Running condensate in moving superfluid

0.2 MB·English

by  E. E. Kolomeitsev

epl draft
Running condensate in moving superfluid
E.E. Kolomeitsev1 and D.N. Voskresensky2
1 Matej Bel University, SK-97401 Banska Bystrica, Slovakia
52 National Research Nuclear University (MEPhI), 115409 Moscow, Russia
1
0
2
n PACS 26.60.-c–Nuclear matter aspects of neutron stars
a PACS 67.25.D-–4HeSuperfluidphase
J PACS 74.20.-z–Theories and models of superconducting state
4 PACS 03.75.Kk–Bose-Einstein condensates, dynamicproperties

Abstract –A possibility of the condensation of excitations with non-zero momentum in moving
]
s superfluid media is considered in terms of the Ginzburg-Landau model. The results might be
a applicabletothesuperfluid4He,ultracoldatomicBosegases,varioussuperconductingandneutral
g
systems with pairing, like ultracold atomic Fermi gases and the neutron component in compact
-
t stars. The order parameters, theenergy gain, and critical velocities are found.
n
a
u
q
.
t
a
m Introduction. – A possibility of the condensation of or they can be produced by interactions among particles
rotons in the He-II, moving in a capillary at zero temper- of the normal subsystem at non-zero temperature. They
-
dature with a flow velocity exceeding the Landau critical canbe alsogeneratedprovidedthe motionis non-inertial,
n
velocity, was suggested in [1]. In [2] the condensation of e.g., in the case of a rotating system.
o
excitations with non-zero momentum in various moving
c In the He-II there exists a branch of roton excita-
[non-relativistic and relativistic media (not necessarily su-
tions [5,6]. The typical value of the energy of the exci-

perfluid) was studied. A possibility of the formation of a
1condensateofzero-sound-likemodeswitha finite momen- tations ∆r = ǫ(kr) at the roton minimum for k = kr de-
v pends on the pressure and temperature. According to [7],
1tum in normalFermi liquids atnon-zerotemperature was forthesaturatedvaporpressure∆ =8.71KatT =0.1K
3further discussed in [3]. The condensation of excitations r
and ∆ = 7.63K at T = 2.10K, and k 1.9 108¯h/cm
7incoldatomicBose gaseswasrecently studiedin [4]. The r r ≃ ·
in the whole temperature interval. (We put the Boltz-
0
works[1,2,4]disregardedanon-linearinteractionbetween
0the “mother” condensate of the superfluid and the con- mann constant kB = 1). An appropriate branch of ex-
. citations may exist also in normal Fermi liquids [3] and
1densate of excitations. The condensation of excitations
0 in cold Bose [4,8] and Fermi [9] atomic gases. In neutral
in superfluid systems at finite temperatures, i.e., in the
5 Fermiliquidswiththesingletpairing,characterizedbythe
presenceofanormalsubsystem,alsohasnotbeenstudied
1 pairing gap ∆, there exist [10] the low-lying Anderson-
v:yet. Bogoliubov mode of excitations with ǫ(k) = kvF/√3 for
Xi The keyidea is as follows[1,2]. Whena medium moves k → 0 and ǫ(k) → 2∆ for large k (k ∼< 2pF), and the
Schmid mode ǫ(k) 2∆. Here p is the Fermi momen-
rasawholewithrespecttoalaboratoryframewithaveloc- ≃ F
tum, v is the Fermi velocity. In charged Fermi liquids
aity higher than a certain critical velocity, it may become F
withthesingletpairingthereisalsothesuitablelow-lying
energeticallyfavorabletotransferapartofitsmomentum
Carlson-Goldmanmodestartingatzeroenergyforasmall
from particles of the moving medium to a condensate of
momentum and reaching the value ǫ=2∆ for large k.
