We describe the Fermionic Casimir effect in the case of two spherical
superfluid scatterers immersed in a normal Fermi system. It is shown that due
to the focusing property of Andreev reflection this new Casimir-like energy is
significantly enhanced when compared to the case of normal scatterers with
specular reflection.Comment: 7 pages, 4 eps figures, latex (with epl.cls

Bulgac, Aurel

Magierski, Piotr

Wirzba, Andreas

English

arXiv

We describe the fermionic Casimir effect in the case of two spherical superfluid scatterers immersed in a normal Fermi system. It is shown that due to the focusing property of Andreev reflection this new Casimir-like energy is significantly enhanced when compared to the case of normal scatterers with specular reflection

Bulgac, A.

Magierski, P.

Wirzba, A.

Juelich Shared Electronic Resources

arXiv:cond-mat/0406255v2  [cond-mat.supr-con]  27 Sep 2005Europhysics Letters PREPRINTFermionic Casimir effect in case of Andreev reflectionA. Bulgac1, P. Magierski1,2 and A. Wirzba31 Department of Physics, University of Washington - Seattle, WA 98195–1560, USA2 Faculty of Physics, Warsaw University of Technology - ul. Koszykowa 75, 00-662Warsaw, Poland3 Institut fu¨r Kernphysik, Forschungszentrum Ju¨lich - D-52425 Ju¨lich, GermanyPACS. 03.65.Sq – Semiclassical theories and applications.PACS. 74.45.+c – Proximity effects; Andreev effect; SN and SNS junctions.PACS. 74.81.-g – Inhomogeneous superconductors and superconducting systems.Abstract. – We describe the Fermionic Casimir effect in the case of two spherical superfluidscatterers immersed in a normal Fermi system. It is shown that due to the focusing propertyof Andreev reflection this new Casimir-like energy is significantly enhanced when compared tothe case of normal scatterers with specular reflection.Introduction. – If two or more voids (or hard-wall scatterers) are formed in a normaluniform Fermi system, effective Casimir-like interactions between such voids appear [1,2]. Forexample, two spherical voids of radius a with distance between their centers R in a Fermimedium at zero temperature and with Fermi wave vector kF would interact with each otherwith the effective potential energyEC(R) ≃ −νdeg~2k2F2ma22piR(R− 2a)j1[2kF(R− 2a)] , (1)where νdeg is the spin-isospin degeneracy factor. Here j1(x) is the spherical Bessel functionand m the fermion mass. There is an illuminating explanation and easy derivation of thisexpression based on semi-classical physics — periodic orbits theory in particular. One canshow that the energy density of an infinite and otherwise homogeneous Fermi system is affectedby the presence of the two spherical voids and that the main contribution to this energy densitychange is determined by the “almost standing wave” created between the two scatterers. Aswas shown in refs. [1,2], this argument can be extended, in principle, to an arbitrary numberof scatterers of arbitrary shapes. Volovik has arrived at similar conclusions by studying 1-dimensional systems [3] and he refers to this energy as the Mesoscopic Casimir energy orforce. In his discussion of this new type of Casimir energy, Volovik stresses that this energyis governed by the “microscopic trans-Planckian physics,” unlike the usual Casimir energy,which is determined by the long-wave physics.The Fermionic Casimir energy appears because of the Dirichlet boundary conditions atthe voids, which semi-classically correspond to specular reflections. If, however, the boundaryseparates a normal Fermi system from a superfluid Fermi system, the specular reflection isc© EDP Sciences2 EUROPHYSICS LETTERSφ S1 S2 θ N Fig. 1 – Two superfluid grains S1 and S2 immersed in a normal Fermi gas N, φ is the angle atcenter specifying the position of the emerging ray on the first disk/sphere and θ the angle made byits momentum with the normal.replaced with Andreev reflection [4]. A particle at the Fermi energy in the normal part ofthe Fermi system, impinging on the boundary between the normal and superfluid system,is retro-reflected as a hole, irrespective of the shape of the boundary. This rather simplebut amazing fact has apparently