Bound states in a superfluid vortex: A detailed study along the BCS-BEC crossover
BBound states in a superfluid vortex: A detailed study along the BCS-BEC crossover
S. Simonucci,
1, 2
P. Pieri,
1, 2, ∗ and G. Calvanese Strinati
1, 2, 3, † School of Science and Technology, Physics Division,Universit`a di Camerino, 62032 Camerino (MC), Italy INFN, Sezione di Perugia, 06123 Perugia (PG), Italy CNR-INO, Istituto Nazionale di Ottica, Sede di Firenze, Firenze 50125 (FI), Italy
The bound states that can occur in a superfluid vortex have recently called for attention owingto the capability of detecting them experimentally. However, a detailed theoretical account forthe presence of these vortex bound states is still lacking, for all temperatures in the superfluidphase and couplings along the BCS-BEC crossover. Here, we fill this gap and present a systematictheoretical study based on the Bogoliubov-de Gennes equations for the bound states that occurover the two characteristic (inner and outer) spatial ranges in which the extension of a superfluidvortex can be partitioned. It is found that the total number of bound states decreases from theBCS (weak-coupling) side of the crossover toward the intermediate-coupling region where they arestill present, whereas the bound states disappear upon entering the BEC (strong-coupling) side. Ascaling relation is also obtained that connects the number of bound states in the inner spatial rangeof the vortex to the depth and width of the vortex itself. A criterion is finally provided in termsof the local density of states, to distinguish where a given fermionic superfluid is located in thecoupling-temperature phase diagram of the BCS-BEC crossover.
I. INTRODUCTION
Recently, interest has arisen in the internal structureof vortices in type-II superconductors, owing to the pres-ence of bound states in the vortex core region of a Y123superconductor that were detected experimentally [1].The relevance of this finding has been highlighted [2],as an indication that the superconducting state of high-temperature superconducting materials should be welldescribed by the conventional BCS pairing theory [3], inspite of the fact that the normal state of these materialsis highly unconventional.In Ref. [1] the experimental findings were supported bytheoretical calculations, based on the approach to the so-lution of the Bogoliubov-de Gennes (BdG) equations [4]developed in Ref. [5], to come up with the local density-of-states as the relevant quantity to be compared withthe experimental data. Although these calculations con-sidered the effects of a realistic band structure as well asthe influence of disorder and nearby vortices on a givenvortex, they were limited to zero temperature and to asingle value of the inter-particle coupling. To understandin a more complete fashion the properties of bound statesin superconducting vortices, it would then be desirable tocharacterize them also as a function of temperature andinter-particle coupling, at least for the simplest case of anisolated vortex. Purpose of this paper is to address thisaspect of the problem, thereby complementing to someextent the theoretical information provided in Ref. [1].The occurrence of bound states in a superconductingvortex was originally proposed in Ref. [6] by solving theBdG equations. That calculation was restricted to zero ∗ [email protected] † [email protected] temperature and to the case of weak coupling for which∆ (cid:28) E F , where ∆ is the bulk value of the gap pa-rameter away from the center of the vortex and E F theFermi energy. In more modern language, this would cor-respond to the BCS (weak-coupling) limit of the BCS-BEC crossover [7]. Later works, that considered the self-consistent solution of the BdG equations for a vortexline, have either relaxed the restriction to zero tempera-ture but still at weak coupling [8], or spanned the wholeBCS-BEC crossover but still at zero temperature [9, 10].To obtain a more complete information on the propertiesof bound states in a superconducting vortex, however, itwould be worthwhile to address both their coupling and temperature dependences at the same time.From a technical point of view, in Refs. [8–10] the studyof a single vortex was performed in terms of the BdGequations by constraining the vortex into a cylinder ofradius R c with infinite walls, such that all single-particlewave functions were to vanish at R c . In this way, how-ever, all single-particle energies turn out to be discrete,in such a way that no sharp distinction there exists be-tween truly bound states with finite spatial extension andunbound continuum states past a well-defined threshold.This limitation was later removed in Ref. [11], where theBdG equations were solved by introducing suitable freeboundary conditions through which the vortex profile ismatched to its asymptotic behavior at a large distancefrom the center. In this way, a clear distinction can bemarked between bound (localized) states and unboundstates extending to infinity. By this approach, the tem-perature dependence of the healing length ξ associatedwith the size of the vortex could be determined even fortemperatures quite close to the critical temperature T c at which ξ diverges. In Ref. [11] this analysis was carriedout from weak to strong coupling across the whole BCS-BEC crossover, to determine not only the profile of thegap parameter but also those of the local number density a r X i v : . [ c ond - m a t . s up r- c on ] A p r and current. In addition, in Ref. [11] an advanced regu-larization procedure was developed for the self-consistentgap equation, which permits to increase considerably theaccuracy of the numerical calculations.In the present paper, we take advantage