Bose excitations with a non-zero momentum k = 0. It
6
would happen, if the spectrum of excitations is soft in Below,we study a possibilityofthe condensationofex-
someregionofthe momenta. Whetherthesystemismov- citations in the state with a non-zero momentum in mov-
ing linearly with a constant velocity or it is resting, is in- ing media in the presence of the superfluid subsystem.
distinguishable according to the Galilei invariance. Thus, The systems of our interest are neutral bosonic superflu-
there should still exist a physical mechanism allowing to ids,suchasthesuperfluid4HeandcoldBoseatomicgases,
produce excitations. The excitations can be created near andsystemswiththeCooperpairing,liketheneutronliq-
a wall, which singles out the laboratory reference frame, uid in neutron star interiors, cold Fermi atomic gases or
p-1
E.E. Kolomeitsev and D.N. Voskresensky
chargedsuperfluids,aspairedprotonsinneutronstarinte- functions of t with non-integer α and β. When the con-
riors and paired electrons in metallic superconductors. In tributions of long-rangefluctuations are completely taken
contrast to previous works we take into account that the intoaccounttheGinzburgparameterFeqξ3 mustbecome
GL
superfluid subsystem and the bosonic excitations should independent of the temperature. Since Feq t2α−β and
be describedin termsofthe verysamemacroscopicwave- ξ t−α/2, we obtain α/2 β = 0. UsGiLng∝the experi-
∝ −
function. Also, our consideration is performed for non- mentalfact that the specific heat of the He-II contains no
zerotemperature, i.e., the presence of the normalcompo- power divergence at T T , we get 2α β 2 = 0.
c
→ − −
nent is taken into account. From these two relations we find α = 4/3, β = 2/3.
Other parameters of He-II at the saturated vapor pres-
Ginzburg-Landau functional. – We start with ex- sure are [6]: T =2.17K, a /T4/3 =1.11 10−16erg/K4/3,
pression for the Ginzburg-Landau (GL) free-energy den- c 0 c ·
b /T2/3 =3.54 10−39erg cm3/K2/3. This parameteriza-
sityofthe superfluidsubsysteminits restreferenceframe ti0oncholds for 10·−6 <t<0·.1,but for roughestimates can
for the temperature T <T , [5,6]:
c
be used up to t = 1. E.g., using eq. (1), with the helium
c 1 atom mass m = 6.6 10−24g we evaluate the He-II mass-
FGL[ψ]= 0 ¯h ψ 2 a(t) ψ 2+ b(t)ψ 4, density as ma /b · 0.3g/cm3, which is of the order of
2 | ∇ | − | | 2 | | 0 0
≃
a(t)=a tα, b(t)=b tβ, t=(T T)/T . (1) the experimental value ρHe =0.15g/cm3 at P =0.
0 0 c c
−
Moving cold superfluid. – We consider now a su-
Here Tc is the critical temperature of the second-order perfluid moving with a constant velocity ~v parallel to a
phase transition, and c0, a0 and b0 are phenomenological wall. The latter singles out the laboratory frame and an
parameters. When treatedwithin the mean-fieldapproxi- interactionof the fluid with the wall may lead to creation
mation,thefunctionalFGL shouldbeananalyticfunction of excitations in the fluid. We start with the case of van-
oft. Then,fromtheTaylorexpansionofFGL intitfollows ishingtemperatures. The wholemediumissuperfluidand
that α=1, β =0. The width of the so-called fluctuation the amplitude of the order parameter can be related to
region, wherein the mean-field approximation is not ap- the particle density ρ = mn = m ψ 2 = ma /b , ψ
s in 0 0 in
plicable, is evaluated from the Ginzburg criterion: in this is the order parameter in the absen|ce o|f the excitations
region of temperatures in the vicinity of Tc, long-range (“in”-state). The energy of the medium in the laboratory
fluctuations of the order parameter are mostly probable, frame is E =mnv2/2 b n2/2.
