The possibility of the condensation of excitations with non-zero momentum in rectilinearly moving and rotating superfluid bosonic and fermionic (with Cooper pairing) media is considered in terms of a phenomenological order-parameter functional at zero and non-zero temperature. The results might be applicable to the description of bosonic systems like superfluid $4He$, ultracold atomic Bose gases, charged pion and kaon condensates in rotating neutron stars, and various superconducting fermionic systems with pairing, like proton and color-superconducting components in compact stars, metallic superconductors, and neutral fermionic systems with pairing, like the neutron component in compact stars and ultracold atomic Fermi gases. The order parameters of the “mother” condensate in the superfluid and the new condensate of excitations, corresponding energy gains, critical temperatures, and critical velocities are found.

## 1. Introduction

The possibility of the condensation of rotons in superfluid helium (He-II) moving in a capillary at zero temperature with a flow velocity exceeding the Landau critical velocity $vcL$ was suggested in Ref. [1]. In Ref. [2], the condensation of excitations with non-zero momentum in various relativistic and non-relativistic cold media moving with velocity exceeding $vcL$ was studied further with the help of the effective Lagrangian for the complex scalar field, which describes Bose excitations in the medium. The Landau critical velocity is determined by the minimum of $\u03f5(k)/k$ at finite momentum $k$, where $\u03f5(k)$ is a branch of the spectrum of Bose excitations. Possible manifestations of the phenomenon in the bulk of He-II, rotating neutron stars with and without pion condensate, nuclei at high angular momentum, and heavy-ion collisions were discussed. A similar effect can also occur in a normal Fermi liquid with a zero-sound branch in the spectrum of particle–hole excitations [3,4]. When the velocity of the Fermi liquid exceeds the Landau critical velocity related to this branch, the number of excitations should grow exponentially with time and in the course of their interactions they may form a Bose condensate with a finite momentum. This possibility was studied in Ref. [3] for a moving Fermi liquid at finite temperature. Various consequences of the phenomenon in application to nuclear systems were announced. In Ref. [5], the results of Ref. [1] for He-II in a capillary were extended to He-II in a bulk. The condensation of excitations in cold atomic Bose gases moving with a flow velocity exceeding $vcL$ was considered in Ref. [6]. The role of a Bose condensate of zero-sound-like excitations with non-zero momentum in the description of the stability of $r$ modes in rapidly rotating pulsars was discussed in Ref. [7].

Below, we study the possibility of the condensation of excitations in a state with non-zero momentum in moving media in the presence of a superfluid subsystem. The systems of interest are neutral bosonic superfluids, such as superfluid $4He$, cf. [1,8–11], cold Bose atomic gases, cf. [6,12–14], inhomogeneous $K\xaf0$ condensates in neutron stars, cf. [15,16], charged bosonic superfluids like $\pi +$ and $\pi \u2212$ and $K\u2212$ condensates with $k\u22600$ in neutron stars, cf. [2,15–17], and various Fermi systems with Cooper pairing, like the neutron superfluid in neutron star interiors, cf. [18], cold Fermi atomic gases, cf. [19], neutron gas in neutron star crusts, cf. [20], or charged superfluids, as paired protons in neutron star interiors, cf. [18], paired quarks in color-superconducting regions of hybrid stars, cf. [21], and paired electrons in metallic superconductors, cf. [22,23].

The key idea of the phenomenon is the following [1,2]: When a medium moves as a whole with respect to a laboratory frame with a velocity higher than $vcL$, it may become energetically favorable to transfer part of its momentum from particles of the moving medium to a Bose condensate of excitations (CoE) with a non-zero momentum $k\u22600$. This would happen if the spectrum of excitations is soft in some region of momenta. References [1,6] studied the condensation of excitations at $T=0$ assuming the conservation of flow velocity. Alternatively, we consider systems with other conditions, assuming the conservation of momentum (or angular momentum for rotating systems) as in Ref. [2]. We consider bosonic and fermionic superfluid systems moving initially with a flow velocity above $vcL$ both for $T=0$ and $T\u22600$ (the latter case has not yet been considered in the mentioned references), taking into account a back reaction of the CoE on the “mother” condensate of the superfluid.

