arXiv:1505.03884v1 [nucl-th] 14 May 2015 iud o aiu ausadmmnu eairo the of parameter behavior Landau Fermi momentum scalar and normal values various cold for the liquids in fluctuations) (density-density fulfilled. are parameters Landau inequal- the some of values if only the stable on in are ities liquids shown Fermi phenomeno- has that [7] several Pomeranchuk Ref. by parameters. low- the Landau described theory logical liquid are Fermi devel- the excitations was In systems lying 6]. ap- nuclear [5, Migdal liquid of by Fermi oped description The the [2–4]. to textbooks proach in see [1], Landau gis arn na -aesae nidcdp-wave temperatures induced low An very at possible state. stable be s-wave therefore, to an pairing is, particle-particle in system the pairing the in against and repulsive interaction is an channel that assume We h eprtr ftesse ssalbtabove that but assumption small the is by system the precluded of be temperature can the [8], Refs. see ei,lk oaignurnsas f es 1–5.For moving [10–15]. Refs. other cf. and stars, gases, − neutron atomic occur rotating like may cold media, phenomena He-II, Similar moving Lan- [9]. in the velocity above velocity condensation critical the a dau with liquids to Fermi momentum lead moving non-zero in may a with this excitations superfluid- consequence zero-sound-like of for a spectrum As condition the necessary that ity. Landau means at minimum the This a satisfies momentum. possesses non-zero spectrum a the of velocity phase ini iltdta eut nepnnilbidpo the of buildup exponential in results that violated is tion iemdsadhdoyai oe.U onwi was it now to Up for zero-sound- that modes. of thought hydrodynamic growth and the modes against like unstable is spectrum 1 nti okw td o-yn clrectto modes excitation scalar low-lying study we work this In by up built was liquids Fermi normal of theory The For oeacu ntblt n oecnesto fsaa qu scalar of condensation Bose and instability Pomeranchuk f < f 0 0 < > xiain r apd for damped, are excitations 0 ewrs ula atr em iud on-ieexcita sound-like liquid, Fermi matter, 71.45.-d Nuclear 71.10.Ay, Keywords: 21.65.-f, numbers: PACS clrmd xiain sivsiae o aiu value various parameter for Landau investigated is excitations scalar-mode iud oigwt vrrtclvlcte,poie na an provided st parameter velocities, T a Landau overcritical in parameter. with excitations moving Landau zero-sound-like homogeneou liquids scalar the a the of in of condensation occur formati the dependence may the momentum condensation by the The tamed on bo be modes. a may scalar performing instability the after Pomeranchuk derived the is that modes excitation Pomeranchuk (a scalar excitations the scalar-mode of growth a against hs eoiyo h pcrmo h eosudaqie mi a acquires sound zero the of − spectrum the on velocity phase h odtoswl efudwe the when found be will conditions the 0 1 esuyecttosi omlFrilqi ihalclsca local a with liquid Fermi normal a in excitations study We f < .INTRODUCTION I. f 0 0 < < − 2 hr r nydme xiain,adfor and excitations, damped only are there 0 h ehnclsaiiycondi- stability mechanical the 1 ainlRsac ula nvriy(Eh) 149Mosc 115409 (MEPhI), University Nuclear Research National f 0 ntepril-oechannel. particle-hole the in 1 f f ae e nvriy K941Bnk ytia Slovakia Bystrica, Banska SK-97401 University, Bel Matej 0 0 ( k ntepril-oecanl For channel. particle-hole the in ) .E Kolomeitsev E. E. > 0. f 0 Dtd coe 4 2018) 14, October (Dated: < T < T − T the 1 c ,p c 1 . ,p , n .N Voskresensky N. D. and es 1,2] ed oapsiiiyo h at hs in phase 0 pasta densities the for of crusts possibility see star a interaction, neutron to Coulomb the leads the 20], of [19, see inclusion Refs. instability, The chemical [18]. a Ref. by mechanical a accompanied system is many-component instability a In density. tion hc ih eraie tcrancniin.We conditions. certain at for realized that be demonstrate modes might zero-sound-like unstable which of description ternative n ntemmnu eedneo h aduparam- Landau the of eter dependence momentum depend- the state on inhomogeneous in ing in re- occur either the may homogeneous by condensation the stabilized The is self-interaction. scalar amplitude pulsive the condensate of The condensate Bose field. static a of accumulation an