of the aboveprocedures developed in Ref. [11] and concentrate ouranalysis on the bound-state part of the spectrum for anisolated vortex embedded in an otherwise homogeneousfermionic superfluid of infinite extent. In this context,we will determine the local density of states as a func-tion of energy and of the spatial position from the vor-tex core, for varying temperature in the superfluid phasefrom T = 0 to T = T c and for couplings along the BCS-BEC crossover. We will thus also be able to study howthe total number of bound states evolves as a function oftemperature and coupling, paying additional attention tohow the bound states are distributed over the two char-acteristic spatial ranges in which the vortex (includingits tail) can be partitioned.The main results obtained in this paper are as follows:(i) By relying on the numerical procedure to solve theBdG equations for a fermionic superfluid vortex devel-oped in Ref. [11], an energy threshold is identified thatclearly separates truly bound states from continuumstates. In this way, the counting of bound states for givencoupling and temperature becomes a meaningful process.(ii) In addition, by partitioning the spatial extent of thevortex into an inner and an outer region (to be identi-fied below), the number of bound states that hinge oneach region is determined as a function of energy belowthreshold. It is found that the number of bound statesin the inner region scales in a universal way on the depthand width of the vortex itself.(iii) The number of bound states is found to be non-negligible even in the intermediate-coupling region of thecrossover, where the Cooper pair size becomes compara-ble with the inter-particle distance. To the extent thatthis small size is compatible with the occurrence of high-temperature superconductivity [12], there is thus no apriori reason to associate the occurrence of bound statesin a vortex with the superconductivity being of the con-ventional BCS type as recently asserted [2].(iv) Finally, it is proposed that the shape of the localdensity of states vs energy, taken about the center of thevortex, can serve as a guide to distinguish where a givenfermionic superfluid stands in the coupling-temperaturephase diagram of the BCS-BEC crossover. In this con-text, one should recall that the local density of states hasaccurately been measured with a scanning tunneling mi-croscope also in high-temperature superconductors [13],as well as in more complex SNS structures made withconventional superconductors [14].The plan of the paper is as follows. Section II recallsthe main features of the solution of the BdG equationsdrawn from Ref. [11] for an isolated vortex embeddedin an infinite superfluid. Emphasis is there given to theoccurrence of an inner and outer region with differentspatial behaviors of the vortex profile where the bound states can reside. Section III presents the numerical re-sults for the number of bound states and the local densityof states, from which the scaling relation and the crite-rion for spotting the position along the BCS-BEC phasediagram mentioned above are derived. Section IV givesour conclusions. Appendix A discusses the asymptoticprofile of an isolated vortex in the context of the Gross-Pitaevskii equation, which holds for the composite bosonsthat form in the BEC limit of the crossover. II. FORMAL ASPECTS
We begin by briefly recalling the method used inRef. [11] to solve the BdG equations in the presence of anisolated vortex embedded in an otherwise infinite homo-geneous superfluid, as a function of both coupling acrossthe BCS-BEC crossover and temperature.The BdG equations read [4]: (cid:18) H ( r ) ∆( r )∆( r ) ∗ −H ( r ) (cid:19) (cid:18) u ν ( r ) v ν ( r ) (cid:19) = ε ν (cid:18) u ν ( r ) v ν ( r ) (cid:19) . (1)Here, H ( r ) = −∇ / m − µ where m is the fermion massand µ the chemical potential (we set (cid:126) = 1 troughout).The local gap parameter ∆( r ) in Eq. (1) is determinedvia the self-consistent condition:∆( r ) = − v (cid:88) ν u ν ( r ) v ν ( r ) ∗ [1 − f F ( ε ν )] (2)where f F ( (cid:15) ) = ( e (cid:15)/ ( k B T ) + 1) − is the Fermi function attemperature T ( k B being Boltzmann constant) and v isthe bare coupling constant of the contact interaction. Asuitable regularization of the inhomogeneous gap equa-tion (2) was implemented in Appendix B of Ref. [11],by drawing elements from the derivation of the Gross-Pitaevskii equation [15] for composite bosons that formin the BEC limit, which was obtained in Ref. [16] start-ing from the BdG equations. In the process, the barecoupling constant v that enters Eq. (2) gets replacedby the (dimensionless) coupling parameter ( k F a F ) − ,where k F = (3 π n ) / is the Fermi wave vector as-sociated with the (bulk) number density n and a F isthe scattering length of the two-fermion problem. Thiscoupling parameter enables one to span the whole BCS-BEC crossover [7]. In practice, the crossover betweenthe BCS and BEC regimes is exhausted within the range − (cid:46) ( k F a F ) − (cid:46) +1 about the unitary limit (UL)where ( k F a F ) − = 0.When looking for bound states inside a vortex like inthe present context, coupling values on the BCS sideof unitarity should primarily be considered, since it willturn out that the number of bound states rapidly van-ishes upon entering the BEC side of unitarity. In addi-tion, on the BEC side of the crossover at finite tempera-ture it would be necessary to include pairing fluctuationsbeyond mean field [7], a task which is beyond the pur-poses of the present paper. On the other hand, on theBCS side of the crossover we shall find it necessary toextend the calculations down to the (numerically ratherdemanding) coupling value ( k F a F ) − = − .