The work is organized as follows. In Sect. 2, we construct the phenomenological order-parameter functional for the description of the CoE coupled with the mother condensate in the superfluid moving linearly with a flow velocity exceeding $vcL$. Section 3 is devoted to a description of cold moving superfluids. Section 4 studies peculiarities of the two-fluid motion in warm superfluids in the presence of a CoE. In Sect. 5, we discuss a particular role of vortices. Some numerical estimates valid for fermion superfluids in the BCS limit and for He-II are performed in Sect. 6. Section 7 describes the CoE in rotating systems with application to rapidly rotating pulsars. Section 8 contains concluding remarks.

## 2. Order-parameter functional for moving fluid

In the spirit of the Landau phenomenological theory of a second-order phase transition, the free-energy density of a superfluid subsystem in its rest frame can be expanded in the order parameter $\psi $ for temperatures $T\u2264Tc$, where $Tc$ is the critical temperature of the second-order phase transition [9,10]:

For $0<t=1\u2212T/Tc\u226a1$, the coefficients $aT$ and $bT$ can be expanded as [10] $aT=a0t\alpha $ and $bT=b0t\beta $, and $cT$ is usually assumed to be constant, $cT=c0$. Within the mean-field approximation from the Taylor expansion of $FL$ in $t\u226a1$, it follows that $\alpha =1$, $\beta =0$. The width of the fluctuation region, wherein the mean-field approximation is not applicable, is evaluated with the help of the Ginzburg [10] and Ginzburg–Levanyuk [24,25] criteria. For ordinary metallic superconductors the fluctuation region proves to be usually very narrow and the mean-field approximation then holds for almost any temperature below $Tc$, except for a tiny vicinity of $Tc$. Thus, for $t\u226a1$, neglecting the mentioned narrow fluctuation region, one may use $\alpha =1$, $\beta =0$. For He-II, fluctuations prove to be important for all temperatures below $Tc$, cf. [10]. Using the experimental fact that the specific heat of He-II has no power divergence at $T\u2192Tc$, we get $\alpha =4/3$ and $\beta =2/3$, which coincides with phenomenological findings [10].

Consider a system at a finite temperature consisting of normal and superfluid parts undergoing rectilinear motion parallel to a wall. The wall singles out the laboratory frame with respect to which the motion is defined. Interactions between particles in normal fluid may lead to the creation of excitations. The mechanisms of excitation production depend on the specifics of problems, and will be discussed below in Sects. 4, 5, and 7.

We assume that the superfluid moves with an initial velocity $v\u2192$ with respect to the wall, and additionally the excitations can carry some net momentum, $j\u2192n$, with respect to the superfluid. Then one can define an average velocity of the excitations with respect to the superfluid component $w\u2192$. With respect to the wall, the excitations have the average velocity $v\u2192n=w\u2192+v\u2192$. The motion of the superfluid as a whole with velocity $v\u2192$ relative to the reference frame of the wall can be described by introducing the phase of the condensate field $\psi =|\psi |ei\varphi $ with $v\u2192=\u210f\u2207\varphi /m.$

We can write the variational functional for the condensate field in the standard form of the two-fluid model [11]:

When the speed of the flow $v$ exceeds the Landau critical velocity,

For the description of the CoE with the given frequency $\u03f5(k0)$, the functional (1) must be supplemented by the functional $Fex[\psi ]$ involving higher-gradient terms so that the variation of the Fourier transform of the full functional reproduces the excitation frequency

We suppose that, when the CoE is formed (we shall call it a “fin”-state), the initial momentum density is redistributed between the fluid and the CoE:

In the presence of the CoE, the resulting order parameter $\psi fin$ is the sum of the mother condensate, $\psi $, and the CoE, $\psi \u2032$, $\psi fin=\psi +\psi \u2032$. The volume-averaged free-energy density of the system with the CoE, $F\xaffin=F\xafL[\psi fin]+F\xafex[\psi fin]$, can be written as