iudpoe ob stable. be to proves liquid ntesaa hne ndpnec on dependence in liquid channel Fermi scalar one-component the a in in excitations of spectrum est utain.I yrdnmctrstecondi- the terms hydrodynamic tion In fluctuations. density hs rniinmgtocr[6 7 o h aynden- baryon the for liquid-gas 17] 0 the [16, sities function neglected) occur matter might a is nuclear transition as symmetric interaction phase form isospin Coulomb Waals the the der In (if van volume. a the has of pressure construction the like Maxwell liquid-gas if the by mixed determined a state in stationary results bub- instability liq- Fermi and spinodal one-component droplets a uid In of exponential instability). mixture an (spinodal aerosol-like bles to the lead of would growth This imaginary. becomes et Vw td oeacu ntblt for instability Pomeranchuk In effec- study modes. an we scalar suggest IV self-interacting Sect. and the for interaction Lagrangian local tive the bosonize we n ugs oe osblt fteocrec fthe of occurrence the of possibility novel a suggest and in,Pmrnhkisaiiy oecondensation Bose instability, Pomeranchuk tions, h e on fti oki htw ugs nal- an suggest we that is work this of point key The h oki raie sflos nSc.I estudy we II Sect. In follows. as organized is work The n oetmdpnec ftescalar the of dependence momentum and s f f f poraemmnu eedneo the of dependence momentum ppropriate 0 f 0 t ihannzr oetmi Fermi in momentum non-zero a with ate 0 0 ( . > ntblt) neetv arninfor Lagrangian effective An instability). 3 k oiainpoeue edemonstrate We procedure. sonization < < n .I h rsneo h odnaeteFermi the condensate the of presence the In ). iu tannzr oetm For momentum. non-zero a at nimum h odtosaefudwe the when found are conditions the 0 0 a neato.Setu fbosonic of Spectrum interaction. lar − − ∼ < no ttcBs odnaeof condensate Bose static a of on h pcrmbcmsunstable becomes spectrum the 1 mle httesedo h rtsound first the of speed the that implies 1 rihmgnossaerelying state inhomogeneous or s n e ecnie osblt of possibility a consider we hen ∼ < 0 . 7 2 n f 0 w Russia ow, 0 where , nai em liquid Fermi a in anta < − ntblt a eutin result may instability 1 n 0 stencersatura- nuclear the is . 3 f n 0 0 ( k ∼ < .I et III Sect. In ). n ∼ < f 0 0 . 7 < n 0 − . 1 2 static Bose-condensate which leads to a stabilization of in Eq. (5) can be expressed in terms of dimensionless the system. In Sect. V we consider condensation of scalar scalar and spin Landau parameters excitations in moving Fermi liquids with repulsive inter- ˜ ′ 2 ω ′ actions. Concluding remarks are formulated in Sect. VI. f(~n , ~n)= a N0Γ0 (~n , ~n) , ′ 2 ω ′ g˜(~n , ~n)= a N0Γ1 (~n , ~n) , (6) II. EXCITATIONS IN A FERMI LIQUID where the normalization constant is chosen as in applica- tions to atomic nuclei [6, 21, 22] with the density of states A. Landau particle-hole amplitude. at the Fermi surface, N0 = N(n = n0), taken at the nu- ∗ mF(n) pF(n) clear saturation density n0 and N(n)= π2 . Such Consider the simplest case of a one-component Fermi a normalization is at variance with that used, e.g., in liquid of non-relativistic fermions. As discussed in the Refs. [1–4]. Their parameters are related to ours defined Introduction, we assume that the system is stable against in Eq. (6) as f = N f/N˜ 0 and g = N g/N˜ 0. pairing. The particle-hole scattering amplitude on the The Landau parameters can be expanded in terms of Fermi surface obeys the equation [2–4, 6] the Legendre polynomials P (~n ~n ′), l · ′ ω ′ ′ ′ Tph(~n , ~n; q)= Γ (~n , ~n) f˜(~n , ~n)= f˜ P (~n ~n ) , (7) l l · ω ′ ′′ ′′ ′′ Xl b + Γ (~n , ~n b) ph(~n ; q) Tph(~n , ~n; q) ~n ′′ , (1) h L i and the similar expression exists for the parameterg ˜. where ~n andb ~n ′ are directions of fermionb momenta before The Landau parameters f˜ ,g ˜ can be directly related and after scattering and q = (ω,~k) is the momentum 0,1 0,1 to observables [4]. For instance, the effective quasiparti- transferred in the particle-hole channel. The brackets cle mass is given by [2, 3] stand for averaging over the momentum direction ~n ∗ mF 2 1 dΩ~n =1+ a N Γ0(cos θ)cos θ =1+ f1 , (8) ... ~n = ... , (2) m 3 h i Z 4 π F where the bar denotes the averaging over the azimuthal and the particle-hole propagator is and polar angles. The positiveness of the effective mass +∞ +∞ is assured by fulfillment of the Pomeranchuk condition 2 dǫ dpp f1 > 3. Note that the traditional normalization of the ph(~n; q)= G(pF+) G(pF−) , (3) − L Z 2πi Z π2 Landau parameters (6) depends explicitly on the effective −∞ 0 mass m∗ through the density of states N. Therefore, it is instructive to rewrite Eq. (8) using the definition in ~ where we denoted pF± = (ǫ ω/2,pF ~n k/2) and pF Eq. (6) stands for the Fermi momentum.