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

Self-consistent mean field methods based on phenomenological Skyrme effective interactions are known to exhibit spurious spin and spin-isospin instabilities both at zero and finite temperatures when applied to homogeneous nuclear matter at the densities encountered in neutron stars and in supernova cores. The origin of these instabilities is revisited in the framework of the nuclear energy density functional theory and a simple prescription is proposed to remove them. The stability of several Skyrme parametrizations is reexamined.


I. INTRODUCTION
The self-consistent mean-field method with Skyrme effective interactions has been very successful in describing the structure and the dynamics of medium-mass and heavy nuclei [1].
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.

II. SKYRME FUNCTIONALS
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 introduce the isospin index t = 0, 1 for isoscalar and isovector quantities respectively. Isoscalar quantities (also written without any subscript) are sums over neutrons and protons (e.g. ρ 0 = ρ = ρ n + ρ p ) while isovector quantities are differences between neutrons and protons (e.g. ρ 1 = ρ n − ρ p ). The Skyrme functional E Sky is then given by The spin current vector is defined by J tκ = µ,ν ε κµν J t,µν , where ε κµν is the Levi-Civita tensor. The so-called "time-even" part E even t ("time-odd" part E odd t ) contains only even (odd) densities and currents with respect to time reversal.
The coupling "constants" C ρ t and C s t generally depend on the isoscalar density ρ = ρ n +ρ p as follows Moreover, local gauge invariance [33,34] imposes the following relations Historically the type of functionals given by Eqs.(2a)-(2c) were obtained from the Hartree-Fock approximation using effective zero-range interactions of the Skyrme type [1,6] v i,j = t 0 (1 + x 0 P σ )δ(r r r ij ) + 1 2 where r r r ij = r r r i − r r r j , r r r = (r r r i + r r r j )/2, p p p ij = −ih(∇ ∇ ∇ i − ∇ ∇ ∇ j )/2 is the relative momentum, P σ is the two-body spin-exchange operator. The relations between the coupling constants in Eqs. (2b) and (2c) and the parameters of the effective force in Eq. (5), can be found for instance in Appendix A of Ref. [1].
Kutschera and Wójcik [14] pointed out that for some Skyrme forces not only is the ground state of NeuM polarized, but also the energy density of polarized NeuM decreases with increasing density. However such a catastrophic ferromagnetic collapse is ruled out by neutron star observations. The origin of this singular behavior can be traced back to the parameters t 2 and x 2 of the Skyrme force. In particular, the authors of Ref. [14] found that in order to prevent a ferromagnetic collapse of NeuM, the parameters of Skyrme forces must satisfy the following inequality This constraint was taken into account to construct the Saclay-Lyon Skyrme parametrizations [35], which were fitted with the parameter x 2 = −1. These forces which were specifically developed for astrophysics, have been widely used in neutron star studies. However it has been found that these forces predict various transitions to spin-ordered phases in nuclear matter [10,12,13,17,18] even though Eq. (6) was enforced. Actually this is a general feature of standard Skyrme forces [7,8]. We will now reexamin this issue in the framework of the nuclear EDF theory.

III. STABILITY OF NUCLEAR MATTER
Let us consider the case of static uniform (possibly polarized) infinite isospin asymmetric nuclear matter. The Skyrme energy density, Eq. (2a), thus reduces to Let us choose the spin-quantization axis so that the only non-vanishing components of the spin density s q s q s q and the spin kinetic density T q T q T q are along the z-axis. For brevity we will simply write s q and T q instead of s qz and T qz . In the following we will neglect the anisotropies induced by the polarization of matter [36]. Introducing the density ρ qσ of nucleons with spins σ =↑, ↓ and the kinetic density of polarized nucleons defined by the spin density and the spin kinetic density can now be expressed as In fully polarized NeuM with all spins up (ρ = ρ n↑ ), Eq. (7) reduces to If the energy density is calculated from a Skyrme force in the Hartree-Fock approximation, we find so that Eq. (11) reduces to Eq. (12) is a consequence of the Pauli exclusion principle and the zero range of the Skyrme interaction. Actually as will be shown elsewhere, Eq. (12) must still be satisfied for nuclear EDFs that are not obtained from effective forces in order to prevent self-interactions. The constraint of Kutschera and Wójcik [14], Eq. (6), can thus be more generally written as If this inequality is violated, E pol NeuM decreases with increasing density thus leading to a ferromagnetic collapse.
