. Specifically, the unitary limit of Section 4.3 and the Tan contact of Section 4.5 have been of most interest in other reviews owing to the widespread recent interest in these topics, which have been treated in Refs. Zwerger2012 (); Randeria2014 (); Pitaevskii2016 (); Zwerger2016 (). The FanoFeshbach resonances of Section 4.1, which are at the heart of the interactioninduced crossover, have been discussed in Refs. Chin2010 (); Gubbels2013 (); Randeria2014 (); Pitaevskii2016 (); Zwerger2016 (). Polarized Fermi gases (considered here in Sections 2.6 and 3.7) have also been treated in Refs. Giorgini2008 (); Radzihovsky2010 (); Chevy2010 (); Zwerger2012 (); Gubbels2013 (); Pitaevskii2016 (). The topic of the singleparticle spectral function and pseudogap of Section 3.3 can be found discussed also in Refs. Chen2005 (); Randeria2014 (); Zwerger2016 (). Also some aspects of pairing in nuclear systems discussed in Section 5 can be found in the reviews of Refs. Brink2005 (); Dean2003 (); Gezerlis2014 (). Finally, Refs. Pitaevskii2016 () and Gezerlis2014 () cover also some introductory material treated here in Sections 1.2, 2.1, 2.4, 3.4, 4.7, 4.8, and 5.1. However, it should be remarked that no other reference thus far has emphasized the aspects of the BCSBEC crossover common to ultracold atoms and nuclear physics as we have done here.
1.2 The BCS wave function and its BEC limit
The starting point is the BCS groundstate wave function, of the form Bardeen1957 (); Schrieffer1964 ()
(1) 
In this expression, is the vacuum state, is a fermionic creation operator for wave vector and spin projection , and and are probability amplitudes given by:
(2) 
where ( being the fermion mass and the chemical potential) and where is the BCS order parameter (sometimes referred to as the superconducting gap) here taken at zero temperature. For simplicity, we have assumed that is independent of the wave vector, as it is the case for a contact interaction. More generally, in the expression of will be replaced by the wavevector (and temperature) dependent value . [Throughout this paper, we use units where the reduced Planck constant and the Boltzmann constant are set equal to unity.]
The BCS wave function (1) has the important property that it is the vacuum to the socalled quasiparticle operators (see Eq. (28) below and Ref. Ring1980 ()), that is, . This relation facilitates considerably the evaluation of expectation values. One can readily show that is the occupation number which goes over to the Fermi step in the limit as tends to zero, and that is the socalled anomalous density (known also as the “pairing tensor” in nuclear physics) which characterizes the BCS wave function.
With reference to the BCSBEC crossover, it has long been known that the BCS wave function (1) contains the BoseEinstein condensation of composite bosons as a limit. This is because, upon setting , the expression (1) can be rewritten in the form (see, e.g., Ref. Ring1980 ()):
(3) 
since owing to Pauli principle. Here, the operator contains fermion pairs but it is not a truly bosonic operator, to the extent that the commutator is not a cnumber but explicitly contains the fermionic operators . However, under some circumstances one may consider that for all practical purposes, provided for all of physical relevance. As a consequence, represents a bosonic coherent state (that is, a BoseEinstein condensate) with a nonvanishing brokensymmetry average .
It is clear from Eq. (2) that the condition can be satisfied for all provided the fermionic chemical potential becomes large and negative. This condition can be achieved when a boundstate with binding energy occurs for the twobody problem in vacuum with positive scattering length , and the coupling parameter becomes large such that . In this limit, approaches the value , which amounts to saying that all fermions are paired up in tight (composite) bosons with a vanishing residual interaction among the bosons. This result for the BEC limit of the fermionic chemical potential can directly be obtained from the meanfield gap equation (at zero temperature), which in the case of a wavevector dependent interaction reads:
(4) 
where now . (In the case of a contact potential, and reduce, respectively, to the coupling constant of Eq. (7) and to the constant gap given by the gap equation (8) below.) Using , , and Eq. (2), one can rewrite Eq. (4) in the form:
(5) 
Provided for all , Eq. (5) is just the Schrödinger equation for the relative motion of two particles of equal mass which are mutually interacting via the potential . The negative eigenvalue of this equation thus corresponds to (minus) the twobody binding energy as stated above.