quite unexpected consequences. Let us imagine now thatwe have two spherical superfluid grains embedded in a normal Fermi system. One mightthen naively expect that there is an infinite number of stable (classical) periodic trajectoriesbetween two such objects. This situation has to be contrasted with the case of specularreflection, when only one unstable trajectory between the “noses” of the two spheres exist.The latter trajectory is responsible for the change in the density of states, which ultimatelyresults in the Fermionic Casimir energy discussed in refs. [1–3]. One can now expect that aninfinite number of stable periodic trajectories would lead to a very large change in the densityof states if two superfluid grains are embedded into a normal Fermi system. From this pointof view the fact that two spherical superfluid scatterers do not generate a chaotic dynamicsis initially quite a surprise. Our goal here is to determine the character and magnitude ofthe Fermionic Casimir energy in the case of two spherical superfluid scatterers embedded in anormal Fermi system. The result we obtain can easily be generalized to scatterers of arbitraryshapes and to any number of scatterers in a leading order approximation, which we shalldefine more carefully below. We would like to mention here that even though other aspectsof the physics of superfluid grains of various sizes embedded in a normal Fermi system havebeen addressed before (see refs. [5, 6] and references therein), the existence and nature of theFermionic Casimir energy was never considered. Various other aspects of the quantum chaosin so-called Andreev billiards/stadiums have been considered by many authors [7], but spatialconfigurations of the type we shall study here have not. Our interest stems partially fromthe physics of neutron stars, where such structures are expected to exist [1,8]. Moreover, thepossible relevance of Andreev reflection was mentioned in connection with the appearance ofQCD-color-superfluid “grains” in the deconfined quark-gluon phase in the core of neutron [9]or of quark stars, and also in connection with the observed cosmological asymmetry betweenmatter and anti-matter and the origin of dark matter [10].Two superfluid grains. – To set the stage, let us consider the geometry illustrated in fig. 1.In the Andreev approximation any straight line joining two arbitrary points on the surface ofthe two superfluid grains leads to a classical periodic orbit. A particle (p) leaving the surfaceof one grain, hits the second grain somewhere and it is reflected exactly backward as a hole(h) and it thus returns to the initial point on the first grain, where it now transforms, afterA. Bulgac et al.: Fermionic Casimir effect 3−40 −30 −20 −10 0 10 20 30 40−8−6−4−202468φφ−θFig. 2 – The Poincare´ surface of section for the case of two superfluid circular/spherical grainsembedded in a normal Fermi gas (in two dimensions), separated by a distance four times larger thentheir respective radii (R=6a). The ratio of the particle and hole momenta is here kp/kh=1.5. Onlystable/elliptic trajectories parameterized by the angle φ (in degrees), where the particle originates onthe surface of the disk S1, and by the angle φ− θ (in degrees) between the momentum of the particleand a parallel of the center-to-center axis are shown. The relatively small values of φ − θ show thatmost trajectories run in a tube around the line joining the two centers.another Andreev reflection, into a particle. This is strictly correct if the particle has exactlythe Fermi energy. Then the energy of both particle and hole states are equal. Otherwisethe relation between the angles of the incoming particle/hole and the reflected hole/particleresembles Snell’s law:kp sin θp = kh sin θh ,k2p2+k2h2= 2µ = k2F , (2)where µ is the Fermi energy and kp > kF, kh < kF denote the fermion momentum in a particlestate and a hole state, respectively (we shall use units: ~2 = m = 1). Note that the energiesof the particle and hole states are symmetric with respect to the Fermi energy. Moreover,the incident and reflected rays are on the same side of the normal to the surface