i.e. their probability is W ∼ e−FGeqLVfl/T ∼ 1, where FGeqL Whenthinespeedofth−e flo0w,v,exceedsthe Landaucrit-
is the equilibrium value, Vfl ξ3 is the minimal volume ical velocity, vL, near the wall there may appear excita-
∼ c
of the fluctuation of the order parameter, the coherence tions with the momentum k =k and the energy ǫ(k ) as
0 0
length ξ is the typical length scale characterizing the or- calculated in the rest frame of the superfluid, where the
der parameter. momentum k corresponds to the minimum of the ratio
0
For metallic superconductors the fluctuation region ǫ(k)/k. For instance, for He-II the spectrum ǫ(k) is the
proves to be very narrow and the mean-field approxima- standardphonon+rotonspectrum,normalizedasǫ(k) k
∝
tion holds for almost any temperatures below Tc, except for small k. The appearance of a large number of excita-
a tiny vicinity of Tc. Thus, neglecting this narrow fluctu- tions motivates us to assume that for v > vcL in addition
ation region one may use α = 1, β = 0 in (1). For the to the mother condensate the excitations may formanew
fermionic systems with the singlet pairing, in the weak- condensatewiththe momentumk =k =0,(“fin”-state).
0
6
coupling (BCS) approximationthe parameters can be ex- The momentum k should be found now from minimiza-
0
tracted from the microscopic theory [5]: tion of the free energy. As we have noticed before, in
previous works [1,2,4] it was assumed that the conden-
c0 =1/2m∗F, a0 =6π2Tc2/(7ζ(3)µ), b0 =a0/n, (2) sateofexcitationsdecouplesfromthe mothercondensate.
Nowwearegoingtotakeintoaccountthatthecondensate
∗ ∗
wherem standsfortheeffectivefermionmass(m m
intheweFak-couplinglimit),n=p3/(3π2¯h3)istheFpa≃rticlFe of excitations is described by the very same macroscopic
F wave-functionasthemothercondensate. Thentheresult-
numberdensity,andthefermionchemicalpotentialisµ
ingorderparameterψ isthesumofthecontributionsof
ǫ = p2/(2m∗). The function ζ(x) is the Riemann ζ≃- fin
F F F the mother condensate, ψ, and the condensate of excita-
function and ζ(3) = 1.202. The values of the parameters ′ ′
tions, ψ , i.e., ψ =ψ+ψ . We choosethe simplest form
fin
are obtained for t 1. With these BCS parameters we
of the order parameter for the condensate of excitations,
≪
have ψ 2 =ntandthepairinggap∆=T 8π2t,see[11].
| | c 7ζ(3) ψ′ =ψ′e−i(ǫ(k0)t−~k0~r)/h¯, (3)
For He-II the fluctuations prove to be imqportant for all 0
temperatures below Tc [6]. Including long-range fluctua- with the amplitude ψ0′, being constant for the case of the
tions, the coefficients of the Ginzburg-Landau functional homogeneoussystemthat weconsider. The particle num-
arenowrenormalized: duetoadivergency(logarithmicor ber conservation yields
power-law-like)of the specific heat at the critical temper-
n= ψ+ψ′ 2 = ψ 2+ ψ′ 2+... . (4)
atureTc,quantitiesaandbineq.(1)becomenon-analytic | | | | | 0|
p-2
Running condensate in moving superfluid
Hereellipsesstandforthespatiallyoscillatingterm,which Inmovingsuperfluidsthereexistexcitationsofthetype
vanishes after the averagingover the system volume. of vortex rings. The energy of the vortex is estimated as
Astheparticledensity,theinitialmomentumdensityis ǫvort = 2π2¯h2 ψ 2Rm−1ln(R/ξ), see [6,12], and the mo-
| |
redistributed in our case between the fluid and the con- mentum is pvort = 2π2¯hψ 2R2, m here is the mass of
| |
densate of excitations: the pair for systems with pairing, and the mass of the
boson quasiparticle in bosonic superfluids, e.g., the mass
ρs~v =(ρs−m|ψ0′|2)~vfin+(~k0+m~vfin)|ψ0′|2. (5) of the 4He atom in case of the He-II, R is the radius of
the vortex ring, and ξ c1/2a−1/2t−α/2 is the minimal
The energy of the moving matter in the presence of the ∼ 0 0
lengthscaleassociatedwiththemothercondensate. Thus,
condensate of excitations is
v =ǫvort/pvort =h¯(Rm)−1ln(R/ξ), is the Landau criti-
c1