More generally, as we shall see below, it is the fermionic chemical potential that takes the key role of the driving field which enables the system to pass from the BCS to the BEC limits of the BCSBEC crossover. As an illustration, Fig. 1 shows the occupation number at zero temperature for different (from positive to negative) values of the chemical potential , which correspond to increasing values of the gap parameter . Note that only when the curves have an inflection point at , which highlights the presence of an underlying Fermi surface even for a system with attractive interparticle interaction. As a consequence, when becomes negative, the Fermi sea gets completely dissolved and the occupation number becomes quite small for all .
1.3 Pairing correlations
The BCS wave function (1), or its equivalent form (3), treats all fermion pairs on the same footing to the extent that a single wave function is assigned to each pair. This meanfield type of approach is appropriate to describe a system of fermions with a mutual attractive interaction when the interparticle correlations extend much beyond the average interparticle distance, in such a way that many pairs are contained within the size of a given pair and different pairs strongly overlap with each other. This is definitely the case for the (BCS) weakcoupling limit, to which the BCS theory of superconductivity was originally meant to apply Bardeen1957 (). When considering the BCSBEC crossover, however, the range of the interparticle correlations can decrease down to the size of the bound pair, which, in turn, can be much smaller than the interparticle distance. Under these circumstances, the gas of composite bosons becomes dilute and one accordingly anticipates that pairing fluctuations beyond mean field can acquire a major role, especially at finite temperature when they are accompanied by thermal fluctuations.
Figure 2 shows the evolution throughout the BCSBEC crossover of the zerotemperature pair coherence length (dashed line) calculated for the BCS ground state (1). This quantity provides information about the intrapair correlation established between fermions of opposite spins Pistolesi1994 () and, at the meanfield level, corresponds to the spatial extension of the order parameter . Since can, in turn, be interpreted as a wave function as shown by Eq. (5), the r.m.s. radius of the pairs can be obtained from the expression:
(6) 
where is the Fourier transform of . [For later convenience, Fig. 2 shows also the corresponding evolution of the healing length (full line), that instead provides information about the interpair correlation Pistolesi1996 () and requires the inclusion of pairing fluctuations beyond mean field  see Section 3.2]. In Fig. 2, the average interparticle distance is used as the unit for , where is the Fermi wave vector related to the particle density via the relation . In Fig. 2 and throughout this paper, the quantity plays the role of the coupling parameter of the theory. Depending on the sign of , this parameter ranges from characteristic of the weakcoupling BCS regime when , to characteristic of the strongcoupling BEC regime when , across the value at unitarity when diverges. In practice, the “crossover region” of most interest, to be discussed throughout this paper, turns out to be approximately limited to the interval .
Pairing fluctuations play a particularly important role in the normal phase above the critical temperature where the order parameter vanishes (the generalization of the BCS approach to finite temperature will be considered in Section 2). This is because, a “local” order is expected to survive above if the system is fluctuating, even though the longrange order characteristic of the superfluid phase is lost. These considerations have led people to associate pairing fluctuations with the occurrence of a pseudogap above (as seen, for instance, by a depression of the singleparticle density of states about the chemical potential), in a similar way to what occurs in the superfluid phase below when the order parameter is instead nonvanishing. Such an analogy (between the order parameter below and the pseudogap above ) has been carried to the point that even the pseudogap state occurring in copperoxide superconductors has been interpreted from the point of view of a BCSBEC crossover scenario Chen2005 ().
In this context, it can be useful to consider the similarity between the pseudogap physics resulting from pairing fluctuations above and the persistence of (damped) spin waves, which are present over regions of limited extent in the normal phase of ferromagnetic (or antiferromagnetic) materials when a strict longrange order is absent Mook1973 (). Through this analogy, pseudogap phenomena in a Fermi system with an attractive interparticle interaction are attributed to the persistence of a “local pairing order” above the superfluid temperature , which occurs even though the (offdiagonal) longrange order is absent. This local order, which is built up by pairing fluctuations above , makes the singleparticle excitation spectrum to resemble that below , although with a frequency broadening due to the decay of the local excitations which cannot propagate over long distances in the absence of longrange order.
In addition, it is relevant to point out that, when raising the temperature and approaching from below, pairing correlations turn out to persist over a finite distance even within mean field, without the need of considering pairing fluctuations beyond mean field. Indeed, explicit calculations of the temperature dependence of the pair correlation function within mean field show that maintains a finite value at Marsiglio1990 (), a result which remains true irrespective of coupling across the BCSBEC crossover Palestini2014 ()). This remark, too, points out the importance of including pairing fluctuations in the normal phase above , in order to obtain a meaningful finite value of even when approaching from above. Otherwise, would be discontinuous when passing from to , since above fermions become noninteracting within the BCS (meanfield) approximation. We shall return to this point more extensively in Section 3.2.