at the pointof incidence, as opposed to the case of specular reflection. Consequently, one can show that,unlike in the case of normal reflection, the disks/spheres now have a focusing effect.Instead of trying to seek various possible periodic trajectories we simply generated aPoincare´ surface of section. Only those orbits are stable for which the separation betweenthe centers of the grains satisfies the following constraint, derived by requiring that the eigen-values of the monodromy matrix have to be inside or on the unit circle in the complex plane:R2a≤kpkp − kh, (3)where we have assumed that |kp−kh|/kF ≪ 1. If this constraint is not satisfied the orbits havea hyperbolic/unstable character. In the case of equality the eigenvalues of the monodromymatrix are equal to (minus) one, the orbits are parabolic/marginally stable in character andelliptic/stable otherwise.Exact expression of the density of states. – The structure of the classical phase spacebriefly described here would apparently imply a rather complex quantum mechanical dynam-4 EUROPHYSICS LETTERSics. As we shall show, however, the most important features of the density of states in suchsystems can be understood and described quite accurately in terms of a simplest periodicorbit and its repetitions. In order to determine the change in the local density of states weshall construct explicitly the scattering states and determine the phase shifts by extendingthe formalism used in case of normal systems [2, 11, 12] to Bogoliubov–de Gennes equations,(T− µ ∆(r)∆∗(r) −(T− µ))(u(r)v(r))= E(u(r)v(r)), (4)where T is the kinetic energy operator, E is the quasi-particle energy, u(r) and v(r) arethe particle and hole spatial components of the quasi-particle wave function and ∆(r) is thepairing field, assumed to be constant inside the two scatterers, i.e. |∆(r)| = ∆, and vanishingoutside. For the sake of simplicity of the analysis we assume that the mean field vanisheseverywhere and that the two scatterers have the same radius a. Physically it makes senseto discuss many fermion systems for which kFa ≫ 1. One can distinguish further betweensmall and large superfluid grains, depending on whether the coherence length ξ = kF/pi∆ issmaller or larger than the radius of the grain a. The pairing fields in the two grains need notto have the same phase, which for the sake of simplicity of the argument we consider to beconstant over the grain. As in ref. [2], we shall use Krein’s formula [13], generalized to thecase of Bogoliubov–de Genes equations. Krein’s formula expresses the change in the densityof states through the phase shifts, and represents a generalization of a much simpler formuladue to Beth and Uhlenbeck [14]. We thus obtain in the 2-dimensional case (the 3-dimensionalcase has a very similar structure):δg(E) =d δN(E)dE, δN(E) = −1piIm ln det(M) , (5)M =... ... ... ...... Mmm′ Mmm′+1 ...... Mm+1m′ Mm+1m′+1 ...... ... ... ... ,Mmm′ =(11 (−1)m+1Amm′(−1)m+1Bmm′ 11),Amm′ =(tmpp(a,E, µ,∆)H(1)m−m′(kpR) tmph(a,E, µ,∆)H(2)m−m′(khR)tmhp(a,E, µ,∆)H(1)m−m′(kpR) tmhh(a,E, µ,∆)H(2)m−m′(khR)),Bmm′ =(tmpp(a,E, µ,∆)H(1)m−m′(kpR) tmph(a,E, µ,∆)eiφ∆H(2)m−m′(khR)tmhp(a,E, µ,∆)e−iφ∆H(1)m−m′(kpR) tmhh(a,E, µ,∆)H(2)m−m′(khR)),with k2p = 2(µ + E), k2h = 2(µ − E) and where m,m′ = 0,±1,±2, · · · are angular momenta.The angle φ∆ is the relative phase of the pairing fields in the two grains and 11 is the identitymatrix. The matrix elements tmpp ≡ tmpp(a,E, µ,∆), . . . are equal to half of the T -matrixelements for the scattering of a particle/hole of angular momentum m off a single sphericalsuperfluid grain (centered at the origin of the coordinate system), and the ordinary Hankelfunctions H(1)m−m′(kpR), . . . determine the propagation between the grains.Asymptotic approximations. – Even though these formulas are already suitable to per-form explicit calculations, one can take advantage of the fact that kFR ≫ 1 and obtainsignificantly simpler expressions by retaining only the leading term in an asymptotic expan-sion of the corresponding Hankel functions. One thus arrives at the following much moreA. Bulgac et al.: Fermionic Casimir effect 50 20 40 60 80 100 120 140−202T pp0 20 40 60 80 100 120 140−101T hh0 20 40 60 80 100 120 140−200002000E [1/a2]T phFig. 