Semiclassical Theory of Bardeen-Cooper-Schrieffer Pairing-Gap Fluctuations
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Semiclassical Theory of Bardeen-Cooper-Schrieffer Pairing-Gap Fluctuations
H. Olofsson, S. ˚Aberg, and P. Leboeuf Mathematical Physics, LTH, Lund University, P.O. Box 118, S-221 00 Lund, Sweden Laboratoire de Physique Th´eorique et Mod`eles Statistiques, CNRS,Bˆat. 100, Universit´e de Paris-Sud, 91405 Orsay Cedex, France
Superfluidity and superconductivity are genuine many-body manifestations of quantum coherence.For finite-size systems the associated pairing gap fluctuates as a function of size or shape. We providea theoretical description of the zero temperature pairing fluctuations in the weak-coupling BCS limitof mesoscopic systems characterized by order/chaos dynamics. The theory accurately describesexperimental observations of nuclear superfluidity (regular system), predicts universal fluctuationsof superconductivity in small chaotic metallic grains, and provides a global analysis in ultracoldFermi gases.
PACS numbers: 74.20.Fg,05.45.Mt,74.78.Na
A microscopic theory of superconductivity based onpairing was set up in 1957 by Bardeen, Cooper and Schri-effer [1]. These theoretical ideas were subsequently ap-plied to finite systems by Bohr, Mottelson and Pines todescribe ground-state superfluid properties of atomic nu-clei [2]. Today pairing effects are central in a broad rangeof quantum systems, including neutron stars, metallicgrains, atomic gases, nuclei, etc [3, 4, 5, 6]. As the sys-tem size diminishes, finite–size effects become importantand lead to corrections with respect to the bulk homo-geneous behavior. Of particular interest is the influenceof the discreteness of the single–particle quantum energylevels. In connection with superconductivity, its impor-tance was initially emphasized by P. W. Anderson [7],who pointed out that superconductivity in small metal-lic grains should disappear when the single–particle meanlevel spacing becomes of the order of the pairing gap. Thevalidity of this criterion was qualitatively confirmed ex-perimentally in the 90’s [8]. Another consequence of thediscreteness of the energy levels is the appearance of fluc-tuations as a parameter is varied. There is at least oneclear experimental evidence of fluctuations of the pairinggap in superfluid systems, through the odd-even stagger-ing of nuclear masses as a function of the mass number.Our purpose here is to present, within a mean–fieldapproximation, a theory of the pairing–gap fluctuationsvalid in the weak coupling BCS limit for arbitrary ballis-tic potentials. Our method is based on periodic orbit the-ory, which has been successful in describing mesoscopicfluctuations of thermodynamic and transport propertiesin many–body systems [9]. The results allow for a de-tailed calculation of the fluctuations in specific systems.In particular, they provide an accurate description of thepairing fluctuations in nuclei (cf Fig. 2). We also focuson statistical properties, which are shown, generically,to be non-universal. The analysis leads to a global andcomplete picture of the typical size of the fluctuations interms of properties of the corresponding classical system,namely regular or chaotic dynamics.Our starting point is the mean field BCS equation for the pairing gap ∆ [1],2 G = Z L − L ρ ( ε ) dε √ ε + ∆ , (1)where G fixes the strength of the pairing (seniority) in-teraction, ρ ( ε ) is the single–particle level density, and wehave put the Fermi energy to zero. The energy cut off ± L is given by the physical conditions, that are often relatedto the determination of the force strength G . Followingsemiclassical approaches, we divide the pairing gap aswell as the single-particle density of states in a smoothpart and a fluctuating part, ∆ = ¯∆ + e ∆ and ρ = ¯ ρ + e ρ ,respectively. In