2 BCS mean field
2.1 The homogeneous infinite matter case for a contact interaction
The BCS theory of superconductivity considers an effective attractive interaction between otherwise free fermions of two different species (conventionally referred to as spin and ), which are embedded in a continuous medium at a distance Bardeen1957 (); Schrieffer1964 (). This attractive interaction, albeit weak, is responsible for the formation of Cooper pairs in the medium Cooper1956 (). In condensed matter (like in metallic superconductors), detailed knowledge of the form of the attractive interaction is not required for most purposes and one may accordingly consider the simple form of a “contact” (zerorange) potential , where is a negative constant. This model interaction fully applies to ultracold Fermi gases (at least in the presence of a broad FanoFeshbach resonance as discussed in Section 4.1), but does not apply to nuclear systems discussed in Section 5 for which a finiterange interaction should in principle be retained. In order to reduce the numerical difficulties, however, even in the treatment of nuclear superfluids the finiterange interaction is often approximated by a zerorange one, which requires one to introduce a densitydependent coupling constant (see, e.g., Ref. Garrido2001 ()). This approach has further been elaborated (for instance, in Refs. Bulgac2002 (); Yu2003 ()) with the introduction of an efficient regularization scheme for finite systems (as discussed in Section 2.4). Here, like in most part of the paper, we consider the case of equal populations , while the new features arising under the more general condition will be explicitly considered in Section 2.6.
The price to pay for the use of a contact interaction is that, when dealing with a homogeneous system, integrals over the wave vector may diverge in the ultraviolet since the Fourier transform is a constant (in what follows, we shall consider a threedimensional system). This difficulty can be simply overcome by introducing a cutoff, given for metallic superconductors by the Debye frequency and for nuclei by a phenomenologically adjusted cutoff energy that reflects the range of the effective force. For ultracold gases, on the other hand, one can exploit the fact that a similar divergence affects also the twobody problem in vacuum, whereby the fermionic scattering length is obtained from the relation SadeMelo1993 ():
(7) 
with an ultraviolet cutoff . This regularization procedure entails the limits and to be taken simultaneously, in such a way that is kept at the desired value. Replacing the parameter with the physical quantity is especially relevant in the context of ultracold gases, where the natural cutoff given by the inverse of the interaction range would be orders of magnitude larger than all wave vectors of physical interest and where can be experimentally controlled Regal2003a (). As we shall see in Section 3.1, the regularization (7) is also of help when dealing with the manybody problem based on this twobody interaction, to the extent that it greatly reduces the number of manybody diagrams that survive in the limit .
In this way, the BCS meanfield equations for the order (gap) parameter at the temperature and for the density become Schrieffer1964 ():
(8)  
(9) 
where is the Fermi function and . Note that the quantity that enters Eqs. (8) and (9) plays the dual role of the order parameter (which is nonvanishing between and ) and of the minimum value of the singleparticle excitation energy at when . [The extension to finite temperature of the gap equation (4) with a finiterange interaction will be considered in Section 5.1  cf. Eq. (115).]
For a weak attractive interaction, is small and negative such that . This limit characterizes what is called the “conventional” BCS theory Bardeen1957 (). In this limit, the Fermi surface of noninteracting fermions is only slightly perturbed by the presence of the interaction, and the chemical potential at coincides with the Fermi energy . This is actually the limit where the meanfield approximation is expected to work best, to the extent that the size of a Cooper pair is much larger than the average interparticle distance and a large number of pairs is contained within the size of Schrieffer1964 ().
2.2 Solution at for a contact potential
For a generic value of the coupling that ranges between the BCS and BEC limits, the values of and at are determined by solving the coupled equations (8) and (9), where one sets since is always positive. Although a numerical solution of the ensuing equations is possible Carter1995 (), it was found independently that the threedimensional integrals over the wave vector occurring in these equations can be expressed either in terms of the complete elliptic integrals Marini1998 () or in terms of the Legendre function Papenbrock1999 (), in such a way that closedform expressions for and are obtained. In Ref. Marini1998 (), the results were expressed in terms of the parameter that ranges from (BEC limit) to (BCS limit), while in Ref. Papenbrock1999 () the inverse of this parameter was used.