It is instructive to rewrite Eq. (11) as where we have introduced the effective mass of a nucleon in a spin state σ defined bȳ with +(−) for spin up (spin down respectively), and M * q is the usual effective mass given bȳ It can be easily seen that in fully polarized NeuM, the effective mass reduces tō so that Eq. (16) coincides with Eq. (14). Setting x 2 = −1 as in the Saclay-Lyon Skyrme forces [35] therefore implies that the effective mass of polarized neutrons is equal to the bare mass.
We have seen that the constraint of Ref. [14] is equivalent to the requirement that the effective mass of polarized neutrons remains always positive. However the ground state of NeuM (and more generally that of isospin asymmetric nuclear matter) could still be polarized as shown below.

A. Landau stability criterion
The stability of unpolarized homogeneous nuclear matter with respect to spin and spinisospin polarizations has been generally addressed using the Landau Fermi-liquid theory (see e.g. Ref. [37]). In this theory, the elementary excitations of the liquid at low temperatures are described in terms of quasiparticles which are in one-to-one correspondence with singleparticle states of the non-interacting Fermi gas. Any small change δñ(k k k) in the distribution function of quasiparticles with wave vector k k k leads to a change δE in the energy density, which can be expressed (up to second order) as where ε(k k k) is the energy of a quasiparticle with wave vector k k k and v(k k k, k ′ k ′ k ′ ) is the residual interaction between quasiparticles with wave vectors k k k and k ′ k ′ k ′ .
In pure NeuM, the residual interaction (neglecting tensor interaction) can be expressed where N is the density of states at the Fermi surface given by with k F = (3π 2 ρ) 1/3 . We have also introduced the Pauli matricesσ σ σ andσ ′ σ ′ σ ′ in order to take into account the spin of the quasiparticles. Small perturbations involve only quasiparticles at the Fermi surface, i.e. with k = k ′ = k F . We can thus expand each term in the residual interaction in Legendre polynomials P ℓ (cos θ) where θ is the angle between k k k and k ′ k ′ k ′ . For instance, where In particular, the condition on G NeuM 0 guarantees that NeuM is stable against small fluctuations of the (isoscalar) spin polarization I σ = s 0 /ρ = (ρ ↑ − ρ ↓ ). This can be seen by expanding the energy per particle up to second order in I σ e(I σ ) ≃ e(0) + 1 2 The first order term vanishes because of the requirement that the unpolarized phase be an equilibrium state.
Using the Skyrme functional, we find Now if the Skyrme functional is fitted to a realistic equation of state of NeuM [13], we find that C ρ 0 + C ρ 1 ≤ 0, which according to Eq. (12) implies that Ferromagnetic instabilities are therefore mainly due to the coupling constants C T t . In order to fulfill the Landau's stability condition G NeuM 0 > −1 at any density, we must have 1 The absence of a ferromagnetic transition in NeuM does not generally forbid the occurence of spin-ordered phases in asymmetric nuclear matter. Let us consider in particular SNM.
