Spin and spin-isospin instabilities in asymmetric nuclear matter at zero and finite temperatures using Skyrme functionals

These interactions have been also widely applied to the description of extreme astrophysical environments such as neutron stars and supernova cores. Actually very soon after Skyrme [2] introduced his eponymous effective interaction, Cameron [3] applied it to calculate the structure of neutron stars. Assuming that neutron stars were made only of neutrons, he found that their maximum mass was significantly higher than the Chandrasekhar mass limit. His work thus brought support to the scenario of neutron star formation from the catastrophic gravitational collapse of massive stars in supernova explosions, as proposed much earlier by Baade and Zwicky [4]. The interior of neutron stars is highly neutron rich but contains also a non-negligible amount of protons, leptons and possibly other particles. However microscopic calculations in uniform infinite nuclear matter using bare nucleon-nucleon potentials have been usually restricted to symmetric nuclear matter (SNM) and pure neutron matter (NeuM). Even though effective interactions are phenomenological, they can provide a convenient interpolation of realistic calculations to determine the equation of state of neutron star cores. Mean-field calculations can be easily extended to finite temperatures and can thus be also used to describe the hot nuclear matter found in supernova cores and protoneutron stars. Moreover, the mean-field method allows a consistent and tractable treatment of both homogeneous matter and inhomogeneous matter (e.g. neutron star crusts [5]) with a reduced computational cost. This opens the way to a unified description of all regions of neutron stars and supernova cores [6].

Nevertheless the application of these effective forces to nuclear matter at high densities has been limited by the occurrence of spurious instabilities [7,8]. In particular, Skyrme forces predict a spontaneous transition to a spin-polarized phase when the density exceeds a critical threshold which depends on the isospin asymmetry [9][10][11][12][13]. Besides, it is found that for some forces the energy density of the spin-polarized phase decreases with increasing density. In this case, the phase transition is accompanied by a catastrophic collapse [14], which is contradicted by the existence of neutron stars (note however that observations alone do not exclude the possibility of a ferromagnetic core inside neutron stars, see for instance Refs. [15,16]). Moreover, the critical density predicted within the Skyrme formalism generally decreases with temperature due to an anomalous behavior of the entropy, which is larger in the spin-ordered phase than in the unpolarized phase [17,18]. This instability can strongly affect the neutrino propagation in hot dense nuclear matter [12,19,20] which is believed to play an important role in the supernova explosion mechanism and in the evolution of protoneutron stars [21]. However, no such spin-polarized phase transition is found by microscopic calculations using realistic nucleon-nucleon potentials. Indeed several calculations based on different methods, such as the lowest-order constrained variational method [22][23][24][25][26], the Brueckner-Hartree-Fock method [27][28][29], the auxiliary field diffusion Monte Carlo method [30] and the Dirac-Brueckner-Hartree-Fock method [31], show that nuclear matter remains unpolarized well above the nuclear saturation density ρ 0 both at zero and finite temperatures.

The prediction of spin-ordering in nuclear matter is one of the main deficiencies of the mean-field method with effective forces. Different extensions of the standard Skyrme force have been recently proposed in order to prevent these phase transitions at zero temperature [12,13]. In this paper, the origin of the spin and spin-isospin instabilities is revisited in the more general framework of the nuclear energy density functional (EDF) theory (see for instance Ref. [32] for a review) and a simpler prescription is proposed to ensure stability of dense nuclear matter for any degree of spin and spin-isospin polarizations and for any temperature. The paper is organized as follows. The Skyrme functionals that we consider here are defined in Section II. Section III is devoted to the discussion about the stability of nuclear matter. Several Skyrme functionals are reexamined in Section IV.

The nuclear EDFs that we consider here are of the form

where E kin is the kinetic energy, E Coul is the Coulomb energy and E Sky is a functional of the local densities and currents (q = n, p for neutron, proton respectively): the density ρ q , the current density j q j q j q , the kinetic density τ q , the spin density s q s q s q , the spin kinetic density T q T q T q and the spin-current tensor J q,µν (see for instance Ref. [1] for precise definitions). It is convenient to int