± The quasiparticle± con- tribution to the full Green’s function is given by m∗ 1 F = . (9) 1 mF mF 1 ∗ f˜ 2 2 3 m (n0) 1 a p pF − F G(ǫ, ~p)= , ξ~p = − ∗ . (4) ǫ ξ~p + i 0 signǫ 2 mF − From this relation we obtain the constraint f˜1 < ∗ ∗ 3 m (n0)/mF for the effective mass to remain positive Here mF is the effective fermion mass, and the param- F eter a determines a quasiparticle weight in the fermion and finite; otherwise the effective mass tends to infinity ˜ ∗ spectral density, 0 < a 1, which is expressed through in the point where f1 =3 mF(n0)/mF. Thus, for the sys- ≤ R −1 tems where one expects a strong increase of the effective the retarded fermion self-energy ΣF (ǫ,p) as a = 1 R − mass, the normalization (6) of the Landau parameters (∂ Σ /∂ǫ)0. The full Green’s function contains also a ℜ F would be preferable. A large increase of the effective regular background part Greg, which is encoded in the fermion mass may be a sign of a quantum phase transi- renormalized particle-hole interaction Γˆω in Eq. (1). tion dubbed in Refs. [23, 24] as a fermion condensation. The interaction in the particle-hole channel can be The latter is connected with the appearance of multi- written as connected Fermi surfaces. As demonstrated in Ref. [25] if ω ′ ω ′ ′ ω ′ ′ such a phenomenon occurs in neutron star interiors, e.g., Γˆ (~n , ~n)=Γ (~n ~n) σ σ0 +Γ (~n ~n) (~σ ~σ) . (5) 0 0 1 at densities close to the critical density of a pion con-
The matrices σj with j = 0,..., 3 act on incoming densation, this would trigger efficient direct Urca cooling ′ processes. On the other hand, right from the dispersion fermions while the matrices σj act on outgoing fermions; σ0 is the unity matrix and other Pauli matrices σ1,2,3 are relation follows that normalized as Trσ σ = 2δ . We neglect here the spin- i j ij m∗ 1 (∂ ΣR/∂ǫ) orbit interaction, which is suppressed for small trans- F F F = − ℜ R 0 , (10) ferred momenta q p . The scalar and spin amplitudes mF 1 + (∂ ΣF /∂ǫp)F ≪ F ℜ 3
0 2 ˜ where ǫp = p /2mF. Thereby, the Landau parameter f1 The amplitudes (11) possess simple poles and logarith- can be expressed via the energy-momentum derivatives mic cuts. For the amplitude Tph,0 the pole is determined of the fermion self-energy. by the equation Below we focus our study on effects associated with the ˜ ω 1 ω k zero harmonic f0 in the expansion of Γ0 as a function of = Φ , . (20) ′ f − v k p (~n~n ). Then the solution of Eq. (1) is 0 F F
′ ′ ′ Similar equation exists in g-channel (with replacement Tph(~n , ~n; q) = Tph,0(q) σ σ0 + Tph,1(q) (~σ ~σ) , 0 f0 g0. Analytical properties of the solution (20) have 1 → b been studied in [27]. Tph,0(1)(q) = ω . (11) 1/Γ0(1) ph(~n; q) ~n Expanding the retarded particle-hole amplitude − hL i R Tph,0(q) near the spectrum branch The averaged particle-hole propagator is expressed through the Lindhard’s function 2ω(k)V 2(k) T R (q) , ph,0 ≈ (ω + i0)2 ω2(k) 2 ω k − ph(~n; q) ~n = a NΦ , , (12) −2 2 ∂Φ hL i − vF k pF V (k)= a N , (21) ∂ω2 ω(k) where we identify the quantity DR(ω, k) = [(ω +i0)2 ω2(k)]−1 2 − z− 1 z− +1 as the retarded propagator of a boson with the dispersion Φ(s, x)= − ln relation ω = ω(k) and the quantity V (k) as the effective 4(z+ z−) z− 1 2 − − vertex of the fermion-boson interaction. z 1 z+ +1 1 + − ln + . (13) For the neutron matter the parameters f = fnn, − 4(z+ z−) z+ 1 2 g = g are the neutron-neutron Landau scalar and − − nn spin parameters. Generalization to the two component Here and below we use the dimensionless variables z± = system, e.g., to the nuclear matter of arbitrary isotopic s x/2, s = ω/kv , x = k/p . For real s the Lindhard’s ± F F composition is formally simple [6]. Then the amplitude function acquires an imaginary part should be provided with four