Scalar quanta in Fermi liquids: zero sounds, instabilities, Bose condensation, and a metastable state in dilute nuclear matter

Abstract: Spectrum of bosonic scalar-mode excitations in a normal Fermi liquid with a local scalar interaction is investigated for various values and momentum dependence of the scalar Landau parameter $f_0$ in the particle-hole channel. For $f_0 >0$ the conditions are found when the phase velocity on the spectrum of the zero sound acquires a minimum at a non-zero momentum. For $-1<f_0 \<0$ there are only damped excitations, and for $f_0<-1$ the spectrum becomes unstable against a growth of scalar-mode excitations. An effective Lagrangian for the scalar excitation modes is derived after performing a bosonization procedure. We demonstrate that the 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. We show that in the isospin-symmetric nuclear matter there may appear a metastable state at a subsaturation nuclear density owing to the condensate. 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 $f_0 (k)\>0$. We also argue that in peripheral heavy-ion collisions the Pomeranchuk instability may occur already for $f_0 >-1$.