The most general form of the residual interaction (neglecting tensor interaction) can be expressed as where N 0 is the density of states at the Fermi surface given by We have also introduced the Pauli matricesτ τ τ ,τ ′ τ ′ τ ′ in order to take into account the isospin of the quasiparticles. As before, we can define dimensionless Landau parameters F ℓ , F ′ ℓ , G ℓ and G ′ ℓ . The Landau's stability conditions are in this case given by The Landau parameters F 0 and F ′ 0 are related to the usual compression modulus and symmetry energy respectively. The conditions on G 0 and G ′ 0 ensure that SNM is stable against small fluctuations of isoscalar and isovector spin densities respectively. These Landau parameters can be expressed in terms of the spin asymmetry coefficient, defined by and the spin-isospin asymmetry coefficient, defined by where Using the Skyrme functional, the Landau parameters G 0 and G ′ 0 are given by The stability of SNM at any density thus requires These two conditions entail Eq. (29). Note that Landau's stability conditions allow one of the coefficients C s t to be negative provided their sum remains positive. Landau's stability conditions, Eqs. (24b),(33c) and (33d), guarantee that the unpolarized state is locally stable (metastable) against small fluctuations of the spin and spin-isospin polarizations. But this criterion does not necessarily imply that the unpolarized state is the ground state, i.e. the state with the lowest energy. In particular, the ground state could still be polarized with finite values of I σ and I στ . Moreover we have only considered so far the two limiting cases of SNM and NeuM. However, the outer core of neutron stars is formed of isospin asymmetric nuclear matter whose composition varies with depth. We thus need a more general stability criterion.

B. General stability criterion
Asymmetric nuclear matter is stable with respect to any degree of spin and spin-isospin polarizations whenever the energy density E pol of the polarized state is larger than the energy density E unpol of the unpolarized state (for a given density ρ). Using Eqs. (7), (17) and (18) we find which for unpolarized matter (i.e. s q = 0, T q = 0) yields with The difference can thus be expressed as where τ pol q = τ q↑ + τ q↓ is the nucleon kinetic density in the polarized phase. The absolute stability of the unpolarized phase can be insured by requiring each term be separately positive so that E pol > E unpol . Now the first term in Eq. (44) is always positive since mechanical stability requires M * q ≥ 0 and the Pauli exclusion principle implies that τ pol q > τ q .
Let us also remark that (s n − s p )(T n − T p ) ≥ 0 because τ qσ increases monotonically with ρ qσ .
The following constraints and therefore guarantee the absence of any spin-ordered phase transitions in asymmetric nuclear matter. It is readily seen from Eqs. (27), (38) and (39) C. Anomalous behavior of the entropy We have seen that the stability of nuclear matter requires that C T t ≥ 0. However, these coefficients cannot take arbitrary values. From Eq. (4), large positive values of C T t translate into large negative values of C J t which, in certain circumstances, can lead to instabilities in finite nuclei whose consequence is a major rearrangement of the single-particle spectrum [44].
We will now show that these coupling constants can be further constrained by requiring the stability of nuclear matter with respect to any degree of spin and spin-isospin polarizations at non-zero temperatures.
It was shown in Refs. [17,18] that not only do Skyrme forces predict a ferromagnetic transition in NeuM above a certain critical density, but worse this density decreases with increasing temperature due to an anomalous behavior of the entropy. This argument can be easily transposed to asymmetric nuclear matter as follows. At low temperatures (compared to nucleon Fermi energies), the difference between the entropy density S pol of the polarized state and the entropy density S unpol of the unpolarized state is approximately given by Now because the polarized phase is more ordered than the unpolarized phase, its entropy according to Boltzmann's definition should thus be lower, i.e. S pol < S unpol as found in realistic calculations [23][24][25]29]. Since this should be true for any isospin asymmetry, we This condition reduces to that of Ref. [17] in the limiting case of fully polarized NeuM.