indices (nn, pp, np and π x pn). However, equations for the partial amplitudes do 2 s , 0 s 1 2 π ≤ x ≤ − x not decouple. For the isospin-symmetric nuclear sys- Φ(s, x)= (1 z2 ) , 1 s 1+ . (14) 4x − 2 2 tem with the omitted Coulomb interaction the situation ℑ 0− , otherwise− ≤ ≤ is simplified since then fnn = fpp and fnp = fpn. In ω For s 1, this case one usually presents Γ0 in Eqs. (5) and (6) as ≫ ω ˜ ˜′ ′ 2 ω ′ ′ 2 Γ0 = (f + f ~τ ~τ)/a N0, and Γ1 = (˜g +g ˜ ~τ ~τ)/a N0, ′ Φ(s, x) 1/(3 z+ z−) . (15) where τi are isospin Pauli matrices. Quantities f and f ≈− ′ (and similarly g and g ) are expressed through fnn and For x 1 the function Φ can be expanded as f as f = 1 (f + f ) and f ′ = 1 (f f ). The am- ≪ np 2 nn np 2 nn np plitudes of these four channels (f, f ′, g,−g′) decouple, cf. s s 1 x2 Φ(s, x) 1+ log − , (16) Ref. [28]. The excitation modes are determined by four 2 2 ′ ≈ 2 s +1 − 12 (s 1) equations similar to Eq. (20), now with 2f0, 2f0, 2g0 − ′ and 2g0. Presence of the small Coulomb potential mod- and if we expand it further for s 1 we get ≪ ifies the low-lying modes determined by Eq. (20) (now with f , f ′ , g and g′ ) only at low momenta [28]. We π x2 s4 x2s2 x4 0 0 0 0 Φ(s, x) 1+ i s s2 . (17) will ignore the Coulomb effects in our exploratory study. ≈ 2 − − 12 − 3 − 6 − 240 We discuss now the properties of the bosonic modes At finite temperatures the function Φ should be re- for different values of the Landau parameter f0. placed by the temperature dependent Lindhard function ΦT calculated in Ref. [26]. Generalization of expan- B. Repulsive interaction (f0 > 0). Zero sound. sion (17) for low temperature case (T ǫF) is given by ≪ For f0 > 0 there exists a real solution of Eq. (20) such 2 π 2 that for k 0 the ratio ωs(k)/k tends to a constant. ΦT (s,x,t)=Φ(s, x) 1 t , (18) → − 12 Such a solution is called a zero sound. The zero sound exists as a quasi-particle mode in the high frequency limit 2 ∗ 2 where t = T/ǫF and ǫF = pF/2mF is the Fermi energy. ω 1/τcol, where τcol ǫF/T is the fermion collision For high temperatures (T ǫ ) we have ≫ ∝ ≫ F time. In the opposite limit it turns into a hydrodynamic (first) sound [29]. At some value k < p the spectrum 2 x2 1 lim ∼ F ΦT (0,x,t)= 1 . (19) branch enters the region with Φ > 0, and the zero sound 3 ℑ 3 t − 6 t − 3 √2πt becomes damped diffusion mode. 4
We search the solution of Eq. (20) in the form 1.5
sound m odes
ωs(k)= k vF s(x), (22) 1.4 where s(x) is a function of x = k/pF which we take in
f =3 2 4 00 1.3 the form s(x) s0 + s2 x + s4 x . The odd powers of x k ≈ F f =2.5 /v
00
are absent since Φ is an even function of x. s Above we assumedℜ that f depends only on ~n~n′. The 1.2
f =2
exact results are however also valid if the Landau parameter f 00 2 f =0
02 s +s (k/p )
2 0 2 F f =1.5 is a very smooth function of x . From now we suppose 00
f =-1.5
1.1
02
f =1
2 00 f0(x) f00 + f02x , (23) ≈ Landau dam ping
1.0 where the parameter f02 is determined by the effective 0.0 0.2 0.4 0.6 0.8 1.0 range of the fermion-fermion scattering amplitude and k/p
F expansion is valid provided f00 f02 for relevant values x < p . According| to| Ref. ≫ [21],| | in the case ∼ F FIG. 1: Spectrum of the sound modes defined by Eq. (20) for of atomic nuclei (n n ), f = f r2 p2 /2, and 0 02 00 eff F various values of the Landau parameters f00 and f02. Solid < < ≃ − 0.5 ∼ reff ∼ 1fm, as follows from the comparison with lines show the results of the numerical solution, dashed lines the Skyrme parametrization of the nucleon-nucleon in- are the quadratic expansions with the parameters s0 and s2 teraction and with the experimental data. For isospin- given by Eqs. (24) and (26). The values of f00 are shown in > symmetric nuclear matter f00 > 0 for n ∼ n0, f00 < 0 for labels. Lines with (without) stars are calculated for f02 = 0 lower densities, and in a certain density interval below (f02 = −1.5α), where α is defined in Eq. (26). The dash- n0, f0 < 1, cf. Refs. [17, 30]. In the purely neutron dotted line shows the border of the imaginary part of the − < Lindhard’s function (13). Below this line the frequency is matter one has 1
0.3 2
unstable f =0
02
f =-1.5
f =2.5
02 ) 00 F
f = 3
00
1 p F
0.2
f =-2
F 00 N/(v
2 f =-1.5
00
(k)/ 0 d
(k) a (k) i 2