In Ref. Marini1998 (), the following results were obtained:
(10)  
(11)  
(12) 
where
(13) 
and
(14) 
In these expressions, and are the complete elliptic integrals of the first and second kind, respectively, and
(15) 
In Ref. Papenbrock1999 (), on the other hand, Eq. (12) was equivalently expressed in the form:
(16) 
where denotes the Legendre function. In addition, analytic expressions for the pair coherence length and the condensate fraction were obtained in Refs. Marini1998 () and Salasnich2005 (), respectively.
Analytic approximations can be readily obtained in the BCS and BEC limits. In the BCS limit where , one obtains in Eq.(15). One can thus approximate , , and , in such a way that:
(17) 
In the BEC limit, on the other hand, and such that . One thus finds:
(18)  
(19)  
(20) 
The full expressions (10), (11), and (12) interpolate smoothly between the above (BCS and BEC) limits through the crossover region . Plots of and as functions of are readily obtained from these equations, as shown in Figs. 3(a) and 3(b). Note, in particular, that the chemical potential: (i) In the BCS limit equals the Fermi energy ; (ii) In the crossover region gradually decreases and eventually changes its sign; (iii) In the BEC limit reaches the asymptotic value . It is just this behaviour of the chemical potential that drives the BCSBEC crossover, as it was anticipated in Section 1.2.
2.3 Extension to finite temperature
At finite temperature, the presence of the Fermi function in the gap equation (8) and density equation (9) makes it impossible to find an analytic solution and one has then to resort to a numerical solution of these equations. An exception occurs, however, upon approaching the critical temperature from below, where analytic results can be obtained both in BCS and BEC limits (see, e.g., Ref. Strinati2000 ()).
At the meanfield level, the critical temperature is defined by the vanishing of the BCS gap parameter . Equations (8) and (9) are thus replaced by:
(21)  
(22) 
In the weakcoupling limit where , Eq. (22) yields the value of the chemical potential of a Fermi gas, which is only slightly smaller than the zerotemperature result provided . Correspondingly, Eq. (21) gives the following expression for :
(23) 
where is Euler’s constant (such that with reference to the result (17) that holds in the same limit). Since in the weakcoupling regime is negative and , Eq. (23) yields consistently with our assumptions.
In the strongcoupling limit where , the role of the two equations (21) and (22) is reversed. If we assume that , we may set in Eq. (21), making it to reduce to the boundstate equation and yielding at the leading order. The same kind of approximation, however, cannot be used in Eq. (22) since has to remain finite. In this equation, we set instead and obtain
(24) 
a result which coincides with the expression of the classical chemical potential at temperature . Using at this point the value as determined from Eq. (21), we can solve the expression (24) iteratively for yielding SadeMelo1993 ():
(25) 
at the leading order in . Although consistently with our assumptions, the expression of given by Eq. (25) diverges in the BEC limit at fixed density, instead of approaching as expected the finite value at which the BoseEinstein condensation of an ideal gas of composite bosons with mass and density occurs ( being the Riemann function of argument ). Physically, the meanfield temperature (25) corresponds to the dissociation of composite bosons, with the factor in the denominator originating from entropy effects. It is thus appropriate to associate the meanfield result (25) with the “pair dissociation temperature” of the composite bosons, which is completely unrelated to the BEC temperature at which quantum coherence is established among the composite bosons. The reason for this failure is that only the internal degrees of freedom of the composite bosons are taken into account by the meanfield expressions (21) and (22), thereby leaving aside the translational degrees of freedom of the composite bosons. To include these, pairing fluctuations beyond mean field need to be considered. The corresponding analysis will be discussed in Section 3.1.
2.4 Bogoliubovde Gennes and HartreeFockBogoliubov equations for inhomogeneous Fermi systems
The BCS wave function (1), for the ground state of a homogeneous Fermi gas with an attractive interparticle interaction, is a variational wave function with an associated gap parameter at zero temperature, which extends to the superfluid phase the HartreeFock approximation for the normal phase by retaining also operator averages of the type that do not conserve the particle number.
