Running condensate in moving superfluid
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n epl draft Running condensate in moving superfluid
E.E. Kolomeitsev and D.N. Voskresensky Matej Bel University, SK-97401 Banska Bystrica, Slovakia National Research Nuclear University (MEPhI), 115409 Moscow, Russia
PACS – Nuclear matter aspects of neutron stars
PACS – He Superfluid phase
PACS – Theories and models of superconducting state
PACS – Bose-Einstein condensates, dynamic properties
Abstract –A possibility of the condensation of excitations with non-zero momentum in movingsuperfluid media is considered in terms of the Ginzburg-Landau model. The results might beapplicable to the superfluid He, ultracold atomic Bose gases, various superconducting and neutralsystems with pairing, like ultracold atomic Fermi gases and the neutron component in compactstars. The order parameters, the energy gain, and critical velocities are found.
Introduction. –
A possibility of the condensation ofrotons in the He-II, moving in a capillary at zero temper-ature with a flow velocity exceeding the Landau criticalvelocity, was suggested in [1]. In [2] the condensation ofexcitations with non-zero momentum in various movingnon-relativistic and relativistic media (not necessarily su-perfluid) was studied. A possibility of the formation of acondensate of zero-sound-like modes with a finite momen-tum in normal Fermi liquids at non-zero temperature wasfurther discussed in [3]. The condensation of excitationsin cold atomic Bose gases was recently studied in [4]. Theworks [1,2,4] disregarded a non-linear interaction betweenthe “mother” condensate of the superfluid and the con-densate of excitations. The condensation of excitationsin superfluid systems at finite temperatures, i.e. , in thepresence of a normal subsystem, also has not been studiedyet.The key idea is as follows [1, 2]. When a medium movesas a whole with respect to a laboratory frame with a veloc-ity higher than a certain critical velocity, it may becomeenergetically favorable to transfer a part of its momentumfrom particles of the moving medium to a condensate ofBose excitations with a non-zero momentum k = 0. Itwould happen, if the spectrum of excitations is soft insome region of the momenta. Whether the system is mov-ing linearly with a constant velocity or it is resting, is in-distinguishable according to the Galilei invariance. Thus,there should still exist a physical mechanism allowing toproduce excitations. The excitations can be created neara wall, which singles out the laboratory reference frame, or they can be produced by interactions among particlesof the normal subsystem at non-zero temperature. Theycan be also generated provided the motion is non-inertial, e.g. , in the case of a rotating system.In the He-II there exists a branch of roton excita-tions [5, 6]. The typical value of the energy of the exci-tations ∆ r = ǫ ( k r ) at the roton minimum for k = k r de-pends on the pressure and temperature. According to [7],for the saturated vapor pressure ∆ r = 8 .
71 K at T = 0 . r = 7 .
63K at T = 2 .
10 K, and k r ≃ . · ¯ h/ cmin the whole temperature interval. (We put the Boltz-mann constant k B = 1). An appropriate branch of ex-citations may exist also in normal Fermi liquids [3] andin cold Bose [4, 8] and Fermi [9] atomic gases. In neutralFermi liquids with the singlet pairing, characterized by thepairing gap ∆, there exist [10] the low-lying Anderson-Bogoliubov mode of excitations with ǫ ( k ) = kv F / √ k → ǫ ( k ) →
2∆ for large k ( k < ∼ p F ), and theSchmid mode ǫ ( k ) ≃ p F is the Fermi momen-tum, v F is the Fermi velocity. In charged Fermi liquidswith the singlet pairing there is also the suitable low-lyingCarlson-Goldman mode starting at zero energy for a smallmomentum and reaching the value ǫ = 2∆ for large k .Below, we study a possibility of the condensation of ex-citations in the state with a non-zero momentum in mov-ing media in the presence of the superfluid subsystem.The systems of our interest are neutral bosonic superflu-ids, such as the superfluid He and cold Bose atomic gases,and systems with the Cooper pairing, like the neutron liq-uid in neutron star interiors, cold Fermi atomic gases orp-1.E. Kolomeitsev and D.N. Voskresenskycharged superfluids, as paired protons in neutron star inte-riors and paired electrons in metallic superconductors. Incontrast to previous works we take into account that thesuperfluid subsystem and the bosonic excitations shouldbe described in terms of the very same macroscopic wave-function. Also, our consideration is performed for non-zero temperature, i.e. , the presence of the normal compo-nent is taken into account.