## 3. Cold superfluid

### 3.1. Bosonic system

At $T\u21920$ the whole medium is superfluid and the amplitudes of the condensates are constrained by the spatially averaged particle number density,

Replacing Eq. (11) in (10) and putting $T=0$, we find the change of the spatially averaged energy density of the system because of the appearance of the CoE, $\delta E\xaf=E\xaffin\u2212E\xafin$,

As follows from Eq. (16), the CoE appears in a second-order phase transition since $d\delta E\xafdv|vcL=0$ but $d2\delta E\xafdv2|vcL\u22600$. The amplitude of the CoE (14) grows with the velocity, whereas the amplitude of the mother condensate decreases. The value $|\psi |2$ vanishes when $v=vc2$, the second critical velocity, at which $|\psi \u20320|2=n\xaf$ according to Eq. (11). The value $vc2$ is evaluated from (14) as

### 3.2. Fermionic system

As shown in Refs. [19,20,30], in fermionic systems with pairing there may exist bosonic modes with suitable spectra, supporting quasiparticle excitations with energy $\u22432\Delta $ and momentum $k0\u22432pF$, where $\Delta $ is the pairing gap computed in the rest frame of the superfluid; see Fig. 2 in [19], and Fig. 4 in [20]. For these modes, the Landau critical velocity is

Besides bosonic excitations there exist fermionic ones with spectrum $\u03f5f(p)=\Delta 2+vF2(p\u2212pF)2$. Stemming from the breakup of Cooper pairs, the fermionic excitations are produced pairwise and the corresponding (fermion) Landau critical velocity is $vc,fL=minp\u21921,p\u21922[(\u03f5f(p1)+\u03f5f(p2))/|p\u21921+p\u21922|]$. The latter expression reduces to [31]

For $T\u21920$, the fermionic excitations are produced near the wall and move, therefore, with respect to the superfluid with the velocity $\u2212v\u2192$.^{1} Hence, the change of the energy density due to the Cooper pair breaking can be calculated as

Thus we can conclude that the creation of the condensate of bosonic excitations with finite momentum in moving cold fermionic systems with pairing leading to a reduction of the flow velocity is energetically more profitable than the breaking of Cooper pairs and the decrease of the pairing gap.

## 4. Warm superfluid, two-fluid motion

Only for a very low $T$ can the normal component be neglected. For a higher temperature, the normal subsystem serves as a reservoir of particles for the formation of the mother and daughter condensates, whose amplitudes are now to be chosen by minimization of the free energy of the system. Therefore, minimizing (10), we now vary $\psi $ and $\psi \u20320$ independently and find

From Eqs. (8) and (25) we find for $v>vcL$ and $\chi T<1$ the resulting velocity of the flow.

Substituting the order parameters from Eq. (25) to Eq. (10), we find for the averaged free-energy density gain owing to the appearance of the CoE

At finite temperature the dynamics of the CoE amplitude can be determined from the equation [34]

When a fluid flowing with $v>vcL$ at $T>T\u02dcc(v)$ is cooled down to $T<T\u02dcc(v)$, it consists of four components: the normal excitations, the superfluid, the vortices, and the CoE, all moving rigidly with $vfin<vcL$ (if $\chi T>0$). If the system is then rapidly re-heated to $T>T\u02dcc(v)$, the superfluid component, the vortices, and the CoE vanish and the remaining normal fluid consists of two fractions: one still moving with $vfin(T\u02dcc)<vcL$, owing to conservation of the momentum, and the other one, originating from the melted CoE, with mass equal to $ma(T\u02dcc)/(2b\u2032T,k0(T\u02dcc))$, moving with a higher velocity until a new equilibrium is established. This may show one possibility for how one could identify the formation of the CoE experimentally.

Note that for fermion superfluids at $T\u22600$ after the CoE is formed, the flow velocity $vfin<vc,fL$, for $v\u2212vcL>4tvcL/9$ (the estimate is done for $\chi T=3b0\rho \xaf/k02$), and hence the Cooper pair breaking does not occur, whereas the condensate of Bose excitations is preserved.