0, in orderto recover what would be expected on the basis of thestandard BCS approach.For an isolated vortex with unit circulation and di-rected along the z axis, the spatially dependent gap pa-rameter is written in cylindrical coordinates as ∆( r ) =∆( ρ, ϕ, z ) = ∆( ρ ) e iϕ with ∆( ρ ) real. Correspondingly,the eigenfunctions of the BdG equations (1) take theform: u ν ( r ) = u ν r ,(cid:96),k z ( ρ ) e i(cid:96)ϕ √ π e ik z z √ π (3) v ν ( r ) = v ν r ,(cid:96),k z ( ρ ) e i ( (cid:96) − ϕ √ π e ik z z √ π (4)where ν r is the radial quantum number, (cid:96) is an integer(both positive and negative), and { u ν r ,(cid:96),k z ( ρ ) , v ν r ,(cid:96),k z ( ρ ) } are real functions for the bound states of primary inter-est here. Within this approach, for given values of (cid:96) and k z the BdG equations are numerically integrated out-wards, starting from ρ = 0 with suitable indicial condi-tions, up to a maximum value R out (values ranging from k F R out = 60 for ( k F a F ) − = 0 up to k F R out = 1500for ( k F a F ) − = − . ρ = R out by using suitable linear combina-tions of Bessel, Neumann, and Hankel functions, in termsof which the solutions of the BdG equations can be ex-pressed for ρ ≥ R out . This procedure offers a definiteadvantage, in that it avoids the common practice of con-straining the vortex in a cylinder with infinite walls at ρ = R out . For this reason, this procedure allows one toclearly distinguish truly bound states (which decay ex-ponentially for ρ (cid:29) R out ) from continuum states (withoscillating behavior extending up to infinity). In prac-tice, we have considered values of (cid:96) not smaller than 200and values of | k z | up to 3 k F .In this way, for given k z and irrespective of the value of (cid:96) , a first energy threshold between bound and continuumstates occurs at the (temperature-dependent) value ∆ recovered by ∆( ρ ) deep in the bulk region when ˜ µ > µ = µ − k z / m ), while a second threshold occursat (cid:112) ∆ + ˜ µ when ˜ µ <
0. Accordingly, for 0 < ε < ∆ only bound states can be found, for ∆ < ε < (cid:112) ∆ + ˜ µ bound states embedded in the continuum (with differentvalues of k z ) can also be found, and for (cid:112) ∆ + ˜ µ < ε only continuum states occur.Once the eigenfunctions { u ν r ,(cid:96),k z ( ρ ) , v ν r ,(cid:96),k z ( ρ ) } havebeen determined together with the corresponding eigen-values { ε ν r ,(cid:96),k z } according to the above procedure, onecan obtain the local density of states given by: N ( ρ ; E ) = (cid:90) + ∞−∞ dk z (2 π ) (cid:88) ν r ,(cid:96) (cid:2) | u ν r ,(cid:96),k z ( ρ ) | δ ( E − ε ν r ,(cid:96),k z )+ | v ν r ,(cid:96),k z ( ρ ) | δ ( E + ε ν r ,(cid:96),k z ) (cid:3) (5) which has dimensions of [volume × energy] − . This quan-tity, which is of primary experimental interest [13, 14],contains contributions from both bound and continuumstates [17]. We are now in a position to calculate N ( ρ ; E )as a function of coupling along the BCS-BEC crossoverand of temperature in the superfluid phase. In addi-tion, by our approach, we can also distinguish the sepa-rate contributions to N ( ρ ; E ) from bound and continuumstates.It is further relevant to point out that, contrary to acommon assumption that the radial vortex profile ap-proaches asymptotically the bulk value ∆ in an ex-ponential way [5, 18], the gap parameter ∆( ρ ) has in-stead a long tail with a power-law dependence of thetype ∆ (cid:16) − ζ ρ (cid:17) . This property, which went appar-ently unnoticed in the literature, was found to be validfor all couplings throughout the BCS-BEC crossover bythe detailed numerical analysis carried out in Ref. [11].As shown in Appendix A, this property can be also cap-tured analytically in terms of the Gross-Pitaevskii equa-tion [15], to which the BdG equations have been shown toreduce in the BEC (strong-coupling) limit [16]. It is actu-ally in the more limited radial range k − F ≤ ρ (cid:46) R v onlythat the vortex profile has an exponential dependence,from which a characteristic (temperature and couplingdependent) coherence length ξ can be determined, where R v marks the position of the maximum value of the radialcurrent [11]. As is turns out from the numerical analysisof Ref. [11] (and confirmed by from the present one), thetwo length scales ζ and ξ coincide with each other withinnumerical error for all