E = ρsvfi2n +(ǫ(k )+ǫ ) ψ′ 2 a ψ 2+ b0n2 , calvelocityforthe vortexproduction,wherenow Ris the
fin 2 0 bind | 0| − 0| | 2 distanceoftheorderofthesizeofthesystem. Forv >v
c1
(6) the vortexringsarepushedoutofthemedium,iftheden-
sity profile has even slight inhomogeneity. Note that for
wherethe energy ofthe excitationǫ(k) shouldbe counted
spatially extended systems the value v is usually lower
fromthe binding energyofa particle inthe condensate at c1
than the Landau critical velocity vL. The flow moving
rest ǫ = ∂E (v = 0)/∂n = b n = a . Replacing c
eq.(5)biinnd(6)andinusingeq.(4)we−exp0resst−he0changeofthe with the velocity v for vc1 ≤ v ≤ vc2 may be consid-
ered as metastable, since the vortex creation probability
volume-averagedenergy density owing to the appearance
is hinderedby alargepotentialbarrierandformationofa
of the condensate of excitations, δE¯ =E¯ E¯ , as
fin− in vortex takesa long time [13]. Note that alreadyfor v just
δE¯ =(ǫ(k0)−k0v)|ψ0′|2+k02|ψ0′|4/(2ρs). (7) stilcigehstlmyaeyxcbeeecdoimnge vsuc1ffi,ctiheentnlyumlabrgere oafndthtehepirrodinutceerdacvtioorn-
′
Minimizing this functional with respect to ψ we obtain forcesthenormalandsuperfluidcomponentstomoveasa
0
the condensate amplitude solid,withthesamevelocity,evenifinitiallytheyhavehad
different velocities. In the exterior regions of the vortex
ψ′ 2 =(ρ /k )(v vL)θ(v vL), vL =ǫ(k )/k . (8)
| 0| s 0 − c − c c 0 0 cores the superfluidity still persists and our consideration
of the condensation of excitations in the velocity interval
From (5) we find that because of condensation of excita-
vL <v <v is applicable.
tions with k = 0 the flow is decelerated to the velocity c c2
6
v = vL. The volume-averaged energy density gain due In the presence of the condensate of excitations the
fin c
to the appearance of the condensate of excitations is density becomes spatially oscillating around its averaged
value. For a weak condensate, i.e., v vL vL, we
δE¯ =−12ρs(v−vcL)2θ(v−vcL). (9) find perturbatively δn = n−n¯ ≈ 2√|n|ψ−0′|cco|s(≪(ǫ(kc0)t−
~k ~r)/¯h). Such a density modulation predicted in [1] was
Minimization of δE¯ with respect to k gives the condi- 0
0 reproducedinthenumericalsimulationofthesupercritical
tion dvL/dk = 0, which is exactly the condition for the
c 0 flow in He-II using the realistic density functional [14].
standard Landau critical velocity. The condensate of ex-
The above consideration holds for any superfluid with
citationsappearsby a second-orderphase transition. The
the conserved number of particles in the mother conden-
amplitude of the condensate of excitations (8) growswith
sate plus the condensate of excitations, e.g., for the cold
the velocity, whereas the amplitude of the mother con-
densate decreases. The value ψ 2 vanishes when v =v , Bose gases,if the spectrum of the over-condensateexcita-
c2
the second critical velocity, a|t |which ψ′ 2 = n accord- tions is such that ǫ(k0)/k0 has a minimum at k =k0 =0,
| 0| as has been conjectured in [4]. 6
ing to eq. (4). The value of the second critical velocity
v is evaluated from (8) as v = vL+k /m. When the In case of the Fermi systems with pairing, for
c2 c2 c 0
mother condensate disappears, at v = v , the excitation the bosonic modes (Anderson-Bogoliubov, Schmid and
c2
spectrum is cardinally reconstructed, and the superfluid- Carlson-Goldmanones) with the excitation energy 2∆,
≃
itydestructionoccursasafirst-orderphasetransition. We cf. [10], the maximum momentum, up to which the mode
assume that for v > v the excitation spectrum has no is not yet destroyed, is k 2p , cf. [9,15]. Hence,
c2 0 F
≃
low-lying local minimum at finite momentum. Then the for these modes the Landau critical velocity is vL
amplitude ψ′ 2 jumps from n to 0 and δE¯ jumps from ∆/p , and for v > vL there is a chance for the concden≃-
δE¯(v ) = | ρ0k|2/(2m2) to 0 at v = v . The case, when satioFn of the Bose ecxcitations as we considered above.