But it can be seen from Eq. (50) that such solutions cannot exist for all densities and must therefore be excluded. Inserting Eq. (49) in Eq. (51a) using Eq. (18) yields The terms in ρ q ′ always satisfy the above inequalities. This is a consequence of the positivity of M * q for any density and isospin asymmetry which requires that C τ 0 +C τ 1 ≥ 0 and C τ 0 −C τ 1 ≥ 0, as can be seen from Eq. (18). The conditions (52a) and (52b) can be ensured for any density ρ q by imposing that the associated terms be respectively negative and positive leading to On the other hand, Eq. (51b) implies Combining these inequalities with Eqs. (45b), we arrive at the following restrictions Eqs. (55) guarantee that asymmetric nuclear matter remains unpolarized at finite temperature T since the free energy density of the polarized phase, defined by F pol = E pol − T S pol , is always higher than the free energy density F unpol = E unpol − T S unpol of the unpolarized phase.

Conventional Skyrme forces have been shown to predict various spin and spin-isospin
instabilities in nuclear matter [7,9,10,13,14,17,18]. We have seen in the previous section that for a nuclear functional given by Eqs. (2a),(2b) and (2c) the stability of asymmetric nuclear matter at any temperature can be ensured by imposing Eqs. (45a) and (55) [the constraint proposed in Ref. [14], Eq. (6) and more generally Eq. (15), prevents a collapse of polarized NeuM, but does not forbid a ferromagnetic transition]. While the coefficients C s t are generally positive (at least for not too high densities), standard Skyrme forces yield negative values of at least one of the couplings constants C T t . The origin of the instabilities can therefore be traced back to the time-odd terms s t s t s t · T t T t T t , which are related to the timeeven terms µ,ν J t,µν J t,µν due to gauge invariance (4). Since the seminal work of Vautherin and Brink [38], it is commonly taken for granted that the spin-current tensor (which is usually approximated by the spin-current vector J q J q J q ) is small in nuclei, and most Skyrme parametrizations therefore neglect them. We have tested this assumption by computing the HFB energies with and without the J 2 and J 2 q terms (denoted respectively by E HFB and E 0 HFB ) for all even-even nuclei with Z, N > 8 and Z < 110 lying between the proton and neutron drip lines (Note that when the J 2 and J 2 q terms are included, the associated time-odd terms in C T t play a role in the exact treatment of the masses of odd nuclei, but not in the equal-filling approximation [39], which we adopt here, as in all our previous papers).
The differences ∆M ≡ E HFB − E 0 HFB are shown in Fig. 1 for the Skyrme parametrization BSk17 [40,41] which was originally fitted with the J 2 and J 2 q terms, and for SkI2 [42] which was not. The impact of the J 2 and J 2 q terms is quite large, reaching about 20 MeV for the heaviest nuclei. The impact of dropping or including the J 2 and J 2 q terms is logically found to be correlated to the amplitude of the C T t = −C J t coupling constants, especially C T 0 . For instance, in the case of the SLy4 [35] interaction (C T 0 = −17.21 MeV fm 5 ), the HFB energy is affected by no more than 5 MeV, while for SkO [46] (C T 0 = −220.54 MeV fm 5 ) values up to 30 MeV can be reached. Adding or removing the J 2 and J 2 q terms a posteriori without refitting all the parameters of the force can thus lead to significant errors. However in all previous studies of spin and spin-isospin instabilities in nuclear matter [7-14, 17, 18], the time-odd terms s t s t s t · T t T t T t were taken into account whereas the Skyrme forces were generally fitted without the J 2 and J 2 q terms. This treatment not only violates gauge symmetry but also introduces inconsistencies in the residual interaction hence in the Landau parameters (see the discussion in Section III of Ref. [43] and also in Section 5D of Ref. [44]).