couplings and temperatures.Accordingly, in the following we shall find it conve-nient to partition the vortex profile into an “inner” region(where the exponential decay applies) and an “outer”region (where the power-law behavior takes over), theboundary between the two regions being (approximately)taken at ρ ≈ ξ . The counting of the number of boundstates will similarly be partitioned. We are then going todistinguish whether a given bound state with quantumnumbers ( ν r , (cid:96), k z ) belongs to the inner or to the outerregion, by looking at the (dimensionless) partial normal-ization condition: P ν r ,(cid:96),k z ( ξ ) = (cid:90) ξ dρ ρ (cid:2) u ν r ,(cid:96),k z ( ρ ) + v ν r ,(cid:96),k z ( ρ ) (cid:3) . (6)We then attribute the location of the bound state tothe inner (outer) region when P ν r ,(cid:96),k z ( ξ ) exceeds (fallsbehind), say, 0 . . ρ − )power-law tail of the gap parameter, it turns out thatthe vortices residing in the outer region by far exceed innumber those residing in the inner region, although mostof them lie far away from the vortex center.In practice, the bound states in the inner region (andtherefore their number) are numerically much more undercontrol than those in the outer region, especially when FIG. 1. (Color online) The local density of states N ( ρ ; E ) ib , contributed by the inner bound states, is shown for three couplings( k F a F ) − = ( − . , − . , .
0) (from left to right) and two temperatures T = (0 , . T c (from bottom to top). Here, N ( ρ ; E ) ib isin units of k F /E F , ρ in units of k − F , and E in units of E F (where E F = k F / m is the Fermi energy). In the left-bottom panel,the arrows pointing toward the energy scale mark the values ± ∆ of the continuum threshold at the corresponding couplingand temperature (cf. Table I below). the latter ones hinge mostly close to the boundary at R out where the connection with decaying exponentialsis enforced through the boundary conditions. In addi-tion, when several vortices would be packed up togetherto form a vortex lattice, the nature of the bound statesin the remote external part of the outer region of whatwould have been a vortex in isolation is expected to bestrongly modified by the presence of the surrounding vor-tices. For this reason, it may anyway not be worth toinsist in extending the size of the outer region beyondthose values of R out that we have considered in practicefor the numerical calculations. III. NUMERICAL RESULTS
We pass to characterize the presence of bound states inan isolated vortex, for all temperatures in the superfluidphase and couplings along the BCS-BEC crossover, bytaking advantage of our accurate numerical calculationsin terms of the BdG equations following the methodolo-gies developed in Ref. [11]. We shall find that the numberof bound states rapidly decreases on the BEC side of uni-tarity, so that, in practice, only the coupling region fromthe BCS regime up to unitarity will be relevant to ourpurposes. Several features will be highlighted by look-ing at the details of the local density of states, or sim-ply by counting the number of bound states that can beidentified for given coupling and temperature. Our mainfindings are summarized by the following features.
A. Contribution from bound and scatteringstates to the local density of states
One of the main questions to be asked at the outset iswhere the various contributions, in which the local den-sity of states N ( ρ ; E ) of Eq. (5) can be partitioned, arespatially located over the extension of an isolated vor-tex. These contributions can be identified as those corre-sponding to the bound states residing essentially in theinner and outer regions of the vortex, and as that corre-sponding to the scattering states. Here, the partial nor-malization criterion (6) is used to assign a given boundstate either to the inner or outer regions of the vortex(see also subsection III-B below).Accordingly, we express the local density of states N ( ρ ; E ) of Eq. (5) as the sum of three contributions,namely, N ( ρ ; E ) ib , N ( ρ ; E ) ob , and N ( ρ ; E ) sc , where “ib”stands for inner-bound , “ob” for outer-bound , and “sc”for scattering . These three contributions are reportedin Figs. 1, 2, and 3, respectively, as a function of both ρ and E , in each case for three characteristic couplings( k F a F ) − = ( − . , − . , .