c2 − 0 c2
in the absence of the mother condensate the spectrum of Besides bosonic excitations there exist fermionic ones
Bose excitations has a low-lying local minimum at k =0, with the spectrum ǫ (p) = ∆2+v2(p p )2. Stem-
6 f F − F
has been considered in [2,3]. Note that in practice the ming from the breakup of Cooper pairs, the fermionic
p
reconstruction of the spectrum may occur for a smaller excitations are produced pairwise. Therefore, the cor-
velocity than that we have estimated. For example, for responding (fermion) Landau critical velocity is vL =
c,f
fermionic superfluids it always should be ψ′ 2 n, oth- min [(ǫ (p ) + ǫ (p ))/p~ + p~ ]. The latter expres-
| 0| ≪ p~1,p~2 f 1 f 2 | 1 2|
erwise the Fermi sea itself would be destroyed. sion reduces to [16] vL =(∆/p )/(1+∆2/p2v2)1/2. We
c,f F F F
p-3
E.E. Kolomeitsev and D.N. Voskresensky
seethatuptoasmallcorrectionvL vL. ForT 0and normal and superfluid subsystems continue to move with
c,f ≃ c →
0 v/vL 1 1 we find <~p~v >/ρ 2 2vL(v vL)3/2 one and the same velocity ~vfin. In the presence of the
an≤d thece−nerg≪y gain due to the fermio≃n pair bcrea−kingc is condensate of excitations the free energy density becomes
p
δE¯pair = (22πd¯h3p)3(ǫf(p)−p~~v)θ(ǫf(p)−p~~v) Ffin =+(12ǫρ(kvfi2)n++ǫFbind)+ψ′F2G+L[ψ2,b∇(tψ) ψ=20]ψ′ 2+ 1b(t)ψ′(144.)
Z 0 bind | 0| | | | 0| 2 | 0|
≈−2√2ρ(v−vcL)2[v/vcL−1]1/2. (10) To get this expression we used eq. (1) and replaced there
′
ψ with ψ = ψ +ψ . The energy of excitations should
Moreover, for v > vL the pairing gap decreases with in- fin
c,f be counted here from the excitation energy on top of the
creaseofvas[17]∆(v)/∆ 1 (3/2)(v/vL 1)2,reaching
zeenreorgfyorgvai=nv(cL120,f)=is2elvecsLs(Rth≈oagne−r(s9-B)aarnddeetnhceeff−pecrotd[1u8c]t)i.onThoef ∂mFoGtehqLe[ψrc=onψdveeq=n0sa+teψa′t]/r∂e|sψt′d|2etψe′r=m0i=ne−d2bay(te)q..(11), ǫbind =
Now, using the momentum conservation (13) we ex-
the condensate of Bose excitations is energetically more (cid:12)
press~v through~v andget(cid:12)for the changeof the volume-
profitable than the Cooper pair breaking. Since in the fin
averaged free-energy density associated with the appear-
presence of the condensate of excitations v =vL an ad-
fin c ance of the condensate of excitations,
ditionalenergygainduetotheappearanceofthelatteris:
fFoGerqL0(T =v/0v,L∆)−1FGeqL1(T. F=or0,v∆>(v)v)L≈ −th(9e/g8a)iρn(vb−ecvocLm)e2s, δF¯ = 12b(t) |ψ|2−a(t)/b(t) 2+ ǫ(k0)−k0v |ψ0′|2
Feq(T≤=0,∆c )−= ≪3ρ(vL)2/4. c2,f +2b(t)(cid:0)|ψ|2−a(t)/b(t)(cid:1)|ψ0′|(cid:0)2+ 21b(t)|ψ0′|(cid:1)4, (15)
GL − c
Moving superfluid-normal composites. – We where b(t)=b(cid:0)(t)+k02/ρ and w(cid:1)e put~k0 ke~v. We note that
turn now to the case of systems consisting of normal and the normal subsystem serves as a reservoir of particles at
superfluid parts, like He-II at finite temperature, metal- the foremation of the condensates, which amplitudes are
lic superconductors, or neutron star matter. Here, the chosen by minimization of the free energy of the system.