Ginzburg-Landau functional. –
We start with ex-pression for the Ginzburg-Landau (GL) free-energy den-sity of the superfluid subsystem in its rest reference framefor the temperature
T < T c , [5, 6]: F GL [ ψ ] = c | ¯ h ∇ ψ | − a ( t ) | ψ | + 12 b ( t ) | ψ | ,a ( t ) = a t α , b ( t ) = b t β , t = ( T c − T ) /T c . (1)Here T c is the critical temperature of the second-orderphase transition, and c , a and b are phenomenologicalparameters. When treated within the mean-field approxi-mation, the functional F GL should be an analytic functionof t . Then, from the Taylor expansion of F GL in t it followsthat α = 1, β = 0. The width of the so-called fluctuationregion, wherein the mean-field approximation is not ap-plicable, is evaluated from the Ginzburg criterion: in thisregion of temperatures in the vicinity of T c , long-rangefluctuations of the order parameter are mostly probable,i.e. their probability is W ∼ e − F eqGL V fl /T ∼
1, where F eqGL is the equilibrium value, V fl ∼ ξ is the minimal volumeof the fluctuation of the order parameter, the coherencelength ξ is the typical length scale characterizing the or-der parameter.For metallic superconductors the fluctuation regionproves to be very narrow and the mean-field approxima-tion holds for almost any temperatures below T c , excepta tiny vicinity of T c . Thus, neglecting this narrow fluctu-ation region one may use α = 1, β = 0 in (1). For thefermionic systems with the singlet pairing, in the weak-coupling (BCS) approximation the parameters can be ex-tracted from the microscopic theory [5]: c = 1 / m ∗ F , a = 6 π T c / (7 ζ (3) µ ) , b = a /n , (2)where m ∗ F stands for the effective fermion mass ( m ∗ F ≃ m F in the weak-coupling limit), n = p / (3 π ¯ h ) is the particlenumber density, and the fermion chemical potential is µ ≃ ǫ F = p / (2 m ∗ F ). The function ζ ( x ) is the Riemann ζ -function and ζ (3) = 1 . t ≪
1. With these BCS parameters wehave | ψ | = nt and the pairing gap ∆ = T c q π t ζ (3) , see [11].For He-II the fluctuations prove to be important for alltemperatures below T c [6]. Including long-range fluctua-tions, the coefficients of the Ginzburg-Landau functionalare now renormalized: due to a divergency (logarithmic orpower-law-like) of the specific heat at the critical temper-ature T c , quantities a and b in eq. (1) become non-analytic functions of t with non-integer α and β . When the con-tributions of long-range fluctuations are completely takeninto account the Ginzburg parameter F eqGL ξ must becomeindependent of the temperature. Since F eqGL ∝ t α − β and ξ ∝ t − α/ , we obtain α/ − β = 0. Using the experi-mental fact that the specific heat of the He-II contains nopower divergence at T → T c , we get 2 α − β − α = 4 / β = 2 / T c = 2 .
17 K, a /T / c = 1 . · − erg / K / , b /T / c = 3 . · − erg · cm / K / . This parameteriza-tion holds for 10 − < t < .
1, but for rough estimates canbe used up to t = 1. E.g. , using eq. (1), with the heliumatom mass m = 6 . · − g we evaluate the He-II mass-density as ma /b ≃ . / cm , which is of the order ofthe experimental value ρ He = 0 .