0) and two significant temper-atures T = (0 , . T c . [One should, however, be awarethat, due to the progressive importance of pairing fluc-tuations beyond mean field at finite temperature for in-creasing coupling, the data at unitarity and finite tem-perature are expected to be quantitatively less reliablethan those at weaker couplings.]From Figs. 1-3 one notices that the contribution N ( ρ ; E ) ib is mostly spatially concentrated over the innerportion of the vortex, while the contribution N ( ρ ; E ) ob is spread over a much wider spatial region. On the other FIG. 2. (Color online) The local density of states N ( ρ ; E ) ob , contributed by the outer bound states, is shown for three couplings( k F a F ) − = ( − . , − . , .
0) (from left to right) and two temperatures T = (0 , . T c (from bottom to top). Like in Fig. 1, N ( ρ ; E ) ob is in units of k F /E F , ρ in units of k − F , and E in units of E F .( k F a F ) − T = 0 T = 0 . T c T c /E F ( T ) (in units of E F ) for the couplingsand temperatures T = (0 , . T c considered in the panels ofFigs. 1-3. The last column gives the corresponding values ofthe critical temperature T c (in units of E F ). hand, N ( ρ ; E ) sc at low energies above (the temperaturedependent) threshold is depressed in the central regionof the vortex owing to the orthogonality requirement ofquantum mechanical states with different energies, whileat high energies above threshold N ( ρ ; E ) sc is almost uni-formly spread over all space owing to the occurrence ofspatial oscillations with small wavelength. The situa-tion remains qualitatively the same also for increasingtemperature from T = 0 to T = 0 . T c . More markeddifferences appear instead for increasing coupling, from( k F a F ) − = − . k F a F ) − = 0, along whichone notices a progressively enhanced asymmetry betweennegative and positive energies. We shall return below toa more detailed discussion of this asymmetry.From Figs. 1-3 one also notices that, for each couplingand temperature, a sharp threshold for the continuumstates occurs at E = ∆ ( T ) (cf. Fig. 3), and that boundstates “embedded” in the continuum are also found when E > ∆ (cf. Figs. 1-2), although they give a small overallcontribution to N ( ρ ; E ). This can be seen, in particular,in the left-bottom panel of Fig. 1, where the two arrows mark the values ± ∆ of the continuum threshold beyondwhich bound states embedded in the continuum appear.For convenience, the numerical values of ∆ ( T ) relevantto Figs. 1-3 are reported in Table I.We have further verified that, past unitarity upon ap-proaching the BEC regime, there occurs a progressivedepletion of N ( ρ ; E ) ib in the central region of the vor-tex, implying that bound states tend to disappear in thiscoupling regime. In addition, for given coupling this de-pletion is essentially complete for T = 0, while it becomesonly partial upon increasing the temperature. We shallreturn below to a more detailed discussion of this feature. B. Counting the number of bound states in theinner and outer regions
Another relevant piece of information can be obtainedby a simple count of the number of bound states, sep-arately in the inner and outer regions as defined above.To this end, we recall that the bound-state wave func-tions are localized in the x - y plane orthogonal to the z axis of the vortex, so that a whole branch spanned by thewave vector k z is associated with each bound state withgiven angular momentum (cid:96) and radial quantum number ν r . Accordingly, in the following the numbers of boundstates will be given per unit length along the direction ofthe vortex axis, since they will be summed (integrated)over the wave vector k z (as it is also done in Eq. (5)).With this procedure, the line density of bound states N inner and N outer , that reside, respectively, in the innerand outer regions of an isolated vortex, can be obtainedas a function of coupling and temperature.Figure 4 shows both N inner (lower panel) and N outer (upper panel) for three characteristic temperatures T = FIG. 3. (Color online) The local density of states N ( ρ ; E ) sc , contributed by the scattering states , is shown for three couplings( k F a F ) − = ( − . , − . , .