0.1 f =2 V f =-1.2 00
00
-1
f =-.9
00
f =-.2 f =-.5 f =1.5
00 00
00
0.0
stable
0.0 0.2 0.4 0.6 0.8
-2
0.0 0.5 1.0 1.5 2.0 2.5
k/p
F
k/p
F FIG. 2: Coupling constant of zero-sound modes and fermions as a function of the momentum k for various values of the FIG. 3: Solid lines show the purely imaginary solutions of Landau parameters f00 and f02, calculated from Eq. (21). All Eq. (31) for various values of the Landau parameter f00 < curves end at the limiting value of xmax = kmax/pF at which 0. Dashed line indicates positions of the maxima of spectra the spectral branch enters into the region of complexity of the ωd for f00 < −1. Dash-dotted lines show the approximated 2 4 Φ function. spectra ωd(k) ≈ ivFk(sd0+sd2 x +sd4x ) with the parameters from Eqs. (34) and (35). with the phase one vph = ωs/k. The quantity ωs(k0)/k0 coincides with the value of the Landau critical velocity uL More generally, for purely imaginary s = i s˜,s ˜ R, for the production of Bose excitations in the superfluid the Lindhard function can be rewritten as ∈ moving with the velocity u>u . L 2 The effective vertex of the boson-fermion coupling (21) 1 s˜2 + 1 x +1 Φ(is,˜ x)= Φ(˜s, x)= s˜2 1 x2 +1 log 2 is shown in Fig. 2 as a function of x for several val- 4x − 4 2 1 2 s˜ + 2 x 1 ues of f00 and f02. We plot it in the range 0
D. Strong attraction, f00 < −1. Pomeranchuk C. Moderate attraction, −1 < f00 < 0. Diffusons. instability.
Assume f02 = 0. For 1 2 found from the relation for the variation of the fermion ωd(k) 2 1 f00 x = isd(x) i − | | + , (29) contribution to the chemical potential [2] kvF ≈− π f00 12 | | 2 pF(n0) which is valid for 1 f 1 and x 1. We see K = ∗ (1 + f00) , (32) 00 3m (n0) that the solution we− found | | is ≪ purely imaginary≪ and its F dependence on x is very weak. being negative for f < 1. 00 − 6 Solutions of Eq. (31) for f00 < 1 are shown in Fig. 3. limit to the collision-less regime of the zero sound, i.e. We see that there is an interval of−k, 0 III. BOSONIZATION OF THE LOCAL For f00 < 1 not only the compressibility is negative INTERACTION but also the square− of the first sound velocity [2] ∂P p2 The description of a fermionic system with a con- u2 = = F (1 + f ) < 0 . (38) ∂ρ 3m m∗ 00 tact interaction can be equivalently described in terms of F F † ~ bosonic fields φq = p ψpψp+q, here we denote q = (ω, k) Here P denotes the pressure and ρ is the mass density. and p = (ǫ, ~p). InP terms of the functional path inte- Note that in the limit T 0 the isothermal and adiabatic gral the transition to the collective bosonic fields can compressibilities coincide,→ whereas for T = 0 the differ- be performed with the help of a formal change of vari- ence becomes substantial, see Ref. [32]. Also6 note that ables by means of a Hubbard-Stratonovich transforma- the first sound exists in the hydrodynamical (collisional) tion [34, 35]. After this transformation the effective Eu- regime, i.e. for ω τ −1 T 2, which is the opposite clidean action for the system with a repulsive interaction, ≪ col ∝ 7 here f0 > 0, can be written within the Matzubara tech- Then making the replacement (46) in the fermion scat- nique in terms of the bosonic fields as tering amplitude (11) we obtain 2 −1 2 φq φ−q Giˆ φˆq R (a N) f0mB 2 R Tph,0(ω, k)= − VB DB (ω, k) . S[φ]= ω Tr log 1 , (42) R −1 R 2Γ − − β (DB,0) (ω, k) ΣB (ω, k) ≡ Xq 0 h i − (47) ˆ ˆ where G and φ are infinite matrices in fre- Hence, we can identify the vertex of the boson-fermion quency/momentum space with matrix elements interaction [Gˆ]p,p′ = δp,p′ G(ǫ, ~p ) and [φˆ]q,q′ = δq,q′ φ~q. Trace 2 2 2 is taken over frequencies and momenta and includes fac- VB = f0mB/(a N) , (48) tor 2 accounting for the fermion spins. Transformation − to zero temperature follows by the standard replacement and the full retarded Green’s function of the boson β−1 dǫ/(2πi). DR(ω, k) with the retarded self-energy n → B ExpandingP R Eq. (42) up to the 4-th order in φ for T =0 we obtain R 2 ω k ΣB (ω, k)= f0mBΦ , . (49) kvF pF 1 S[φ] sgn(f ) φ (Γω)−1 + a2NΦ(s, x) φ (43) ≈ 2 0 q 0 −q Note that for very large mB that we have assumed, Xq R R DB (ω, k) in (47) differs from Dφ (ω, k) in (45) only by 1 a prefactor. For the attractive interaction, which we are + U(q , q , q , q )φ φ φ φ δ , 1 2 3 4 q1 q2 q3 q4 q1+q2+q3+q4,0 now interested in (f < 0), we have m2 > 0 and V 2 > 0. 