At finite temperature, the “meanfield” character of the BCS gap equation (8) for and of the associated coherence factors and given by the expressions (2), is better captured by an alternative procedure, whereby a meanfield (or HartreeFocktype) decoupling is directly performed at the level of the Hamiltonian by including particle nonconserving averages. Accordingly, one replaces the grandcanonical Hamiltonian by its meanfield approximation as follows
(26) 
where, for simplicity, we have considered the case of a zerorange interaction with strength (the generalization to finiterange interactions will be considered below). The parameter in Eq.(26) thus satisfies the condition
(27) 
where the thermal average is selfconsistently determined through the meanfield Hamiltonian itself. The quadratic form (26) can be readily diagonalized by the canonical transformation
(28) 
where and are constrained by at any given . In terms of the new operators and , the meanfield Hamiltonian (26) becomes (see, e.g., Chapt. 6 of Ref. Ring1980 ())
(29) 
(here, for simplicity, all quantities are assumed to be real). Provided that and are solutions to the equation
(30) 
whereby and , the coefficient of the second term on the righthand side of Eq. (29) vanishes identically, while the coefficient of the first term equals . Correspondingly, the selfconsistent condition (27) for reduces to:
(31) 
with . Note further that the last term of Eq. (29), which can be written as with the help of Eq. (30), corresponds to the (grandcanonical) groundstate energy associated with the BCS state.
What is relevant here is that the above procedure can be readily generalized to situations in which depends on the spatial position , owing, for instance, to the presence of an external potential (like a trapping potential or a barrier), or of a magnetic field, or of the selfconsistent mean field in finite nuclei, which introduce spatial inhomogeneities in the system. In this case, the algebraic matrix equation (30) is replaced by a pair of coupled Schrödingerlike equations for a twocomponent singleparticle fermionic eigenfunction, of the form deGennes1966 ():
(32) 
where contains the external potential (in addition, in the presence of an external vector potential the gradient operator in the kinetic energy is replaced as usual by ). Note that the Hartree term is absent in the diagonal elements of the matrix (32), owing to the use of a contact interparticle interaction which makes this term to vanish in the limit . [In weak coupling, however, one often introduces in a Hartreeliketerm of the form Bruun1999 (); Grasso2003 ().] These equations, known as the Bogoliubovde Gennes (BdG) equations, have to be solved up to selfconsistency for the local gap parameter , which acts like an offdiagonal pair potential and satisfies the local condition:
(33) 
In addition, the eigenfunctions obey the orthonormality condition:
(34) 
where the Kronecker delta on the righthand side is readily generalized to account for continuous eigenvalues. Physical quantities like the local number density, the current density, and the energy density can all be expressed in terms of the eigenfunctions and eigenvalues of Eq. (32) deGennes1966 ().
Note that the complex character of (as well as of the eigenfunctions ) has been restored in Eqs. (32)–(34), to allow for the presence of a particle current. Note also that it is the particlehole mixing characteristic of the BCS pairing to be responsible for the coupling of the two components of the eigenfunctions of Eq. (32), in contrast to the normal Fermi gas where particle and holeexcitations are separately good quasiparticles.
When solving the BdG equations for a contact interparticle interaction with coupling constant (which is the case of ultracold Fermi gases), technical problems arise from the need to regularize the selfconsistency condition (33) for the gap parameter since the sum over therein diverges in the ultraviolet. In the homogeneous case with a uniform gap parameter this regularization can be readily implemented with the help of Eq.(7), by expressing the bare coupling constant that enters the gap equation in terms of the scattering length of the twobody problem [cf. Eq.(8)]. This procedure, however, cannot be utilized when the gap parameter has a spatial dependence, and a new strategy is required. A first generalization of the regularization procedure for fermions on the BCS side of the crossover trapped in a harmonic potential was given in Ref. Bruun1999 () in terms of a pseudopotential method in real space, but it turns out that this method is not easily implemented in numerical calculations. A numerically more efficient regularization procedure was described in Ref. Grasso2003 () for the solution of the BdG equations (32) in a harmonic oscillator basis, which for practical purposes has to be truncated at some energy cutoff . Following Refs. Bulgac2002 (); Yu2003 (), the contribution of states above the cutoff is included within the ThomasFermi approximation. In this way, one obtains the following regularized version of the gap equation:
(35) 
where the numerical solution of the BdG equations (32) is explicitly performed only for eigenvalues up to that appear on the righthand side of Eq.(35). The prefactor plays the role of a (cutoff dependent) effective coupling constant and is given by
(36) 
where
(37) 
The positiondependent wave vectors and are related to the chemical potential and energy cutoff by and . As noticed in Refs. Bulgac2002 (); Grasso2003 (), Eq. (37) can also be used for negative values of by allowing for imaginary values of .