0) (from left to right) and two temperatures T = (0 , . T c (from bottom to top). Like in Figs. 1and 2, N ( ρ ; E ) sc is in units of k F /E F , ρ in units of k − F , and E in units of E F . The white stripe around E = 0 corresponds tothe gap region where no scattering states can occur and the associated local density of states identically vanishes. (0 , . , . T c , with the coupling ( k F a F ) − ranging from − . T = 0 the calculation has been extended to( k F a F ) − = 1 . N inner and N outer rapidly decrease fromthe BCS regime to the UL and quickly vanish once theBEC side past unitarity is reached, thus implying thatthe bound states have no relevance for the compositebosons that form out of fermion pairs in the BEC regime;(ii) For given coupling, N inner increases upon increasingthe temperature, whereby the depth ∆ of the vortex de-creases but at the same time its width ξ increases; (iii)For given coupling and temperature, N outer by far ex-ceeds N inner , although N outer appears to be essentiallyindependent of temperature.It should be remarked that the values of N outer wehave obtained are numerically less reliable that those of N inner , because they depend on the value of the radius R out where the boundary conditions are enforced on thesolutions of the BdG equations (cf. Section II). In thiscontext, we found it numerically too demanding to makethis radius exceed k F R out = 1500 for the smaller couplingwe could reach deep in the BCS regime. For these rea-sons, in the following we shall limit ourselves to furtherexamine the dependence of N inner on ∆ and ξ , leavingaside similar considerations for N outer . C. Symmetry vs asymmetry of the local densityof states about the vortex center
A characteristic feature, that distinguishes whethera homogeneous attractive Fermi gas lies in the weak- coupling (BCS) or the unitary regime, is the degree ofasymmetry of the single-particle spectral function, withthe two peaks of this spectral function being much moresymmetric about zero frequency in the BCS regime thanat the UL [19]. The occurrence of particle-hole asymme-try is a characteristic feature of the BCS-BEC crossover,whereby the underlying Fermi surface gets progressivelymodified when passing from the BCS to the BEC limitsacross UL, as the ratio ∆ /E F increases. Our interesthere is to verify to what extent this characteristic fea-ture is maintained also for the present inhomogeneousproblem, such that a measurement of the local densityof states about the vortex center would be able to detectwhether or not the superfluid properties of the Fermi gasdeviate from those of an ordinary BCS superconductor.As we have already noted while commenting Figs. 1-3, all three contributions to the local density of states N ( ρ ; E ) that we have considered become progressivelymore asymmetric when passing from the BCS to the uni-tary regime. A more quantitative characterization of thisasymmetry can be obtained as follows. Let us considera spatial region in the inner part of the vortex, say, oflinear extension ξ/ ρ = 0(in such a way to concentrate our attention close to thecenter of the vortex). We then look for the behavior ofthe integrated quantity D ( E ) = 2 π (cid:90) ξ/ dρ ρ N ( ρ ; E ) (7)where N ( ρ ; E ) is the local density of states (5). Thequantity (7) has dimensions of [length × energy] − andcontains, in principle, contributions from both bound and (a) N o u t e r / k F (k F a F ) -1 T=0T=0.5T c T=0.9T c (b) N i nn e r / k F (k F a F ) -1 T=0T=0.5T c T=0.9T c FIG. 4. (Color online) The line density of bound states N inner in the inner region (lower panel) and N outer in the outer region(upper panel) of the vortex vs the coupling ( k F a F ) − for threetemperatures: T = 0 (filled circles), T = 0 . T c (open boxes),and T = 0 . T c (stars). Both N inner and N outer have beendivided by k F to make them dimensionless. continuum states, depending on the value of E . In thefollowing, we shall be interested in the energy interval − ∆ ( T ) ≤ E ≤ ∆ ( T ) where ∆ ( T ) is the bulk valueof the gap parameter at temperature T , such that onlybound states contribute to D ( E ) in this energy interval.Plots of D ( E ) vs E are shown in Fig. 5 for threecharacteristic couplings ( k F a F ) − = ( − . , − . , . T = (0 , . , . T c for each coupling (with the exception of the coupling( k F a F ) − = − . T = 0 . T c is lacking). Ir-respective of temperature, drastic changes are seen tooccur in the shape of D ( E ), which becomes progressivelymore asymmetric when passing from ( k F a F ) − = − . k F a F ) − = 0 . D ( E ) overeither the positive or negative portions of the gap region, F a F ) -1 =-4T=0 T=0.5T c D ( E ) E F / k F (k F a F ) -1 =-2T=0 T=0.5T c T=0.9T c (a)(b)(c) E/ Δ (T)(k F a F ) -1 =0T=0 T=0.5T c T=0.9T c FIG. 5. (Color online) The quantity D ( E ) of Eq. (7) (in unitsof k F /E F ) is shown vs E (in units of the bulk gap ∆ ( T ) attemperature T ), for the couplings ( k F a F ) − = − . k F a F ) − = − . k F a F ) − = 0 . T = 0 (full line), T = 0 . T c (dashed line), and T = 0 . T c (dotted line). thereby defining N + = (cid:90) ∆ ( T )0 dE D ( E ) (8) N − = (cid:90) − ∆ ( T ) dE D ( E ) (9)which have dimensions of [length] − . In this way, a suit-able asymmetry parameter ( N + − N − ) / ( N + + N − ) canthen be introduced, which can be analyzed as a functionof coupling and temperature. The results are reportedin Fig. 6. One sees that, essentially for all temperatures,there occurs a steep increase of this asymmetry param-eter, from the value zero in the BCS regime to almostunity at UL, which at T = 0 has reached half the wayat about the coupling ( k F a F ) − = − .