3 – The structure functions Tpp, Thh and Tph for the case of two identical disks with µ = 200/a2and ∆ = 50/a2 and φ∆ = 0. The solid lines are the real parts of the corresponding quantities andthe dashed lines the imaginary parts, respectively. Notice the difference in the magnitudes.transparent expressions:Tph =(∑mtmph)2cosφ∆ ≡ |Tph| exp(iφph) cosφ∆ ,Tpp =(∑mtmpp(−1)m)2≡ |Tpp| exp(iφpp) , Thh =(∑mtmhh(−1)m)2≡ |Thh| exp(iφhh) ,δN(E) ≈4|Tph| cosφ∆pi2√khkpRsin[(kp − kh)R+ φph]−2|Tpp|pi2kpRcos(2kpR+ φpp)+2|Thh|pi2khRcos(2khR− φhh) . (6)The structure functions Tph, Tpp and Thh characterize the properties of the scatterers andare independent of the separation R between them. In fig. 3 we show some typical behaviorof these functions with the quasi-particle energy E. Note the great difference in size betweenthe normal (pp, hh) and Andreev (ph) structure functions, especially when E ≤ ∆.Even though these structure functions are relatively easy to evaluate numerically in thecase of almost any conceivable size grains, it is extremely instructive to go one step furtherand derive an approximate expression for these structure functions themselves. An analysis ofthe structure functions (omitted here) eventually leads us to the following significantly simplerexpression for the change in the density of states valid for E ≪ |∆|:δN(E) = −Im[∫ amin(kp,kh)−amin(kp,kh)dν exp(i(kp − kh)ν22kpkha)]2ei(kp−kh)(R−2a)pi2R√kpkh cos(φ∆) . (7)One can easily recognize the argument of the second exponential as the classical action of theshortest particle-hole periodic orbit. The hole momentum enters with a minus sign since itis anti-parallel to the hole velocity. The squared integral would become a product of similarfactors in the case of two grains of different radii. For E=0 this overall factor becomes4m2max = 4k2Fa2, i.e. the squared number of partial waves undergoing Andreev reflection.6 EUROPHYSICS LETTERS0 20 40 60 80 100−1.5−1−0.500.511.5E [1/a2]δ N(E)Fig. 4 – The density of states variation δN(E) evaluated exactly (solid line), in the approximationof eq. (6) (dashed line) and under a further asymptotic expansion of the cylindrical functions in thestructure functions (dotted line) for two identical disks with R=6a, µ = 200/a2, ∆=50/a2 and φ∆=0.Results and conclusions. – In fig. 4 we compare the change in the density of statesevaluated exactly (see eq. (5)) with the simplified expression of eq. (6) and with a furtherasymptotic approximation of the structure functions on which eq. (7) is based. It is nowobvious that the oscillatory character of the change in the density of states is fully accountedfor by the shortest periodic orbit, quantized as expected by(kp − kh)(R − 2a)− 2 arccos (E/∆) = 2pin , n = 0,±1,±2, · · · . (8)The maxima in δg(E) = dδN(E)/dE correspond exactly to those energies for which the abovecondition is fulfilled. At each Andreev reflection the reflected wave acquires an additionalphase arccos(E/∆), leading to the overall minus sign in eq. (7), valid for the case E ≪ ∆.By using the exact expression (5) or the simplifications (6) or (7), one can now easilyevaluate the interaction energy of spherical superfluid grains in the Fermi sea,ESC(R) = νdeg∫∞0dE Ed δN(E)dE= −νdeg∫∞0dE δN(E) , (9)as function of their separation R. This Casimir energy is completely dominated by the stateswith E ≤ ∆. We thus obtain, by numerical integration, for two spherical three-dimensionalgrainsESC(R) ≈ νdeg~2k2F2mkFa4 cosφ∆2piR2(R− 2a)X , (10)where X = O(1) and positive and very weakly dependent on separation. By comparing witheq. (1), one can show that ESC(R) ≫ |EC(R)| for the typical case kFa ≫ 1. This is easy toaccept now: in the case of two superfluid grains, embedded in a normal Fermi system, there aremany “bound states” (see fig. 2) due to the appearance of a large number of periodic orbits,whereas there exists only a single unstable periodic orbit between two hard spheres [1, 2].In summary, we have constructed both exact and simplified expressions for the calculationof the density of