15 g / cm at P = 0. Moving cold superfluid. –
We consider now a su-perfluid moving with a constant velocity ~v parallel to awall. The latter singles out the laboratory frame and aninteraction of the fluid with the wall may lead to creationof excitations in the fluid. We start with the case of van-ishing temperatures. The whole medium is superfluid andthe amplitude of the order parameter can be related tothe particle density ρ s = m n = m | ψ in | = m a /b , ψ in is the order parameter in the absence of the excitations(“in”-state). The energy of the medium in the laboratoryframe is E in = mnv / − b n / v , exceeds the Landau crit-ical velocity, v L c , near the wall there may appear excita-tions with the momentum k = k and the energy ǫ ( k ) ascalculated in the rest frame of the superfluid, where themomentum k corresponds to the minimum of the ratio ǫ ( k ) /k . For instance, for He-II the spectrum ǫ ( k ) is thestandard phonon+roton spectrum, normalized as ǫ ( k ) ∝ k for small k . The appearance of a large number of excita-tions motivates us to assume that for v > v L c in additionto the mother condensate the excitations may form a newcondensate with the momentum k = k = 0, (“fin”-state).The momentum k should be found now from minimiza-tion of the free energy. As we have noticed before, inprevious works [1, 2, 4] it was assumed that the conden-sate of excitations decouples from the mother condensate.Now we are going to take into account that the condensateof excitations is described by the very same macroscopicwave-function as the mother condensate. Then the result-ing order parameter ψ fin is the sum of the contributions ofthe mother condensate, ψ , and the condensate of excita-tions, ψ ′ , i.e., ψ fin = ψ + ψ ′ . We choose the simplest formof the order parameter for the condensate of excitations, ψ ′ = ψ ′ e − i ( ǫ ( k ) t − ~k ~r ) / ¯ h , (3)with the amplitude ψ ′ , being constant for the case of thehomogeneous system that we consider. The particle num-ber conservation yields n = | ψ + ψ ′ | = | ψ | + | ψ ′ | + . . . . (4)p-2unning condensate in moving superfluidHere ellipses stand for the spatially oscillating term, whichvanishes after the averaging over the system volume.As the particle density, the initial momentum density isredistributed in our case between the fluid and the con-densate of excitations: ρ s ~v = ( ρ s − m | ψ ′ | ) ~v fin + ( ~k + m~v fin ) | ψ ′ | . (5)The energy of the moving matter in the presence of thecondensate of excitations is E fin = ρ s v ǫ ( k ) + ǫ bind ) | ψ ′ | − a | ψ | + b n , (6)where the energy of the excitation ǫ ( k ) should be countedfrom the binding energy of a particle in the condensate atrest ǫ bind = ∂E in ( v = 0) /∂n = − b n = − a . Replacingeq. (5) in (6) and using eq. (4) we express the change of thevolume-averaged energy density owing to the appearanceof the condensate of excitations, δ ¯ E = ¯ E fin − ¯ E in , as δ ¯ E = ( ǫ ( k ) − k v ) | ψ ′ | + k | ψ ′ | / (2 ρ s ) . (7)Minimizing this functional with respect to ψ ′ we obtainthe condensate amplitude | ψ ′ | = ( ρ s /k ) ( v − v L c ) θ ( v − v L c ) , v L c = ǫ ( k ) /k . (8)From (5) we find that because of condensation of excita-tions with k = 0 the flow is decelerated to the velocity v fin = v L c . The volume-averaged energy density gain dueto the appearance of the condensate of excitations is δ ¯ E = − ρ s ( v − v L c ) θ ( v − v L c ) . (9)Minimization of δ ¯ E with respect to k gives the condi-tion d v L c / d k = 0, which is exactly the condition for thestandard Landau critical velocity. The condensate of ex-citations appears by a second-order phase transition. Theamplitude of the condensate of excitations (8) grows withthe velocity, whereas the amplitude of the mother con-densate decreases. The value | ψ | vanishes when v = v c ,the second critical velocity, at which | ψ ′ | = n accord-ing to eq. (4). The value of the second critical velocity v c is evaluated from (8) as v c = v L c + k /m . When themother condensate disappears, at v = v c , the excitationspectrum is cardinally reconstructed, and the superfluid-ity destruction occurs as a first-order phase transition. Weassume that for v > v c the excitation spectrum has nolow-lying local minimum at finite momentum. Then theamplitude | ψ ′ | jumps from n to 0 and δ ¯ E jumps from δ ¯ E ( v c ) = − ρk / (2 m ) to 0 at v = v c . The case, whenin the absence of the mother condensate the spectrum ofBose excitations has a low-lying local minimum at k = 0,has been considered in [2, 3]. Note that in practice thereconstruction of the spectrum may occur for a smallervelocity than that we have estimated. For example, forfermionic superfluids it always should be | ψ ′ | ≪ n , oth-erwise the Fermi sea itself would be destroyed. In moving superfluids there exist excitations of the typeof vortex rings. The energy of the vortex is estimated as ǫ vort = 2 π ¯ h | ψ | Rm − ln( R/ξ ), see [6, 12], and the mo-mentum is p vort = 2 π ¯ h | ψ | R , m here is the mass ofthe pair for systems with pairing, and the mass of theboson quasiparticle in bosonic superfluids, e.g. , the massof the He atom in case of the He-II, R is the radius ofthe vortex ring, and ξ ∼ c / a − / t − α/ is the minimallength scale associated with the mother condensate. Thus, v c = ǫ vort /p vort = ¯ h ( Rm ) − ln( R/ξ ) , is the Landau criti-cal velocity for the vortex production, where now R is thedistance of the order of the size of the system. For v > v c the vortex rings are pushed out of the medium, if the den-sity profile has even slight inhomogeneity. Note that forspatially extended systems the value v c is usually lowerthan the Landau critical velocity v L c . The flow movingwith the velocity v for v c ≤ v ≤ v c may be consid-ered as metastable, since the vortex creation probabilityis hindered by a large potential barrier and formation of avortex takes a long time [13]. Note that already for v justslightly exceeding v c , the number of the produced vor-tices may become sufficiently large and their interactionforces the normal and superfluid components to move as asolid, with the same velocity, even if initially they have haddifferent velocities. In the exterior regions of the vortexcores the superfluidity still persists and our considerationof the condensation of excitations in the velocity interval v L c < v < v c is applicable.In the presence of the condensate of excitations thedensity becomes spatially oscillating around its averagedvalue. For a weak condensate, i.e. , | v − v L c | ≪ v L c , wefind perturbatively δn = n − ¯ n ≈ √ n | ψ ′ | cos(( ǫ ( k ) t − ~k ~r ) / ¯ h ). Such a density modulation predicted in [1] wasreproduced in the numerical simulation of the supercriticalflow in He-II using the realistic density functional [14].The above consideration holds for any superfluid withthe conserved number of particles in the mother conden-sate plus the condensate of excitations, e.g. , for the coldBose gases, if the spectrum of the over-condensate excita-tions is such that ǫ ( k ) /k has a minimum at k = k = 0,as has been conjectured in [4].In case of the Fermi systems with pairing, forthe bosonic modes (Anderson-Bogoliubov, Schmid andCarlson-Goldman ones) with the excitation energy ≃ k ≃ p F , cf. [9, 15]. Hence,for these modes the Landau critical velocity is v L c ≃ ∆ /p F , and for v > v L c there is a chance for the conden-sation of the Bose excitations as we considered above.Besides bosonic excitations there exist fermionic oneswith the spectrum ǫ f ( p ) = p ∆ + v ( p − p F ) . Stem-ming from the breakup of Cooper pairs, the fermionicexcitations are produced pairwise. Therefore, the cor-responding (fermion) Landau