This approach was extended in Ref. Simonucci2013 () to the whole BCSBEC crossover and generic inhomogeneous situations, by combining the introduction of the cutoff in the quasiparticle energies such that with the derivation of the GrossPitaevskii equation for composite bosons in the BEC limit that was done in Ref. Pieri2003 () (cf. Section 2.5). Under the assumption that the energy cutoff is the largest energy scale in the problem (such that , , and , where is the Fermi energy associated with the mean density  conditions that can be somewhat relaxed in the weakcoupling limit), the regularized version of the gap equation reads eventually Simonucci2013 ():
(38) 
The quantities that enter the lefthand side of this equation are defined as follows:
(39) 
The righthand side of Eq.(38), which results from an explicit numerical integration of the BdG equations over the reduced energy range , acts as a source term on the nonlinear differential equation for associated with the lefthand side. By implementing the numerical calculation of the BdG equations for an isolated vortex embedded in a uniform superfluid, it was shown in Ref. Simonucci2013 () that the inclusion of the various terms on the lefthand side of Eq.(38) [from the linear () term, to the linear plus cubic () terms, and finally to the linear plus cubic plus Laplacian () terms] plays an increasingly important role in reducing the total computational time and memory space at any coupling throughout the BCSBEC crossover, by decreasing the value of the cutoff up to which the eigenfunctions of the BdG equations have to be explicitly calculated Simonucci2013 ().
In practice, the BdG equations have been solved numerically up to selfconsistency only for relatively simple geometries since their numerical solution becomes rapidly too demanding due to computational time and memory space. For ultracold Fermi gases, the BdG equations have been solved to account for the spatial dependence of the order parameter in balanced Bruun1999 (); Grasso2003 (); Ohashi2005 () and imbalanced Castorina2005 (); Jensen2007 () trapped gases, as well as for the microscopic structure of a single vortex at zero temperature on the BCS side Nygaard2003 () and across the BCSBEC crossover Sensarma2006 (), and also at finite temperature across the BCSBEC crossover Simonucci2013 (). The microscopic structure of a single vortex for the unitary Fermi gas has been analyzed in Ref. Bulgac2003 () with BdGlike equations resulting from the superfluid localdensity approximation of Ref. Bulgac2002b (), which has later been used also to calculate the profile of the order parameter in balanced trapped gases Bulgac2007 () or in the FFLO phase of imbalanced Fermi gases throughout the BCSBEC crossover Bulgac2008b () (see Section 2.6). In addition, the BdG equations have been used to determine the occurrence of vortex lattices in a rotating trap on the BCS side of the crossover Feder2004 (); Tonini2006 (). A study of the Josephson effect with a onedimensional barrier throughout the BCSBEC crossover at zero temperature Spuntarelli2010 () will be reported in Section 4.6.
In nuclear physics, where in the most sophisticated cases finiterange (instead of zerorange) forces are used in the meanfield and gap equations (a famous example in this context being the Gognyforce Decharge1980 ()), both matrix elements in (32) become nonlocal. The corresponding equations, usually referred to as the HartreeFockBogoliubov (HFB) equations, are thus more general than the BdG equations and read (for simplicity, we do not write down the spin and isospin structure):
(40) 
where
(41) 
and
(42) 
In the above expressions, and are the Hartree and Fock fields, respectively, with the density matrix given by (such that is the number density). In nuclear physics, the phenomenological effective forces are generally density dependent, also in the pairing channel, as indicated in the argument of in Eq.(42).
The nonlocal HFB equations (40), that hold for finite nuclei, are numerically much harder to solve than the local BdG equations (32). Yet, they are almost routinely solved even for deformed and rotating nuclei, in general the results of the calculations being in excellent agreement with experiments (see Refs. Bender2003 (); Vretenar2005 () for reviews). Owing to the high numerical cost to solve these equations, however, approximate local approximations have been developed for the meanfield terms of Eq.(41) (such as the Skyrme Energy Density Functionals (EDF) Erler2011 ()) and for the pairing interaction of Eq.(42).
The need to regularize the gap equation arises also for the nuclear problem. In this context, it is worth mentioning that the renormalization scheme of Eq. (35) discussed above was originally developed in Refs. Bulgac2002 (); Yu2003 () for the nuclear problem, in the case when the gap equation reduces to a local form. In this case, Eqs. (36) and (37) are replaced by an effective coupling constant which takes the form:
(43) 
Here, in contrast to Eqs. (36) and (37), the effective mass and the bare coupling constant are now position dependent through the value of the local density, which can be fitted such that as a function of reproduces the gap obtained with the Gogny force Garrido2001 () in infinite matter. In addition, the wave vectors and