states and the Fermionic Casimir energy in the novel case of two sphericalsuperfluid scatterers immersed in a normal Fermi system. We have furthermore shown andA. Bulgac et al.: Fermionic Casimir effect 7explained that these quantities are significantly enhanced in comparison to the normal casewhere the obstacles are scatterers with hard-wall reflection conditions. Since the change in thedensity of states is dominated by the Andreev reflection, which is almost exactly backward,the case of many grains reduces simply to the sum over all unobstructed pairs of grains. Inparticular, one can expect that the energetics in portions of the neutron stars would be greatlyaffected when the Fermionic Casimir energy is taken into account — more so than was thecase when only normal systems were considered [1, 8]. Generalizations of the present resultsto more complicated geometries appear to be straightforward.∗ ∗ ∗This work was supported in part by DOE grant DE-FG02-97ER41014, by the PolishCommittee for Scientific Research (KBN) under Contract No. 1 P03B 059 27 and by theForschungszentrum Ju¨lich under Contract No. 41445400 (COSY-067). We would like to thankone of the referees for valuable suggestion.REFERENCES[1] Bulgac A. and Magierski P., Nucl. Phys. A, 683 (2001) 695; Erratum: 703 (2002) 892;Phys. Scripta T, 90 (2001) 150.[2] Bulgac A. and Wirzba A., Phys. Rev Lett., 87 (2001) 120404.[3] Volovik G. E., JETP Lett., 73 (2001) 375; The Universe in a Helium Droplet, (ClarendonPress, Oxford) 2003, sect. 29-5.[4] Andreev A. F., Zh. Eksp. Teor. Phys., 46 (1964) 1823 [JETP, 19 (1964) 1228]; de GennesP. G., Superconductivity of Metals and Alloys (Addison-Wesley, Reading) 1998; Blonder G.E., Tinkham M. and Klapwijk T. M., Phys. Rev. B, 25 (1982) 4515.[5] Feigel’man M. V. and Larkin A. I., Chem. Phys., 235 (1998) 107; Spivak B., Zyuzin A.and Hruska M., Phys. Rev. B, 64 (2001) 132502.[6] Shytov A. V., Lee P. A. and Levitov L. S., Usp. Fiz. Nauk, 168 (1998) 222.[7] Kosztin I., Maslov D. L. and Goldbart P. M., Phys. Rev. Lett., 75 (1995) 1735; AltlandA. and Zirnbauer M. R., Phys. Rev. Lett., 76 (1996) 3420; Lodder A. and Nazarov Yu.V., Phys. Rev. B, 58 (1998) 5783; Schomerus H. and Beenakker C. W. J., Phys. Rev. Lett.,82 (1991) 2951; Ihra W. et al., Eur. Phys. J. B, 21 (2001) 425; Adagideli I. and GolddbartP. M., Int. J. Mod. Phys., 16 (2002) 1381; Jacquod Ph. et al., Phys. Rev. Lett., 90 (2003)207004; Korma´nyos A. et al., 70, 2004 (052512) and many other references therein.[8] Bulgac A. et al., Tours Symposium on Nuclear Physics V, edited by M. Arnould et al.,Vol. 704 (AIP Conference Proceedings, Melville, NY) 2004, pp. 483-487 and references therein.[9] Sadzikowski M., Acta Phys. Polon. B, 33 (2002) 1601; Sadzikowski M. and Tachibana M.,Phys.Rev. D, 66 (2002) 045024; Acta Phys. Polon. B, 33 (2002) 4141.[10] Oaknin D. H. and Zhitnitsky A., Phys. Rev. D, 71 (2005) 023519.[11] Henseler M., Wirzba A. and Guhr T., Ann. Phys. (N.Y.), 258 (1997) 286.[12] Lloyd P., Proc. Phys. Soc., 90 (1967) 207; Lloyd P. and Smith P. V., Adv. Phys., 21 (1972)69; Berry M. V., Ann. Phys. (N.Y.), 131 (1981) 163.[13] Krein M. G., Math. Sborn., 33 (1953) 597; Sov. Math.-Dokl., 3 (1962) 707; Birman M. Sh. andKrein M. G., Sov. Math.-Dokl., 3 (1962) 740; Thirring W., Quantum Mechanics of Atomsand Molecules (Springer–Verlag, New York) 1981, sect. 3.6; Birman M. Sh. and Yafaev D.R., St. Petersburg Math. J., 4 (1993) 833.[14] Beth E. and Uhlenbeck G. E., Physica (Utrecht), 4 (1937) 915; Huang K., StatisticalMechanics (John Wiley & Sons, New York) 1987, sect. 10.3.

Fermionic Casimir Effect in Case of Andreev Reflection

We describe the fermionic Casimir effect in the case of two spherical
superfluid scatterers immersed in a normal Fermi system. It is shown
that due to the focusing property of Andreev reflection this new
Casimir-like energy is significantly enhanced when compared to the
case of normal scatterers with specular reflection

Fermionic Casimir Effect in Case of Andreev Reflection

Abstract

Similar works

Full text

Available Versions

Juelich Shared Electronic Resources

EDP Sciences OAI-PMH repository (1.2.0)