critical velocity is v L c, f =min ~p ,~p [( ǫ f ( p ) + ǫ f ( p )) / | ~p + ~p | ] . The latter expres-sion reduces to [16] v L c, f = (∆ /p F ) / (1 + ∆ /p v ) / . Wep-3.E. Kolomeitsev and D.N. Voskresenskysee that up to a small correction v L c, f ≃ v L c . For T → ≤ v/v L c − ≪ < ~p ~v > /ρ ≃ p v L c ( v − v L c ) / and the energy gain due to the fermion pair breaking is δ ¯ E pair = Z p (2 π ¯ h ) ( ǫ f ( p ) − ~p ~v ) θ ( ǫ f ( p ) − ~p ~v ) ≈ − √ ρ ( v − v L c ) [ v/v L c − / . (10)Moreover, for v > v L c, f the pairing gap decreases with in-crease of v as [17] ∆( v ) / ∆ ≈ − (3 / v/v L c − , reachingzero for v = v L c , f = e v L c (Rogers-Bardeen effect [18]). Theenergy gain (10) is less than (9) and the production ofthe condensate of Bose excitations is energetically moreprofitable than the Cooper pair breaking. Since in thepresence of the condensate of excitations v fin = v L c an ad-ditional energy gain due to the appearance of the latter is: F eqGL ( T = 0 , ∆) − F eqGL ( T = 0 , ∆( v )) ≈ − (9 / ρ ( v − v L c ) ,for 0 ≤ v/v L c − ≪
1. For v > v L c , f the gain becomes F eqGL ( T = 0 , ∆) = − ρ ( v L c ) / Moving superfluid-normal composites. –
Weturn now to the case of systems consisting of normal andsuperfluid parts, like He-II at finite temperature, metal-lic superconductors, or neutron star matter. Here, thenumber of particles in the condensate is not conservedeven at v = 0 since a part of particles can be transferredfrom the superfluid to the normal subsystem. The stateof the superfluid subsystem is described by the GL func-tional (1). We assume that the normal and superfluid sub-systems move with the same velocity ~v with respect to alaboratory frame. This means that we consider the mo-tion of the fluid during the time τ shorter than the typicalfriction time, τ normfr , at which the normal component isdecelerated, if the fluid has a contact with the wall.Minimization over the order parameter in the rest frameof the fluid yields | ψ eq v =0 | = a ( t ) /b ( t ) , F eqGL [ ψ eq v =0 ] = − b ( t ) | ψ eq v =0 | / . (11)In the absence of the population of excitation modes thesuperfluid and normal subsystems decouple. In this casethe initial free-energy density of the system is given by F in = ρv / F bind − a ( t ) / (2 b ( t )) . (12)Here ρ is the total (normal+superfluid) mass density, ρ = ρ n ( v = 0) + m | ψ ( v = 0) | , and F bind is a bindingfree-energy density of the normal subsystem in its restreference frame (coinciding with the rest frame of the su-perfluid in the case under consideration). The explicitform of F bind is not of our interest.When the condensate of excitations is formed, the initialmomentum density is redistributed between the fluid andthe condensate of excitations: ρ ~v = ( ρ − m | ψ ′ | ) ~v fin + ( ~k + m~v fin ) | ψ ′ | . (13)Here we assume (as argued below) that after the appear-ance of the condensate of excitations in the form (3) the normal and superfluid subsystems continue to move withone and the same velocity ~v fin . In the presence of thecondensate of excitations the free energy density becomes F fin = ρ v + F bind + F GL [ ψ, ∇ ψ = 0] (14)+( ǫ ( k ) + ǫ bind ) | ψ ′ | + 2 b ( t ) | ψ | | ψ ′ | + b ( t ) | ψ ′ | . To get this expression we used eq. (1) and replaced there ψ with ψ fin = ψ + ψ ′ . The energy of excitations shouldbe counted here from the excitation energy on top of themother condensate at rest determined by eq. (11), ǫ bind = ∂F eqGL [ ψ = ψ eq v =0 + ψ ′ ] /∂ | ψ ′ | (cid:12)(cid:12) ψ ′ =0 = − a ( t ) .Now, using the momentum conservation (13) we ex-press ~v fin through ~v and get for the change of the volume-averaged free-energy density associated with the appear-ance of the condensate of excitations, δ ¯ F = b ( t ) (cid:0) | ψ | − a ( t ) /b ( t ) (cid:1) + (cid:0) ǫ ( k ) − k v (cid:1) | ψ ′ | + 2 b ( t ) (cid:0) | ψ | − a ( t ) /b ( t ) (cid:1) | ψ ′ | + e b ( t ) | ψ ′ | , (15)where e b ( t ) = b ( t ) + k /ρ and we put ~k k ~v . We note thatthe normal subsystem serves as a reservoir of particles atthe formation of the condensates, which amplitudes arechosen by minimization of the free energy of the system.Therefore, we vary δ ¯ F with respect to ψ and ψ ′ indepen-dently. Thus, minimizing (15) we find | ψ ′ | = k (cid:0) v − v L c (cid:1) k /ρ − b ( t ) θ (cid:0) v − v L c (cid:1) θ (cid:0) k /ρ − b ( t ) (cid:1) , (16) | ψ | = (cid:0) a ( t ) /b ( t ) − | ψ ′ | (cid:1) θ ( e T c ( v ) − T ) θ ( v c ( t ) − v ) . The quantity e T c stands for the renormalized critical tem-perature, which depends now on the flow velocity, and v c ( t ) stands for the second critical velocity depending on T . The condition | ψ | = 0 implies the relation between v and T v = v L c + a ( t ) k / (2 b ( t ) ρ ) − a ( t ) / (2 k ) . (17)The solution of this equation for the velocity, v c ( t ), in-creases with the decreasing temperature, and the solutionfor the temperature, e T c ( v ), decreases with increasing v . At T = e T c ( v ) or v = v c ( t ) we have | ψ | = 0 but | ψ ′ | = 0,and for T > e T c ( v ) or for v > v c ( t ) the condensate | ψ ′ | vanishes, if, as we assume, for | ψ | = 0 the spectrum ofexcitations does not contain a low-lying branch. Thus, thesuperfluidity is destroyed at T = e T c ( v ) or v = v c ( t ) in afirst-order phase transition.From (13) and (16) we find for v > v L c and k / ( ρb ( t )) > v fin = v L c − ( v − v L c ) / (cid:0) k / (3 b ( t ) ρ ) − (cid:1) < v L c . (18)Substituting the order parameters from (16) in (15),we find for the volume-averaged free-energy density gainowing to appearance of the condensate of excitations δ ¯ F = − ρ v − v L c ) − b ( t ) ρ/k θ ( v − v L c ) θ ( v c − v ) (19)p-4unning condensate in moving superfluidfor k /ρ > b ( t ) . Thus, for v L c < v < v c the free energydecreases owing to the appearance of the condensate of ex-citations with k = 0 in the presence of the non-vanishingmother condensate. The value of k is to be found from theminimization of eq. (19). As e T c , the condensate momen-tum k gets renormalized and differs now from the valuecorresponding to the minimum of ǫ ( k ) /k . For k /ρ ≫ b the expression (19) for the gain in the volume-averagedfree-energy density transforms at T = 0 into the expres-sion (9) for the gain in the volume-averaged energy den-sity. As in the case of T = 0, for T = 0 the condensate ofexcitations appears at v = v L c in the second-order phasetransition but it disappears at v = v c in the first-orderphase transition with the jumps δ ¯ F ( v c ) ≈ a ( t ) k b ( t ) ρ , | ψ ′ ( v c ) | = a ( t )2 b ( t ) . (20)The dynamics of the condensate of excitations is deter-mined by the equation Γ ˙ ψ ′ = − δ ¯ F /δψ ′∗ . We emphasizethat the above consideration assumes that the formationrate Γ of the condensate of excitations is faster than thedeceleration rate 1 /τ normfr of the normal subsystem.When a homogeneous fluid flowing with v > v L c at T > e T c ( v ) is cooled down to T < e T c ( v ), it consists of threecomponents: the normal and superfluid ones and the con-densate of excitations, all moving with v fin < v L c . If thesystem is then re-heated to T > e T c ( v ), the superfluidityand the condensate of excitations vanish and the remain-ing normal fluid consists of two fractions: one is movingwith v fin ( e T c ) < v L c and the other one, δn = a ( e T c ) / (2 b ( e T c )),is moving with a higher velocity until a new equilibriumis established. Note also that for fermion superfluids at T = 0 after the condensate of excitations is formed theflow velocity v fin < v L c, f , for v − v L c > tv L c /
9, and therebythe Cooper pair breaking does not occur, whereas the con-densate of Bose excitations is preserved.