the weak coupling limit ∆ ≪ L , thesmooth part of the gap is given by the well known solu-tion ¯∆ = 2 L exp( − / ¯ ρG ) (see Ref. [10] for regularizationschemes). The fluctuating part of the density e ρ can beexpressed as [9] e ρ ( ε ) = 2 X p ∞ X r =1 A p,r cos( rS p / ¯ h + ν p,r ) , (2)where the sum is over all primitive periodic orbits p (and their repetitions r ) of the classical underlying ef-fective single-particle Hamiltonian. Each orbit is charac-terized by its action S p , stability amplitude A p,r , period τ p = ∂S p /∂ε and Maslov index ν p,r (all evaluated atenergy ε ). Assuming e ∆ ≪ ¯∆, an equation for e ∆ maybe obtained by multiplying Eq. (1) by ∆, replacing thesingle–particle level density by its semiclassical expres-sion, and expanding up to lowest order in fluctuatingproperties. Assuming moreover ¯∆ ≪ L gives e ∆ = ¯∆¯ ρ X p ∞ X r =1 A p,r Y ( rτ p ) cos (cid:18) rS p ¯ h + ν p,r (cid:19) , (3)where Y ( τ ) = Z ∞−∞ dε cos ( τ ε/ ¯ h ) √ ε + ¯∆ = 2 K ( τ /τ ∆ ) . (4)This equation, where all classical quantities involved areevaluated at Fermi energy, contains detailed informationabout the variations of the pairing gap. Note that e ∆ onlydepends on G and L through ¯∆. K ( x ) is the modifiedBessel function of second kind. Through it, a new char-acteristic “pairing time” associated with the pairing gapis introduced, τ ∆ = h π ¯∆ . (5)Since K ( x ) ∝ exp( − x ) / √ x for x ≫
1, the Bessel func-tion exponentially suppresses all contributions for times τ ≫ τ ∆ (making the sum convergent). The averagepart of the gap ¯∆ is thus playing, in this respect, arole very similar to the temperature in the general the-ory of mesoscopic fluctuations (cf Ref. [11]). In contrast, K ( x ) ∝ − log( x ) for x ≪
1, and short orbits (comparedto τ ∆ ) are logarithmically enhanced.Since the value of the actions depend on the shapeof the mean–field potential, Eq. (3) predicts, generically,fluctuations of the pairing gap as one varies, for instance,the particle number, or the shape of the system at fixedparticle number. The fluctuations result from the inter-ference between the different oscillatory terms that con-tribute to e ∆. The symmetries of the potential and thenature (integrable or chaotic) of the underlying classicalmotion are crucial to understand the interference pat-tern. When the motion is regular (integrable), contin-uous families of periodic orbits having the same action,amplitude, etc, exist. The coherent contribution to thesum (3) of these families of periodic orbits produces largefluctuations. In contrast, in the absence of regularity orsymmetries, incoherent contributions of smaller ampli-tude coming from isolated unstable orbits are expected.Moreover, aside the dependence on the regular or chaoticnature of the single-particle motion, the presence or ab-sence of universality in the statistical properties of thefluctuations will depend on the dominance of short orlong periodic orbits.We will make below an analysis of the predictions ofEq. (3) in the nuclear case, as the neutron number isvaried. Before, and in order to avoid at this stage a de-tailed study of a particular system, we concentrate ona global analysis, namely the typical size or root meansquare (RMS) of the BCS gap fluctuations in a genericmesoscopic system. The second moment of the fluctua-tions may be expressed from Eq. (3) as D e ∆ E = ¯∆ τ H Z ∞ dτ Y ( τ ) K ( τ ) , (6)where τ H = h/δ is Heisenberg time ( δ = ¯ ρ − is the single–particle mean level spacing at Fermi energy), and K ( τ )is the spectral form factor, i.e. the Fourier transform ofthe two-point density–density correlation function [12].The structure of the form factor K ( τ ) is characterizedby two different time scales. The first one, the smallest of the system, is the period τ min of the shortest periodicorbit. The form factor is zero for τ ≤ τ min , and dis-plays non-universal (system dependent) features at times τ min < ∼ τ ≪ τ H . As τ further increases, the function be-comes universal, depending only on the regular or chaoticnature of the dynamics, and finally tends to τ H when τ ≫ τ H . The result of the integral (6) thus depends onthe nature of the dynamics, and on the relative value of τ ∆ with respect to τ min and τ H . According to Ander-son criterion [7], superconductivity exists if ¯∆ > δ (weare not interested here in the ultrasmall regime ¯∆ < δ [13] where the BCS theory fails). Then, ¯∆ > δ implies τ ∆ /τ H = δ/ π ¯∆ ≪
1. Because the Bessel function K ex-ponentially suppresses the amplitude for times τ ≫ τ ∆ ,one can safely ignore the structure of the form factor fortimes of the order or bigger than τ H , and use the so calleddiagonal approximation of K ( τ ) [12]. In the simplest ap-proximation, all the non–universal system–specific fea-tures are taken into account only through τ min [11], andone can write K ( τ ) = 0 for τ < τ min and, for τ ≥ τ min , K ( τ ) = τ H for integrable systems and K ( τ ) = 2 τ forchaotic ones with time reversal symmetry.This finally gives the expressions for fluctuations ofthe pairing gap (normalized to the single–particle meanlevel spacing), σ = rD e ∆ E /δ , assuming regular dynam-ics [14], σ = π δ F ( D ) , (7)and assuming chaotic dynamics, σ = 12 π F ( D ) , (8)where F n ( D ) = 1 − R D x n K ( x ) dx/ R ∞ x n K ( x ) dx . Theargument D = τ min τ ∆ = 2 πg ¯∆ δ (9)is a system dependent quantity inversely proportionalto the dimensionless conductance, g = τ H /τ min , an in-trinsic characteristic of the system independent of thepairing coupling. D can also be viewed as the systemsize divided by the coherence length of the Cooper pair, ξ = ¯ hv F / (2 ¯∆), where v F is Fermi velocity. Equations(7) and (8), which together with Eq. (3) are the main re-sults of this study, show that the variance of the pairinggap is a function of its normalized mean part, ¯∆ /δ , andof the dimensionless conductance, g , as shown in Fig. 1.The monotonic function F n ( D ) has the following limit-ing behaviors, F n ( D ) → D →
0, whereas F n ( D ) → D ≫
1. Thus, in a system charac-terized by large g -value, D →
0. In this case F n ( D ) → ∆/δ σ N u c l e i R e g u l a r s y s t e m s Chaotic systemsg=10 g=100 g=1000 g =
10 g =
100 g = Nano-grainsUltrasmallregime Pairingregime - FIG. 1: Fluctuations of the pairing gap as a function of themean value for mesoscopic systems (log-log scale; all quanti-ties normalized with the single-particle mean spacing). Thedashed (regular, in blue) and dotted (chaotic, in red) curvescorrespond to different values of the dimensionless conduc-tance, g , and the limiting case of g → ∞ is shown by solidlines. The results are valid in the pairing regime ¯∆ /δ > of the gap fluctuations given by the prefactors in Eqs. (7)and (8), that correspond to a pure uncorrelated Poissonsequence and to a GOE random matrix spectrum, respec-tively (the latter, σ = 1 / (2 π ), was obtained previouslyin Ref. [13]). This situation is shown by the solid linesin Fig. 1: purely GOE fluctuations imply a constant am-plitude of the normalized fluctuations of the pairing gap,whereas an increase with ¯∆ /δ is seen for systems withuncorrelated spectra. In contrast, in the generic caseof systems characterized by finite values of g , F n , andtherefore the pairing fluctuations, may significantly devi-ate from universality (cf Fig. 1). Thus, in general, purestatistical models (like GOE) do not provide an adequatedescription of the pairing fluctuations.We shall now apply these results to different physicalsituations. Our first example is a system dominated byregular dynamics, namely ground states of atomic nu-clei, which bring the best experimental data availableat present on the superfluidity of finite Fermi systems.The ground-state superfluidity of atomic nuclei implies amass difference between systems with an even and oddnumber of particles. The connection between the pairinggap and the mass differences is given by the three–pointmeasure ∆ ( M ) = B ( M ) − [ B ( M + 1) + B ( M − / M is the odd neutron N or proton Z number. Inthe presence of other possible interactions, this quantityhas been shown to be a very good measure of pairing cor-relations [15]. ∆ is shown in Fig. 2 for neutrons. Theaverage dependence of the neutron and proton gaps iswell approximated, from experimental data, by¯∆ = 2 . A / MeV , (10) Neutron Number N ∆ [ M e V ] Mass Number A R M S ∆ [ M e V ] ~ FIG. 2: Nuclear pairing gaps for neutrons. The blue dashedline shows average experimental gaps, and isotopic variationwithin one standard deviation is marked by the shaded area.Calculations from the cavity model are shown by the solidred line. Average behavior (Eq. (10)) is shown by solid blackline. Inset: RMS of pairing gap fluctuations versus mass num-ber A . The dots are experimental data for protons and neu-trons. Solid and dashed lines are regular (Eq. (7)) and chaotic(Eq. (8)) predictions, respectively. Data from Ref. [24]. where A = N + Z is the total number of nucleons [16].We notice a rather strong variation around the averagevalue. The A dependence of the RMS of the experimentalpairing fluctuations is shown in the inset of Fig. 2.In order to evaluate the RMS of the pairing fluctua-tions from the theoretical expressions, Eqs. (7) and (8),we need the following estimates of nuclear properties (forone nucleon type): mean level spacing, δ ≈ /A MeV,and dimensionless conductance g ≈ . A / [17]. Thisgives D ≈ . A / , which ranges from 0 .
27 for A = 25to 0 .
33 for A = 250. Though these values clearly setatomic nuclei in the regime τ min /τ ∆ < D = τ min /τ ∆ = 0, so that significant deviations fromuniversality are expected. By inserting these estimatesin Eqs. (7) and (8) the fluctuations are easily evaluated,assuming regular or chaotic dynamics. The resultingcurves are compared to the experimental one in the in-set of Fig. 2. Note, as expected [18], the good agree-ment between the regular dynamics and the experimen-tal curve, either in the overall amplitude as well as in the A –dependence.One may go beyond a statistical description, and useEq. (3) to obtain a detailed description of the fluctu-ations. For that purpose, we assume for the nuclearmean field a simple hard-wall cavity potential. The shapeof the cavity at a given number of nucleons is fixed byminimization of the energy against quadrupole, octupoleand hexadecapole deformations. To simplify, we take N = Z . The periodic orbits of a spherical cavity areused in Eq. (3), with modulations factors that take intoaccount deformations and inelastic scattering [19]. Weset the average of e ∆ to zero, as was done with the exper-imental data (although of interest by itself, we will notconsider its behavior here). In Fig. 2 we compare the the-oretical result e ∆( N ) to the experimental value averagedover the different isotopes at a given N . The agreementis excellent; the theory describes all the main featuresobserved in the experimental curve.Our second example are the superconducting proper-ties of nano-sized clean (ballistic) metallic grains [20],where we may expect the dynamics to be chaotic, seeRef. [4]. The existence of a superconducting gap wasdemonstrated in the regime ¯∆ > δ [8], whereas no gapwas observed when ¯∆ < δ (the transition occurs around N ∼ N is the number of conduction electronsin the grain). The N dependence of the average gap ¯∆ ispoorly understood. We will adopt for grains the thin-filmvalue ¯∆ ≈ . × − eV [4]. The mean level spacingis δ = (2 E F ) / (3 N ) ≈ . /N eV, whereas g ≈ . N / .Eq. (9) gives D ≈ . × − N / , which ranges from0 .
05 to 0 .