## 5. Vortices

Above, we focused our consideration on the cases where either the vortices are absent (as in a narrow capillary [1]) or they leave the system (in open systems), or the presence of vortices supports a common rigid motion of the normal and superfluid components [22] (e.g., as in systems with charged components [35], or in rotating systems, like neutron stars [18]).

In the case of He-II moving in a narrow capillary, vortices do not appear—see [1,5]. For rectilinearly moving superfluids in extended geometry there may appear excitations of the type of vortex rings and other structures [36]. The energy of the ring is estimated [10,11] as $\u03f5vort=2\pi 2\u210f2|\psi |2R$$m\u22121ln(R/\xi )$, and the momentum is $pvort=2\pi 2\u210f|\psi |2R2$, where $R$ is the radius of the vortex ring and $\xi $ is the coherence length, $\xi ~\u210f(cT/aT)1/2$, as estimated above. Thus, $vc1=\u03f5vort/pvort=\u210f(Rtm)\u22121ln(Rt/\xi )$ is the Landau critical velocity for the vortex production, where in the absence of impurities $Rt$ is of the order of the transverse size of the system. For a system of distributed impurities moving together with the fluid, $Rt$ is the typical distance between the defects. Vortices are pinned to the impurities and move together with them and the superfluid. In an open clean system at $v>vc1$ the vortex rings are pushed to infinity by Magnus and Iordanskii forces. Note that for spatially extended systems the value $vc1$ is lower than the Landau critical velocity $vcL$. A flow moving with velocity $v$ for $vc1\u2264v$ may be considered as metastable, since the vortex creation probability is hindered by a large potential barrier and formation of a vortex takes a long time [37,38]. The vortex production rate increases strongly, however, when $v$ approaches $vcL$ [37,38]. For motion in a pipe, the vortices are captured by the pipe wall, forming after a while a stationary subsystem in the frame of the walls. Periodic solitonic solutions of the Gross–Pitaevskii equation were studied in [39]. This situation might be rather similar to that of a mother condensate moving in a periodic potential, produced by the spatial variations of the CoE order parameter [6]. Since in exterior regions of the vortices the superfluidity persists, our consideration of the condensation of excitations for $vcL<v$ is applicable. Note that in He-II under a high external pressure $vcL$ decreases and at some conditions becomes lower than $vc1$, see [40], and in the interval $vcL<v<vc1$ there are no vortices but the CoE may appear.

In superconducting systems vortices, if formed, are involved in a common motion with the superconducting subsystem due to the appearance of a tiny London field [35] distributed throughout the medium that supports the condition $w=0$.

In rotating superfluids, vortices appear at rotation frequency $\Omega >\Omega c1=\u210fmR2ln(R/\xi )$, where for the spherical system $R$ is the size of the system (transversal size for the cylindrical system), and their number grows with an increase of $\Omega $. When the density of vortices becomes sufficiently large, they form a lattice, cf. [22], thereby forcing the superfluid and normal components to move as a rigid body, i.e. with $w\u21920$.

## 6. Estimates for fermionic and bosonic superfluids

We now apply the expressions derived in the previous sections to several practical cases.

### 6.1. Fermionic superfluid

Consider a fermion system with the singlet pairing. In the weak-coupling (BCS) approximation the parameters of the functional (1) can be extracted from the microscopic theory [9]:

With parameters (31) we estimate $b0\rho \xaf/k02=3\Delta 2/(8vF2pF2)$ and $a0/k0=3\Delta 2/(4vFpF2),$ where $\rho \xaf\u2243n\xafmF$. We see that if $b\u2032T,k0~b\u2033T,k0~bT$ one gets $0<\chi T=3bT\rho \xaf/k02\u226a1$, since the latter inequality is reduced to the inequality $\Delta \u226a\u03f5F$, which is well satisfied. In this limit, $|\psi 0\u2032|2$ given by Eq. (25) gets the same form as Eq. (14). The resulting flow velocity after condensation of excitations, (27), is lower than $vcL$ but close to it.