4 X 0 B B {qi} For repulsive interaction instead of the bare scalar boson the bare vector boson would be an appropriate choice, where the effective field self–interaction is given by the cf. [37], or we may come back to the formalism given by function Eqs. (42) – (44). 1 The spectrum follows from the solution of the Dyson U(q1, q2, q3, q4)= 2i GpGp+q R −1 2 − 4! 1 equation 0 = [DB (ω, k)] mB(1 + f0Φ(s, x)) for P(1X,...,4) Xp m ω, k and coincides with≈ the − solution of Eq. (20). B ≫ G G . (44) The boson spectral function is given by × p+q1 +q2 p+q1+q2+q3 R AB(ω, k)= 2 DB (ω, k), Here the sum P runs over the 4 permutations of 4 mo- − ℑ R 2 menta qi. AsP we will show, Eq. (43) is applicable also 2 Tph,0(ω, k)= VB AB(ω, k) . (50) for f < 0 after the replacement φ iφ that results in − ℑ 0 → the appearance of the prefactor sgn(f0) in Eq. (43). The general analysis of Eq. (43) for arbitrary external IV. AVOIDING OF THE POMERANCHUK momenta was undertaken in Ref. [36]. The first term INSTABILITY BY A BOSE CONDENSATION in Eq. (43) (quadratic in φ) can be interpreted as the inversed retarded propagator of the effective boson zero- A. Condensation of a scalar field for f0 < −1. sound-like mode: Consider a Fermi liquid with local scalar interaction (DR)−1(ω, k)= sgn(f )[(Γω)−1 + a2NΦ(s, x)] . (45) φ − 0 0 f0 < 1. We assume that the bosonization procedure of the interaction,− Eqs. (46) and (47), is performed. Accord- To describe modes for an attractive interaction, f0 < 0, ing to the perturbative analysis in Sect. II D, for f0 < 1 it is instructive to re-derive the expression for the mode there are modes which grow with time. The growth of hy-− propagator using the approach proposed in Ref. [37]. The drodynamic modes (first sound), cf. Eqs. (39) and (40), ω local four-fermion interaction Γ0 can be viewed as an results in the formation of a mixed phase. Besides the hy- interaction induced by the exchange of a scalar heavy drodynamic modes the zero-sound-like modes grow with boson with the mass mB, time, cf. Eqs. (36) and (37). We study now the oppor- tunity that the instability of the zero-sound mode may ω 2 ω Γ0 mB ω 2 R result in a formation of a static condensate of the scalar Γ0 2 2 2 Γ0 mB DB,0(ω, k) , →−(ω + i0) k mB ≡− field. This leads to a decrease of the system energy and − − (46) to a rearrangement of the excitation spectrum on the ground of the condensate field. R We will exploit the simplest probing function describ- where DB,0 is the bare retarded Green’s function of the 2 ing the complex scalar field of the form of a running wave heavy boson and we assumed that mB is much larger then typical squared frequencies and momenta, ω2, k2, in −iωct+i~kc~r the problem. ϕ = ϕ0e , (51) 8 with the condensate frequency and momentum (ωc,~kc) which corresponds to the gain in the energy density and the constant amplitude ϕ0. The choice of the struc- 4 ture of the order parameter is unimportant for our study 2 ω˜ (kc) 2 EB = N 2 θ( ω˜ (kc)) , (59) of the stability of the system. − kc − 2λ 1+ 2p2 Guided by the construction of the full Green’s function F of the effective boson field, see Eq. (47), the effective where θ(x)=1 for x> 0 and 0 otherwise. Lagrangian density for the condensed field (51) can be written as, cf. Ref. [22, 38], B. Bose condensation in a homogeneous state. L = D−1(ω , k ) ϕ 2 1 V 4Λ(ω , k ) ϕ 4 . (52) ℜ B c c | 0| − 2 B c c | 0| 2 The energy density of the condensate is given then by If the minimum of the gapω ˜ is realized at kc = 0, e.g., the standard relation it is so for f02 0, the energy density EB is minimized ≥ for ωc = kc = 0. Then, the field amplitude and the E = ω∂L/∂ω L. B − condensate energy density are Fully equivalently, this Lagrangian density can be written 2 2 zf 2 zf in the form suggested by expansion (43), now applied for φ0 = N 2 θ(zf ) , EB = N θ(zf ) , (60) the running wave classical field, after the field redefinition a λ − 2λ m ϕ = a N/ f φ , where we used the same notation zf as in Eq. (34). The B 0 | 0| 0 p Bose condensate is formed for f00 < 1. L = D−1(ω , k ) φ 2 1 Λ(ω , k ) φ 4 . (53) − ℜ φ c c | 0| − 2 c c | 0| For the case of non-zero temperature, T = 0, in the mean field approximation one should just replace6 Φ Here and in Eq. (52) the quantity Λ(ω , k ) represents the c c Φ , where the fermion step-function distributions are re-→ self-interaction amplitude of condensed modes which cor- T placed to the thermal distributions, and from Eq. (18) responds to the ring diagram with four fermion Green’s for T ǫF one recovers the critical temperature of the functions (44), Λ(ωc, kc)=6U(qc, qc, qc, qc). As we ≪ MF − − condensation T /ǫF = 12 zf /π valid for 0 0.2 The latter equation shows that if originally we have f00 < 1, the renormalized interaction 1 < fren,00 < 1/2. Thus,− in the presence of the boson− condensate the− Fermi liquid is free from the Pomeranchuk instability. homogeneous BC stable Fermi liquid 0.0 Using the renormalized parameter fren,00 we can cal- culate how the fermion chemical potential changes in the 02 f presence of the condensate. We use here the standard expression from Ref. [6] for the variation of the chemi- cal potential with the particle density in the Fermi liquid -0.2 theory inhomogeneous BC 2ǫF,0 fren,00 δµF = δn + δn , (66) 3n0 N -0.4 -2.0 -1.5 -1.0 -0.5 0.0 2 ∗ here ǫF,0 = pF(n0)/2mF(n0). The energy density can be f µF 00 obtained from the relation E = 0 µFdn. The compressibility of the systemR becomes now FIG. 4: Phase diagram of a Fermi liquid with a scalar interac- tion in the particle hole channel described by the momentum δµF 2 2 K = n0 + KB dependent coupling constant f0(k/pF)= f00 + f02k /pF. De- δn n0 pending on the values of parameters f00 and f02 the Fermi 2 p (n0 ) liquid can be either in a stable state or in states with homo- = F 1+ f + K , (67) ∗ ren,00 B geneous or inhomogeneous static Bose condensates (BC) of 3mF(n0) scalar modes. Possibility of the development of the spinodal where the first term in the second line is always positive instability is suppressed. 1+ f = ( f 1)/(2 f 1) > 0 , (68) ren,00 | 00|− | 00|− for f < 1 under consideration. The second term is and a finite momentum kc are determined by relations 00 − 2 d 2 the condensate term: ω˜ (kc) 0 , dk ω˜ (kc) = 0 . The critical condition for the appearance≤ of the condensate can be deduced from 2 d EB 2 2 2 the expansion of f0 ω˜ (k) in (57) for x xm: KB = n0 2 . (69) | | ∼ dn n0 f00 2 (f02 12 ) Similarly, the square of the first sound velocity in the 1+ f0(x)Φ(0, x) 1+ f00 +3 − ≈ f00 presence of the condensate becomes (f02 + 20 ) 2 f + f00 f f00 2 ∂P pF 02 20 2 2 2 2 02 12 u = = (1 + f ) > 0 , (70) [x xm] , xm =6 − . (71) ∗ ren,00 − 12 − f00 ∂ρ 3mFmF f02 + 20 whereas in the absence of the condensate it was (1 + For f < 1 this function has always a negative mini- ∝ 00 − f00) < 0, cf. Eq. (38). mum at x = 0, the second and deeper minimum appears Recall that u2 is negative within the spinodal region at finite x, if which exists, if the pressure has a van-der-Waals form. cr Thus, the formation of the scalar Bose condensate, sug- f02 f = f00/12 . (72) ≤ 02 gested here, might compete with the development of the For 1 V. CONDENSATE OF BOSE EXCITATIONS −1 2 2 WITH NON-ZERO MOMENTUM IN A MOVING Λ=Λ(ω(k0), k0) + (Z0 (k0)) k0/ρ . (81) FERMI LIQUID For ω = 0,e Λ(0, k0) is calculated explicitly, cf. Eq. (55). Note that above equations hold also for Λ = 0. Let us apply the constructed above formalism to the The condensate of excitations is generated for the ve- analysis of a possibility of condensation of zero-sound-like locity of the medium exceeding the Landau critical ve- excitations with a non-zero momentum and frequency in locity, u>uL = ω(k0)/k0, where the direction of the a moving Fermi liquid. The main idea was formulated in condensate vector ~k coincides with the direction of the Refs. [9–15]. When a medium moves in straight line with 0 velocity, k~ ~u, and the magnitude k is determined by the velocity u>u = min[ω(k)/k] (where the minimum 0 0 L the equationk ω(k )/k = dω(k )/dk. The gain in the is realized at k = k = 0), it may become energetically 0 0 0 0 energy density after the formation of the classical con- favorable to transfer a6 part of the momentum from the densate field with the amplitude φ and the momentum particles of the moving medium to a condensate of col- 0 k0 is then lective Bose excitations