number of particles in the condensate is not conserved Therefore,we vary δF¯ with respect to ψ and ψ0′ indepen-
even at v = 0 since a part of particles can be transferred dently. Thus, minimizing (15) we find
from the superfluid to the normal subsystem. The state k v vL
of the superfluid subsystem is described by the GL func- ψ′ 2 = 0 − c θ v vL θ k2/ρ 3b(t) , (16)
| 0| k2/ρ 3b(t) − c 0 −
tional(1). Weassumethatthenormal and superfluidsub- 0 (cid:0) − (cid:1)
systems move with the same velocity ~v with respect to a |ψ|2 = a(t)/b(t)−2|ψ(cid:0)0′|2 θ(T(cid:1)c((cid:0)v)−T)θ(vc2((cid:1)t)−v).
laboratory frame. This means that we consider the mo-
The quan(cid:0)tity T stands for (cid:1)the renormalized critical tem-
tionofthefluidduringthetimeτ shorterthanthetypical c e
friction time, τnorm, at which the normal component is perature, which depends now on the flow velocity, and
fr
v (t) stands foer the secondcriticalvelocity depending on
decelerated, if the fluid has a contact with the wall. c2
T. The condition ψ 2 =0 implies the relation between v
Minimizationovertheorderparameterintherestframe
| |
and T
of the fluid yields
|ψveq=0|2 =a(t)/b(t), FGeqL[ψveq=0]=−b(t)|ψveq=0|4/2. (11) v =vcL+a(t)k0/(2b(t)ρ)−3a(t)/(2k0). (17)
The solution of this equation for the velocity, v (t), in-
Intheabsenceofthepopulationofexcitationmodesthe c2
creaseswith the decreasingtemperature,andthe solution
superfluid and normal subsystems decouple. In this case
forthetemperature,T (v),decreaseswithincreasingv. At
the initial free-energy density of the system is given by c
T = T (v) or v = v (t) we have ψ 2 = 0 but ψ′ 2 = 0,
F =ρv2/2+F a2(t)/(2b(t)). (12) and focr T > T (v) ocr2efor v > v (|t)|the conden|sa0t|e 6ψ′ 2
in bind− c c2 | 0|
vanishees, if, as we assume, for ψ 2 = 0 the spectrum of
Here ρ is the total (normal+superfluid) mass density, | |
excitationsdoeesnotcontainalow-lyingbranch. Thus,the
ρ = ρ (v = 0)+m ψ(v = 0)2, and F is a binding
n bind superfluidity is destroyed at T = T (v) or v = v (t) in a
| | c c2
free-energy density of the normal subsystem in its rest
first-order phase transition.
reference frame (coinciding with the rest frame of the su- From(13)and(16)wefind forve>vL andk2/(ρb(t))>
c 0
perfluid in the case under consideration). The explicit
3 the resulting velocity of the flow
form of F is not of our interest.