Estimates for fermionic superfluids and He-II. –
We apply now expressions derived in the previous sectionto several practical cases. First, we consider a fermionsystem with pairing. With the BCS parameters (2) we es-timate b ρ/k = 3∆ / (8 v p ) and a /k = 3∆ / (4 v F p ) , where ρ ≃ nm F . We see that inequality k /ρ ≫ b isreduced to inequality ∆ ≪ ǫ F , which is well satisfied.In this limit | ψ ′ | given by eq. (16) gets the same formas eq. (8). The resulting flow velocity after condensa-tion, (18), is lower than the Landau critical velocity butclose to it, v fin ≃ v L c − v L c ) ( v − v L c ) / (8 v ) .Since for the BCS case we have α = 1, β = 0, eq. (17)for the new critical temperature is easily solved, for v > v L c e T c T c = 1 − k b ( v − v L c ) a ( k /ρ − b ) ≈ − v − v L c v F . (21)We also estimate the maximal second critical velocity as v max c ≃ v L c + v F .We turn now to the bosonic supefluid – helium-II. Mak-ing use of the values of the GL parameters presented above v c2 v c2 | ’ | /| (v=0)| v/v Lc v fin /v Lc | | /| (v=0)| t=0.1 t=0.5 Fig. 1: Condensate amplitudes | ψ | and | ψ ′ | , eq. (16), andthe final flow velocity v fin , eq. (18), in superfluid He plot-ted as functions of the flow velocity for various temperatures, t = ( T c − T ) /T c . Vertical arrows indicate the values of thesecond critical velocity v c . Velocities are scaled by the val-ues of the Landau critical velocities v L c ( t = 0 .
5) = 59 m / s and v L c ( t = 0 .
1) = 55 m / s, and the condensates are normalized tothe condensate amplitude in the superfluid at rest, eq. (11). and taking into account that we deal with the rotonicexcitation, i.e. , k ≃ k r and ǫ ( k ) ≃ ∆ r , we estimate, k / ( b ρ ) ≃
47 , v L c ( T → ≃
60 m / s , a /k ≃ / s . Using the results of [7] the temperature dependence of v L c can be fitted with 99% accuracy as v L c ( T ) /v L c (0) ≃ − . e − . / ˜ t + 200˜ te − / ˜ t , where ˜ t = T /T c . Using theseparameters we evaluate the condensate amplitudes andthe final flow velocity as functions of temperature and de-pict them in Fig. 1. The condensate of excitations appearsat v = v L c in a second-order phase transition. For v > v L c the amplitude of the condensate | ψ ′ | ( | ψ | ) increases (de-creases) linearly with v . The closer T is to T c , the steeperthe change of the condensate amplitudes is. The final ve-locity of the flow, which sets in after the appearance of thecondensate of excitations, decreases with the increase of v .For He-II, we have α = 4 / β = 2 / v > v L c : e T c T c = 1 − " k b ρ − s k b ρ − k a h v − v L c i / ≈ − .