with the momentum k0. The condensation may occur, if in the spectrum branch ω(k) −1 2 1 4 δE = Z0 (k0)[uk0 ω(k0)]φ0 + 2 Λφ0 . (82) there is a region with a small energy at sufficiently large − − momenta. The amplitude of the condensate field is founde by mini- As in Ref. [11], we consider a fluid element of the mization of the energy. From (82) one gets medium with the mass density ρ moving with a non- uk ω(k ) relativistic constant velocity ~u. The quasiparticle energy 2 −1 0 0 φ0 = Z0 (k0) − θ(u uc) . (83) ω(k) in the rest frame of the fluid is determined from the Λ − dispersion relation The resulting velocity of the mediume becomes −1 Dφ (ω, k)=0 . (76) (u uc)θ(u uc) ℜ ufin = uc + − − ) . (84) 1 + [Z−1(k )]2k2/(Λ(ω(k ), k )ρ We continue to exploit the complex scalar condensate 0 0 0 0 0 field described by the simplest running-wave probing For a small Λ, we have ufin = uc + O(Λ). function, cf. Eq. (51), and the Lagrangian density (53), For the repulsive interaction f0 > 0 there is real zero- 2 but now for the condensate of excitations. sound branch of excitations ωs(k) kvF(s0 + s2 x + 4 ≈ The appearance of the condensate with a finite mo- s4 x ), where the parameters si depend on the cou- ~ mentum k0, frequency ω = ω(k0) and an amplitude ϕ0 pling constants f00 and f02 according to Eqs. (24), (26), leads to a change of the fluid velocity from ~u to ~ufin, as and (27). As shown in Sect. II B the ratio ωs(k)/(kvF) it is required by the momentum conservation has a minimum at k0 = pF s2/(2s4) provided f02 is 2 − 2 2 smaller than fcrit,02 = f00p/[12(s0 1) ]. The Lan- ρ~u = ρ~u + ~k Z−1 φ2 , (77) − − fin 0 0 0 dau critical velocity of the medium is equal to uL/vF = | | 2 ωs(k0)/(vFk0) s0 s /(2s4). The quasiparticle weight ~ −1 2 ≈ − 2 where k0Z0 φ0 is the density of the momentum of the of the zero-sound mode (78) is condensate of| the| boson quasiparticles with the quasipar- 2 ticle weight −1 a N ∂Φ(s, x) Z0 (k0)= ω k k . (85) k v ∂s s( 0) , 0 0 F vFk0 pF −1 ∂ −1 Z0 (k0)= Dφ (ω, k) > 0 . (78) h∂ω ℜ iω(k0),k0 The amplitude of the condensate field (83) can be written as If in the absence of the condensate of excitations the 2 u uL energy density of the liquid element was Ein = ρu /2, 2 −1 ϕ0 = Z0 (k0)k0 − θ(u uL) . (86) then in the presence of the condensate of excitations, Λ − e 11 The energy density gained owing to the condensation of the zero-sound-like modes by performing bosonization of the excitations is a local fermion-fermion interaction. We argue that the 2 Fermi liquid with f0 < 1 might become stable owing 2 −1 2 (u uL) to appearance of the static− (homogeneous or inhomoge- δE = k0(Z0 (k0)) − θ(u uL) . (87) − 2Λ − neous) Bose condensate of the scalar quanta. Properties of the novel condensate state are investigated. For a small λ, Eqs. (86), (87) simplifye as In the future it would be important to search for a pos- (u u ) sibility of realization of this phenomenon in some Fermi ϕ2 = ρ − L θ(u u ), 0 −1 L liquids. It would be interesting to perform a careful com- Z0 (k0)k0 − ρ parative study of the possibilities of the Bose condensa- δE = (u u )2θ(u u ) . (88) − 2 − L − L tion and the ordinary spinodal instability appearing at the first-order phase transitions in the Fermi systems. For this, an explicit expression for the energy density VI. CONCLUDING REMARKS functional for a specific system, e.g. for the nuclear mat- ter, is required. Such a programme might be realized We described the spectrum of scalar excitations in nor- within the relativistic mean-field model. Similarly, we mal Fermi liquids for various values of the Landau param- expect a stabilization of the Fermi liquid with a strong attractive spin-spin interaction g < 1 by a conden- eter f0 in the particle-hole channel and for different mod- 0 − els of its momentum dependence. For f0 > 0 we found a sate of a virtual boson field at certain conditions. These condition on the momentum dependence of f0(k) when questions will be considered elsewhere. the zero-sound excitations with a non-zero momentum can be produced in the medium moving with the veloc- ity larger than the Landau critical velocity. Such ex- citations will form an inhomogeneous Bose condensate. 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