5. This quantitythus provides one with a quick clue for the closeness ofthe superfluid Fermi gas to the UL, by looking at thedistribution of the bound states about the vortex center.From the experimental results available from Ref. [1],no clear evidence for the occurrence of particle-hole ( N + - N - ) / ( N + + N - ) (k F a F ) -1 T=0 T=0.5T c T=0.9T c FIG. 6. (Color online) Asymmetry parameter, defined interms of the quantities (8) and (9), as a function of the cou-pling ( k F a F ) − for three different temperatures: T = 0 (filledcircles), T = 0 . T c (open boxes), and T = 0 . T c (stars). asymmetry can apparently be extracted. One has toconsider, however, that in Ref. [1] the presence of boundstates in vortices was evidenced in a high- T c cuprate su-perconductor for the first time, through a delicate sub-traction procedure of a significant background. Addi-tional detailed measurements of density-of-states spectrawould then be required to evidence a particle-hole asym-metry over and above this subtraction. D. A scaling relation
In Fig. 4, N inner was found to depend separately oncoupling and temperature. However, both the depth ∆ (or bulk value of the gap parameter) and the width ξ ofthe vortex also depend on coupling and temperature. Itwill then be interesting to eliminate this double depen-dence and express N inner directly as a function of a single variable obtained by combining ∆ and ξ , irrespective ofthe corresponding values of coupling and temperature.Through this attempt, we have found that the product∆ ξ is the appropriate scaling variable on which the linedensity of bound states N inner effectively depends.Accordingly, we have organized the numerical values N inner from Fig. 4 into the single plot shown in Fig. 7,where N inner is reported vs ∆ ξ over a double-log scaleso as to put on equal footing quite different sets of data(spanning four decades, from 10 − to 10 ). Here, ξ is isunits of the inverse of the Fermi wave vector k F and ∆ in units of the Fermi energy E F , such that the variable∆ ξ of Fig. 7 is dimensionless.The resulting linear scaling dependence of N inner vs∆ ξ (which is evidenced in the figure by the straightline N inner /k F = 0 .
263 ∆ ξ ) appears quite remarkable,to the extent that essentially all data with individuallyquite different values of ∆ and ξ fall on this straightline. We have further verified that this scaling relation N i nne r / k F Δ ξ Δ ξ FIG. 7. (Color online) Scaling plot showing the values of theline density of bound states N inner (in units of k F ), obtainedfor given coupling and temperature, reported as a function ofthe variable ∆ ξ , where the values of ∆ and ξ are obtainedat given coupling and temperature. For each coupling valuelisted in the figure, the values of N inner correspond to the threetemperatures T = (0 , . , . T c (with the exception of thecoupling ( k F a F ) − = − . T = 0 . T c is also added, and of the coupling ( k F a F ) − = − . T = 0 . T c is missing). between N inner and ∆ ξ holds also when the counting ofbound states residing in the inner region is enlarged, byshifting ξ , e.g., to 2 ξ in Eq. (6), provided that the samereplacement ξ → ξ is also made in the variable ∆ ξ onwhich N inner depends. IV. CONCLUDING REMARKS
In this paper, we have given a detailed account aboutthe number, energy location, spatial location, and shapeof the large number of bound states that are presentwithin an isolated vortex embedded in an otherwise ho-mogeneous superfluid. We have done this by an accuratesolution of the BdG equations based on the methodsdeveloped in Ref. [11], for a dilute Fermi gas spanningthe BCS-BEC crossover with the temperature rangingfrom zero to the superfluid critical temperature. In thisway, we have been able to provide a criterion for locatingwhere a given Fermi gas lies along this crossover, in termsof the asymmetry between positive and negative energiesof the local density of states. From our numerical calcu-lations we were also able to extract a universal scalingrelation, that relates the number of bound states in theinner region of the vortex with the depth and width ofthe vortex itself.We have further verified that bound states occur inthe interior of a vortex for couplings on the BCS side ofunitary, and rapidly disappear when entering the BECregime past unitarity. Consistently with this theoreticalfinding, a clear experimental finding for the occurrence ofbound states in a vortex in a superconducting material [1]can indeed be greeted as a signature that the superfluidphase of that material may be described by the conven-tional BCS theory [2]. Yet, it is also known from theanalysis of Ref. [11] that