bind
Whenthecondensateofexcitationsisformed,theinitial vfin =vcL−(v−vcL)/ k02/(3b(t)ρ)−1 <vcL. (18)
momentum density is redistributed between the fluid and
Substituting the order(cid:0)parameters from(cid:1) (16) in (15),
the condensate of excitations:
we find for the volume-averaged free-energy density gain
ρ~v =(ρ m ψ′ 2)~v +(~k +m~v ) ψ′ 2. (13) owing to appearance of the condensate of excitations
− | 0| fin 0 fin | 0|
ρ (v vL)2
Here we assume (as argued below) that after the appear- δF¯ = − c θ(v vL)θ(v v) (19)
ance of the condensate of excitations in the form (3) the −21−3b(t)ρ/k02 − c c2−
p-4
Running condensate in moving superfluid

for k2/ρ > 3b(t). Thus, for vL < v < v the free energy
0 c c2
decreasesowingtotheappearanceofthecondensateofex- 1.0 L
citations with k = 0 in the presence of the non-vanishing vfin/vc
6
mothercondensate. Thevalueofk istobefoundfromthe
0 0.8 2 2
minimization of eq. (19). As T , the condensate momen- | |/| (v=0)|
c
tcuomrreksp0ognedtisnrgetnoortmhealmiziendimaunmdedoifffǫe(rks)n/kow. Fforormk2t/hρe va3lube 0.6 | ’0|2/| (v=0)|2
0 ≫ 0
the expression (19) for the gain in the volume-averaged
0.4 t=0.1
free-energy density transforms at T = 0 into the expres- t=0.5
sion (9) for the gain in the volume-averaged energy den-
0.2
sity. As in the case of T =0, for T 6=0 the condensate of vc2 vc2
excitations appears at v = vL in the second-order phase
c
0.0
transition but it disappears at v = v in the first-order
c2 0 1 2 3 4 5
L
phase transition with the jumps v/vc
δF¯(vc2)≈ a82b2(t()tk)ρ02, |ψ0′(vc2)|2 = 2ab((tt)). (20) Fthige.fi1n:alCflonowdenvsealotecitaymvpfilint,udeqes. (|ψ18|2),ainnds|uψp0′e|2rfl,ueiqd. 4(1H6e),palontd-
ted as functions of the flow velocity for various temperatures,
The dynamics of the condensate of excitations is deter- t = (Tc −T)/Tc. Vertical arrows indicate the values of the
mined by the equation Γψ˙′ = δF¯/δψ′∗. We emphasize second critical velocity vc2. Velocities are scaled by the val-
that the above consideration as−sumes that the formation ues of the Landau critical velocities vcL(t=0.5) =59m/s and
rate Γ of the condensate of excitations is faster than the vcL(t = 0.1) = 55m/s, and the condensates are normalized to
deceleration rate 1/τnorm of the normal subsystem. the condensate amplitudein thesuperfluid at rest, eq. (11).
fr
When a homogeneous fluid flowing with v > vL at
c
T >T (v)iscooleddowntoT <T (v),itconsistsofthree
c c and taking into account that we deal with the rotonic
components: the normalandsuperfluidones andthe con-
excitation, i.e., k k and ǫ(k ) ∆ , we estimate,
0 r 0 r
sdyesntseameteisofthexencitraet-ihoenast,eadlltomTovi>ngeTwc(ivth),vtfihne<supvceLr.fluIfidtihtye kU02s/in(bg0ρth)e≃re4s7u,ltsvcLo(f≃T[7→] th0e) ≃tem60pmer/a≃stu,rea0d/ekp0e≃nde1n6cme/sof.
and the condensate of excitations vanish and the remain- vL can be fitted with 99% accuracy as vL(T)/vL(0)
ing normal fluid consists of two freactions: one is moving 1c 0.7e−2.14/t˜+200t˜e−8/t˜, where t˜= T/Tc. Usincg thes≃e
c
withv (T )<vL andtheotherone,δn=a(T )/(2b(T )), −
fin c c c c parameters we evaluate the condensate amplitudes and
is moving with a higher velocity until a new equilibrium
the finalflowvelocityas functions oftemperature andde-
is establisehed. Note also that for fermion sueperfluides at picttheminFig.1. Thecondensateofexcitationsappears
T = 0 after the condensate of excitations is formed the at v = vL in a second-order phase transition. For v > vL
tflhoew6 Cvoeolopceirtypavifirnb<reavkcL,ifn,gfodroves−novtcLo>cc4utrv,cLw/h9e,raenadstthheerceobny- theamplcitudeofthecondensate|ψ0′|2 (|ψ|2)increases(dec-
creases)linearly withv. The closerT is to T , the steeper
c
densate of Bose excitations is preserved.
the change of the condensate amplitudes is. The final ve-
Estimates for fermionic superfluids and He-II. – locityoftheflow,whichsetsinaftertheappearanceofthe
We apply now expressions derived in the previous section condensateofexcitations,decreaseswiththeincreaseofv.