05 ( v/v L c ( T c ) − / . (22)The mother condensate | ψ | vanishes when v reachesthe value of the second critical velocity v c , which de-pends on the temperature as v c ≈ v L c ( t ) + (363 t / − . t / )m / s. At v = v c the superfluidity disappears in afirst-order phase transition. The corresponding energy re-lease can be estimated from (20) as δF ( v c ) ≈ a b t / ≃ . t / ( T c ∆ C p ), where ∆ C p = 0 . · erg / (cm K) is thespecific heat jump at T c [6]. Rotating superfluids. Pulsars. –
The novel phasewith the condensate of excitations may also exist in rotat-p-5.E. Kolomeitsev and D.N. Voskresenskying systems. Excitations can be generated because of therotation. Presence of friction with an external wall or dif-ference between velocities of the superfluid and the normalfluid are not necessarily required to produce excitations.Now we should use the angular momentum conservationinstead of the momentum conservation. The structure ofthe order parameter is more complicated than the planewave [2]. With these modifications, the results, which weobtained above for the motion with the constant ~v , con-tinue to hold. The value of the critical angular velocityΩ c ∼ v c /R is very low for systems of a large size R .In the inner crust and in a part of the core of a neutronstar, protons and neutrons are paired in the 1S state ow-ing to attractive pp and nn interactions. In denser regionsof the star interior the 1S pairing disappears but neu-trons might be paired in the 3P state [19]. The chargedsuperfluid should co-rotate with the normal matter with-out forming vortices, this results in the appearance of atiny magnetic field ~h = 2 m p ~ Ω /e p (London effect) in thewhole volume of the superfluid, m p ( e p ) is the proton mass(charge) [19]. This tiny field, being < ∼ − G for the mostrapidly rotating pulsars, has no influence on the relevantphysical quantities and can be neglected.With the typical neutron star radius, R ∼
10 km, andfor ∆ ∼ MeV typical for the 1 S nn pairing, we esti-mate Ω c ∼ − Hz. For Ω > Ω c the neutron starcontains arrays of neutron vortices with regions of thesuperfluidity among them, and the star as a whole ro-tates as a rigid body. The vortices would cover thewhole space, only if Ω reaches unrealistically large valueΩ vort c ∼ Hz. The most rapidly rotating pulsar PSRJ1748-2446ad has the angular velocity 4500 Hz [20]. Thevalue of the critical angular velocity for the formation ofthe condensate of excitations in the neutron star matteris Ω c ∼ Ω L c ≃ ∆ / ( p F R ) ∼ Hz for the pairing gap ∆ ∼ MeV and p F ∼
300 MeV /c at the nucleon density n ∼ n ,where n is the density of the atomic nucleus, and c is thespeed of light. The superfluidity continues to coexist withthe condensate of excitations and the array of vortices un-til the rotation frequency Ω reaches the value Ω c > Ω L c , atwhich both the condensate of excitations and the superflu-idity disappear completely. From eq. (17) with the BCSparameters we estimate Ω c ∼ v c /R < ∼ Hz. Thus,in the detected rapidly rotating pulsars the condensate ofexcitations might coexist with superfluidity.
Conclusion. –
In this letter we studied a possibilityof the condensation of excitations with k = 0, when a su-perfluid flows with a velocity larger than the Landau crit-ical velocity, v > v L c . We included an interaction betweenthe “mother” condensate of the superfluid and the conden-sate of excitations and considered the superfluid at zeroand finite temperatures. We assumed that the superfluidand normal components move with equal velocities. Inpractice it might be achieved for v > v c , where v c is thecritical velocity of the vortex formation. We found thatthe condensate of excitations appears in a second-order phase transition at v = v L c and the condensate amplitudegrows linearly with the increasing velocity. Simultaneouslythe mother condensate decreases and vanishes at v = v c ,then the superfluidity is destroyed in a first-order phasetransition with an energy release. For v L c < v < v c theresulting flow velocity is v fin ≤ v L c , whereby the equality isrealized for T = 0. We argued that for the fermion super-fluids the condensate of bosonic excitations might be sta-ble against the appearance of fermionic excitations fromthe Cooper-pair breaking. Finally, we considered conden-sation of excitations in rotating superfluid systems, suchas pulsars. Our estimates show that superfluidity mightcoexist with the condensate of excitations in the rapidlyrotating pulsars. ∗ ∗ ∗ The work was supported by Grants No.VEGA 1/0457/12 and No. APVV-0050-11, by “New-CompStar”, COST Action MP1304, and by Polatom ESFnetwork.
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