Since for the BCS case we have $\alpha =1$, $\beta =0$, Eq. (26) for the new critical temperature is easily solved, for $v>vcL$:

### 6.2. Bosonic superfluid: He-II

We turn now to the bosonic superfluid, He-II. In He-II there exists a branch of the phonon–roton excitations [9,10]. The typical energy of the rotonic excitations $\Delta r=\u03f5(kr)$ at the roton minimum $k=kr$ depends on the pressure and temperature. According to [41], for the saturated vapor pressure $\Delta r=8.71\u2009K$ at $T=0.1\u2009K$ and $7.63\u2009K$ at $T=2.10\u2009K$, and $kr\u22431.9\u22c5108\u210f\u2009cm\u22121$ in the whole temperature interval. Other parameters of He-II at the saturated vapor pressure are [10]:

Taking into account that we deal with the rotonic excitation, *i.e.*, $k0\u2243kr$ and $\u03f5(k0)\u2243\Delta r$, we estimate

Using the results of [41], $vcL(T)$ dependence can be fitted with 99% accuracy as

## 7. Rotating superfluids: pulsars

The novel phase with the CoE may also exist in rotating systems. Here, excitations can be generated because of the rotation. Now we should use angular momentum conservation instead of momentum conservation. Also, the structure of the order parameter is more complicated than the plane wave. For the cylindrical geometry a probing CoE function can be taken in the form [2]

In the inner crust and in a part of the core of a neutron star, protons and neutrons are paired in the $1S0$ state owing to attractive $pp$ and $nn$ interactions, cf. [18]. In denser regions of the star interior the $1S0$ pairing disappears, but neutrons might be paired in the $3P2$ state. The charged $pp$ superfluid component should co-rotate with the normal matter. This, as we have mentioned, is due to the appearance of a tiny magnetic field $h\u2192=2mp\Omega \u2192/ep$ (London effect) in the whole volume of the superfluid; $mp$ ($ep$) is the proton mass (charge) [35]. This tiny field, being $\u227210\u22122G$ for the most rapidly rotating pulsars, has no influence on parameters of the star and can be neglected.

With a typical neutron star radius $R~10\u2009km$, and for $\Delta ~MeV$ typical for the $1S0\u2009nn$ pairing, we estimate $\Omega c1~10\u221214Hz$. For $\Omega \u226b\Omega c1$ the neutron star contains arrays of neutron vortices with regions of superfluidity in between them, and the star rotates as a rigid body. The vortices would completely overlap only if $\Omega $ reached the unrealistically large value $\Omega c2vort~1020Hz$. The most rapidly rotating pulsar PSR J1748-2446ad has the angular velocity 4500 Hz [42]. The value of the critical angular velocity for the formation of the CoE in neutron star matter is $\Omega c~\Omega cL\u2243\Delta /(pFR)~102Hz$ for the pairing gap $\Delta ~MeV$ and $pF~300\u2009MeV/c$ at the nucleon density $n~n0$, where $n0\u22430.17\u2009fm\u22123$ is the density of the atomic nucleus, and $c$ is the speed of light. The superfluidity will coexist with the CoE and the array of vortices until the rotation frequency $\Omega $ reaches the value $\Omega c2>\Omega cL$, at which both the CoE and the superfluidity disappear completely. From Eq. (26) with the BCS parameters we estimate $\Omega c2~vc2/R\u2272104Hz$.