to several practical cases. First, we consider a fermion ForHe-II,wehaveα=4/3,β =2/3andtherenormalized
systemwithpairing. Withthe BCSparameters(2)wees- critical temperature determined by eq. (17) is for v >vcL:
timateb ρ/k2 =3∆2/(8v2p2)anda /k =3∆2/(4v p2),
0 0 F F 0 0 F F 3/2
where ρ ≃ nmF. We see that inequality k02/ρ ≫ 3b0 is Tc =1 k02 k04 2k0 v vL
Irnedtuhciesdlitmoitineψq′u2algitiyve∆n b≪y eqǫF.,(1w6h)icghetsisthweelslasmaetisffioerdm. Tec −"6b0ρ −s36b20ρ2 − 3a0h − ci#
| 0| 1 0.05(v/vL(T ) 1)3/2. (22)
as eq. (8). The resulting flow velocity after condensa- ≈ − c c −
tion, (18), is lower than the Landau critical velocity but The mother condensate ψ 2 vanishes when v reaches
close to it, v vL 9(vL)2(v vL)/(8v2). | |
fin ≃ c − c − c F the value of the second critical velocity vc2, which de-
Since for the BCS case we have α = 1, β = 0, eq. (17) pends on the temperature as v vL(t) + (363t2/3
forthenewcriticaltemperatureiseasilysolved,forv >vcL 23.5t4/3)m/s. At v =v the supce2rfl≈uidicty disappearsin−a
c2
T 2k b (v vL) v vL first-orderphase transition. The correspondingenergyre-
Tecc =1− a0(0k020/ρ−−3bc0) ≈1− −vF c . (21) 5le.a9ste4/c3a(nTb∆eCest)i,mwahteerdef∆roCm=(200).7a6s1δ0F7(evrcg2/)(≈cm4837bKa020)ti4s/3th≃e
We also estimate the maximal second critical velocity as c p p ·
specific heat jump at T [6].
vmax vL+v . c
c2 ≃ c F
Weturnnowtothebosonicsupefluid–helium-II.Mak- Rotating superfluids. Pulsars. – The novel phase
inguseofthevaluesoftheGLparameterspresentedabove withthecondensateofexcitationsmayalsoexistinrotat-
p-5
E.E. Kolomeitsev and D.N. Voskresensky
ing systems. Excitations can be generated because of the phase transition at v =vL and the condensate amplitude
c
rotation. Presenceof friction with anexternalwallor dif- growslinearlywiththeincreasingvelocity. Simultaneously
ferencebetweenvelocitiesofthesuperfluidandthenormal the mother condensate decreases and vanishes at v =v ,
c2
fluid are not necessarily required to produce excitations. then the superfluidity is destroyed in a first-order phase
Now we should use the angular momentum conservation transition with an energy release. For vL < v < v the
c c2
instead of the momentum conservation. The structure of resultingflowvelocityisv vL,wherebytheequalityis
fin ≤ c
the order parameter is more complicated than the plane realizedfor T =0. We arguedthat for the fermion super-
wave [2]. With these modifications, the results, which we fluids the condensate of bosonic excitations might be sta-
obtained above for the motion with the constant ~v, con- ble against the appearance of fermionic excitations from
tinue to hold. The value of the critical angular velocity the Cooper-pair breaking. Finally, we considered conden-
Ω v /R is very low for systems of a large size R. sation of excitations in rotating superfluid systems, such
c1 c1
∼
Inthe inner crustandina partofthe coreofa neutron as pulsars. Our estimates show that superfluidity might
star,protonsandneutrons arepairedinthe 1S state ow- coexist with the condensate of excitations in the rapidly
0
ingtoattractiveppandnninteractions. Indenserregions rotating pulsars.
of the star interior the 1S pairing disappears but neu-
0
trons might be paired in the 3P state [19]. The charged
2 ∗∗∗
superfluid should co-rotate with the normal matter with-
out forming vortices, this results in the appearance of a The work was supported by Grants No.
tiny magnetic field ~h = 2m Ω~/e (London effect) in the VEGA 1/0457/12 and No. APVV-0050-11, by “New-
p p
wholevolumeofthesuperfluid,m (e )istheprotonmass CompStar”,COSTActionMP1304,andby PolatomESF
p p
(charge)[19]. This tiny field, being ∼< 10−2G for the most network.
rapidly rotating pulsars, has no influence on the relevant
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p-6