Pomeranchuk instability and Bose condensation of scalar quanta in a Fermi liquid E. E. Kolomeitsev1 and D. N. Voskresensky2 1Matej Bel University, SK-97401 Banska Bystrica, Slovakia 2 National Research Nuclear University (MEPhI), 115409 Moscow, Russia (Dated: October 14, 2018) We study excitations in a normal Fermi liquid with a local scalar interaction. Spectrum of bosonic scalar-mode excitations is investigated for various values and momentum dependence of the scalar Landau parameter f0 in the particle-hole channel. For f0 > 0 the conditions are found when the phase velocity on the spectrum of the zero sound acquires a minimum at a non-zero momentum. For −1 < f0 < 0 there are only damped excitations, and for f0 < −1 the spectrum becomes unstable against a growth of scalar-mode excitations (a Pomeranchuk instability). An effective Lagrangian for the scalar excitation modes is derived after performing a bosonization procedure. We demonstrate that the Pomeranchuk instability may be tamed by the formation of a static Bose condensate of the scalar modes. The condensation may occur in a homogeneous or inhomogeneous state relying on the momentum dependence of the scalar Landau parameter. Then we consider a possibility of the condensation of the zero-sound-like excitations in a state with a non-zero momentum in Fermi liquids moving with overcritical velocities, provided an appropriate momentum dependence of the Landau parameter f0(k) > 0. PACS numbers: 21.65.-f, 71.10.Ay, 71.45.-d Keywords: Nuclear matter, Fermi liquid, sound-like excitations, Pomeranchuk instability, Bose condensation I. INTRODUCTION density fluctuations. In hydrodynamic terms the condi- tion f < 1 implies that the speed of the first sound 0 − The theory of normal Fermi liquids was built up by becomes imaginary. This would lead to an exponential growth of the aerosol-like mixture of droplets and bub- Landau [1], see in textbooks [2–4]. The Fermi liquid ap- proach to the description of nuclear systems was devel- bles (spinodal instability). In a one-component Fermi liq- uid spinodal instability results in a mixed liquid-gas like oped by Migdal [5, 6]. In the Fermi liquid theory the low- lying excitations are described by several phenomeno- stationary state determined by the Maxwell construction logical Landau parameters. Pomeranchuk has shown in if the pressure has a van der Waals form as a function Ref. [7] that Fermi liquids are stable only if some inequal- of the volume. In the isospin symmetric nuclear matter ities on the values of the Landau parameters are fulfilled. (if the Coulomb interaction is neglected) the liquid-gas phase transition might occur [16, 17] for the baryon den- In this work we study low-lying scalar excitation modes < < sities 0.3n0 ∼ n ∼ 0.7n0, where n0 is the nuclear satura- (density-density fluctuations) in the cold normal Fermi tion density. In a many-component system a mechanical liquids for various values and momentum behavior of the instability is accompanied by a chemical instability, see scalar Landau parameter f0 in the particle-hole channel. Ref. [18]. The inclusion of the Coulomb interaction, see We assume that an interaction in the particle-particle Refs. [19, 20], leads to a possibility of the pasta phase in channel is repulsive and the system is, therefore, stable the neutron star crusts for densities 0.3n < n < 0.7n . against pairing in an s-wave state. An induced p-wave 0 ∼ ∼ 0 pairing to be possible at very low temperatures T <Tc,p, The key point of this work is that we suggest an al- see Refs. [8], can be precluded by the assumption that ternative description of unstable zero-sound-like modes which might be realized at certain conditions. We arXiv:1505.03884v1 [nucl-th] 14 May 2015 the temperature of the system is small but above Tc,p. demonstrate that for f0 < 1 instability may result in For f0 > 0 the conditions will be found when the − phase velocity of the spectrum possesses a minimum at an accumulation of a static Bose condensate of the scalar a non-zero momentum. This means that the spectrum field. The condensate amplitude is stabilized by the re- satisfies the Landau necessary condition for superfluid- pulsive self-interaction. The condensation may occur in ity. As a consequence this may lead to a condensation the homogeneous either in inhomogeneous state depend- of zero-sound-like excitations with a non-zero momentum ing on the momentum dependence of the Landau param- in moving Fermi liquids with the velocity above the Lan- eter f0(k). In the presence of the condensate the Fermi dau critical velocity [9]. Similar phenomena may occur liquid proves to be stable. in moving He-II, cold atomic gases, and other moving The work is organized as follows. In Sect. II we study media, like rotating neutron stars, cf. Refs. [10–15]. For spectrum of excitations in a one-component Fermi liquid 1 < f0 < 0 excitations are damped, for f0 < 1 the in the scalar channel in dependence on f0(k). In Sect. III spectrum− is unstable against the growth of zero-sound-− we bosonize the local interaction and suggest an effec- like modes and hydrodynamic modes. Up to now it was tive Lagrangian for the self-interacting scalar modes. In thought that for f0 < 1 the mechanical stability condi- Sect. IV we study Pomeranchuk instability for f0 < 1 tion is violated that results− in exponential buildup of the and suggest a novel possibility of the occurrence of the− 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.