There are many other millisecond pulsars in low-mass X-ray binaries of a typical age $\u2273108\u2009yr$. Thus, in the detected rapidly rotating pulsars the CoE might coexist with superfluidity, which would also affect their hydrodynamical description [43–45]. A possible influence of the CoE on the window of the r-mode instability in the millisecond pulsars was recently studied by us in [7]. Also, a CoE may appear in the presence of a charged pion condensate with a finite momentum in massive neutron stars [17]—see a discussion of an additional slowing down of the pulsar which may arise owing to the presence of the $\pi +$ condensation in [2]. In massive neutron stars there may also exist $K\u2212$ and/or $K\xaf0$ condensates with finite momentum, cf. [15,16]. A similar effect to that on a charged pion condensate may exist on $K\u2212$ and $K\xaf0$ condensates. Another interesting issue is the possibility of the formation of CoEs in color-superconducting regions of rotating hybrid stars. Various CoEs may arise there since pairing gaps between quarks of different colors and flavors may have essentially different values, e.g. in 2SC, 2SC + X, color spin locking, and other possible phases, see [46].

## 8. Conclusion

In this paper we studied the possibility of the condensation of excitations with $k\u22600$, when a superfluid initially flows with respect to a wall with a velocity $v$ larger than the Landau critical velocity $vcL$. Differing from Refs. [1,5,6], which studied bosonic superfluid systems for $T=0$ at a fixed velocity $v$, we considered this phenomenon for bosonic and fermionic superfluid systems both for $T=0$ and $T\u22600$ at the conserving momentum for rectilinear motion (at the conserving angular momentum for a rotation). In the presence of the CoE the final velocity of the superfluid $vfin$ becomes less than $v$. Also, compared to Refs. [1,2,5] we incorporated the interaction between the CoE and the “mother” condensate of the superfluid. We studied the case of $T\u226aTc$, when the normal component can be neglected, and the case of higher $T$, when it serves as a reservoir of particles affecting the formation of the mother condensate and CoE. The latter case was not previously described in the literature.

At finite temperatures we first studied systems where the superfluid and normal components move with respect to each other with a relative velocity $w\u2192$ (the average velocity of excitations with respect to the superfluid component), and then focused on the case of $w=0$. Note that at finite $T$ the mother condensate may exist only for very low values of $w\u2192$ (much less than the Landau critical velocity). In rotating superfluids vortices form a lattice and the system rotates as a rigid body. Also, charged subsystems are forced to move as a whole owing to the London force. These are conditions when indeed one can put $w=0$.

A back reaction of the CoE on the mother condensate proves to be important both for $T=0$ and for $T\u22600$. We found that the CoE appears in a second-order phase transition at $v=vcL$ and the condensate amplitude grows linearly with increasing velocity. Simultaneously, the mother condensate decreases and vanishes at $v=vc2$, then the superfluidity is destroyed in a first-order phase transition with an energy release. For $vcL<v<vc2$ the resulting flow velocity is $vfin<vcL$.

We found that for cold fermion systems with pairing the creation of the condensate of bosonic excitations with finite momentum, leading to a reduction of the flow velocity up to the value of the Landau critical velocity $vcL$, is energetically more profitable than the breaking of Cooper pairs appearing for $v>vc,fL$ ($vcL>vc,fL$) and the decrease of the pairing gap (except the case when the initial velocity $v$ is in a narrow vicinity of the critical point). To the best of our knowledge the possibility of condensation of bosonic excitations with finite momentum in moving fermionic systems with pairing has not yet been considered in the literature. For fermion superfluids at $T\u22600$, after the CoE is formed the flow velocity becomes less than $vc,fL$ and the Cooper pair breaking does not occur, whereas the condensate of Bose excitations is preserved. The CoE appears in the second-order phase transition. The mother condensate decreases and vanishes at $v=vc2(T)$, then the superfluidity is destroyed in a first-order phase transition with an energy release.

We discussed condensation of Bose excitations in rotating superfluids, such as pulsars, and showed that in the existing most rapidly rotating millisecond pulsars superfluidity might coexist with the CoE.

## Acknowledgements

We thank M. Yu. Kagan for detailed discussion of the results. The work was supported by the Ministry of Education and Science of the Russian Federation (Basic part), by Slovak Grant No. VEGA-1/0469/15, and by “NewCompStar,” COST Action MP1304.

## References

^{1}At finite temperatures fermionic excitations are mainly produced inside the pre-existing normal component moving with the velocity $w\u2192$ with respect to the superfluid component.