it is rather the continuum partof the spectrum of a vortex (as obtained by solving theBdG equations) that exhausts, in practice, most part ofthe contribution to physical quantities, such as the profileof the gap parameter and the local number density.In this paper, the analysis has been limited to consid-ering an s -wave order parameter, and the question mayarise about the role of the symmetry of the order pa-rameter on our results. In this context, we may refer tothe results of Ref. [20], where a BdG calculation for anisolated vortex with a d -wave order parameter was re-ported, with the conclusion that the spatial dependenceof the quasi-particle density of states is similar to the onewith an s -wave order parameter, except possibly for thecontribution from the extended states.A further comment is in order on the existence ofbound states in the continuum (usually referred to asBICs) that we have found in an isolated vortex for sym-metry reasons, owing to the different values of the quan-tum k z between these bound states and the continuumstates in which they are embedded. As a consequence,a slight geometric perturbation of the vortex line alongthe z -direction could turn BIC states into decaying res-onances [21]. In this context, it is interesting to mentionthat BIC states have recently gained considerably ex-perimental and technological interest as a generic wavephenomenon which can occur in supercavity lasing [22].On physical grounds, the Caroli-de Gennes-Matriconbound states that are present in the interior of a vor-tex have a similar origin as the Andreev-Saint-Jamesstates that show up as subgap states in the context ofthe Josephson effect [23–25] (for a more recent reviewdealing also with cuprate superconductors, see Ref. [26]).Also for the Andreev-Saint-James states, a systematictheoretical study of their occurrence has been performedthroughout the BCS-BEC crossover by solving the BdGequations at zero temperature [27]. It would then beinteresting to address experimentally the occurrence ofthese subgap states also along the way of the BCS-BECcrossover, in terms of suitable spectroscopic measure-ments [28]. ACKNOWLEDGMENTS
This work was partially supported by the Italian MIURunder Contract PRIN-2015 No. 2015C5SEJJ001.
Appendix A: INTERNAL STRUCTURE OF AVORTEX
As mentioned in Section II, the accurate numerical so-lution of the BdG equations performed in Ref. [11] hasevidenced the feature that the radial profile of the gapparameter has a ( ρ − ) power-law tail for all couplingsthroughout the BCS-BEC crossover (as well as for all temperatures in the superfluid phase). In this Appendixwe show that, in the BEC (strong-coupling) limit of theBCS-BEC crossover, whereby the fermionic BdG equa-tions reduce to the bosonic Gross-Pitaevskii (GP) equa-tion for the composite bosons that form in this limit [16],the ( ρ − ) long-range behavior of the condensate wavefunction Φ( r ) = (cid:113) m a F π ∆( r ) can be determined by sim-ple analytic considerations. Although this result has al-ready been reported for a vortex filament in an almostideal Bose gas described at low temperature by the GPequation [29], the reason to briefly discuss it here is thatits relevance for a vortex in a fermionic superfluid de-scribed by the BdG equations has passed essentially un-noticed in the literature [1, 5, 18].The (time-independent) GP equation reads [15]: (cid:20) − m B ∇ + V ext ( r ) + 4 πa B m B | Φ( r ) | − µ B (cid:21) Φ( r ) = 0(A1)where m B , a B , and µ B are the bosonic mass, scatteringlength, and chemical potential, respectively. [For com-posite bosons built up in terms of superfluid fermionsdescribed by the BdG equations, m B = 2 m , a B = 2 a F ,and µ B = 2 µ + ε where ε = ( ma F ) − is the bindingenergy of the composite bosons [16].]For an isolated vortex filament directed along the z axis, one sets V ext ( r ) = 0 and writes generically thevortex solution in cylindrical coordinates in the formΦ( r ) = √ n e isϕ f ( ρ ), where s is an integer referred toas the topological charge of the flow (we will set s = 1at the end of the calculation). Far away from the vortexaxis, one expects the local bosonic density n ( r ) = | Φ( r ) | to reach its bulk value n , such that µ B = U n where U = πa B m B . Introducing at this point the bosonic heal-ing length ξ B = (2 m B U n ) − / and the rescaled radialvariable η = ρ/ξ B , the GP equation (A1) acquires theform:1 η ddη (cid:18) η df ( η ) dη (cid:19) + (cid:18) − s η (cid:19) f ( η ) − f ( η ) = 0 . (A2)To determine the asymptotic behavior of f ( η ) for η (cid:29) f ( η ) = 1 + Cη γ + · · · (A3)with γ <
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