We have therefore reexamined the stability of several Skyrme parametrizations for which the J 2 and J 2 q terms were not included in the fit: SGII [45], SLy4 [35], SkI1-SkI5 [42], SkO [46] and LNS [47]. The parametrization SGII [45] was constructed in order to improve the Landau parameters G 0 and G ′ 0 and the description of Gamow-Teller resonances in nuclei. The Skyrme Saclay-Lyon forces and especially the parametrization SLy4 [35], have been widely used not only in nuclear physics, but also in neutron-star studies because these forces were constrained to reproduce a realistic neutron-matter equation of state. The SkI [42] forces were all constrained (except for SkI1) to reproduce the isotopic shifts of the root mean square charge radii of neutron rich Pb and Ca nuclei. Forces SkI3 and SkI4 were constructed with non-standard spin-orbit couplings. For the parametrization SkI5, the 16 O ground-state data were excluded from the fit. We have also included the parametrization SkO [46] from the same group. The parametrization LNS [47] was fitted to Brueckner calculations. The Landau parameters in SNM and in NeuM calculated at saturation density ρ 0 , with and without the terms in C T t are shown in Table I. For comparison we have also indicated the predictions from Brueckner-Hartree-Fock calculations in SNM [48] and from realistic calculations based on the renormalization group approach in NeuM [49]. As can be seen in Table I Table II shows the critical densities of the spin-ordered phase transitions according to Landau's stability criterion. It can be seen that dropping the terms s t s t s t · T t T t T t eliminates the instabilities in almost all Skyrme forces. This prescription is also consistent with Eqs. (55) and therefore prevents an anomalous behavior of the entropy thus ensuring the stability of nuclear matter for any temperatures.
Moreover, setting C T t = 0 is the only prescription which guarantees the Landau stability conditions of Eqs. (24b), (33c) and (33d) at any density both for ℓ = 0 and ℓ = 1. Indeed the Landau parameters G 1 , G ′ 1 in SNM and G NeuM 1 in NeuM, are given by Requiring G 1 ≥ −3, G ′ 1 ≥ −3 and G NeuM 1 ≥ −3 for any density thus leads to C T t ≤ 0. Combining these inequalities with Eqs. (40) yields C T t = 0. Adopting these particular values tends to be supported by the following basic sum rules of Landau Fermi liquid theory [51] and Even though Skyrme forces generally violate these sum rules, the prescription C T t = 0 significantly improves the second sum rule as can be seen in Table III. It is quite remarkable that dropping the terms s t s t s t · T t T t T t not only removes all kinds of instabilities in nuclear matter but also improves the internal consistency of the nuclear functional. Nevertheless with this prescription, the Landau parameters G 1 , G ′ 1 and G NeuM 1 all vanish leading to unrealistic effective masses in polarized matter. Indeed, according to Eqs. (17) M * q↑ = M * q↓ = M * q which obviously holds in the limit of vanishing spin polarizations but is otherwise contradicted by realistic calculations [23-25, 29, 31]. In particular, these calculations indicate that in polarized NeuM M n↑ > M n↓ whenever ρ ↑ > ρ ↓ . Imposing the less stringent stability conditions (55) leads to a splitting of effective masses but with a wrong sign. This deficiency calls for further extensions of existing Skyrme functionals.
In the discussion above, we have implicitly adopted the point of view of the nuclear EDF theory [32] that the different terms appearing in Eqs (2b) and (2c) can be a priori considered as independent from each other (apart from the requirements of gauge invariance). It is therefore perfectly legitimate to set C J t = −C T t ≡ 0. However in the framework of effective forces, the coupling constants are uniquely determined by the parameters of the force. In particular, the coefficients C s t and C T t are now given by We have therefore studied the stability of the few Skyrme parametrizations which were fitted with the J 2 and J 2 q terms: SkP [52], SLy5 [35], SkO ′ [46], SkX [53] and BSk17 [40,41]. The parametrization SkP, which was specifically designed to be used both in the particle-hole channel and in the particle-particle channel, is still used nowadays. The forces SLy5 and SkO ′ were fitted following the same protocol as SLy4 and SkO respectively, but they include the contribution of the J 2 and J 2 q terms. The force SkX [53] was constructed in an attempt to improve the description of single-particle energies. BSk17 is the force underlying our nuclear mass model HFB-17, based on the Hartree-Fock-Bogoliubov method [40,41] Table V. For those few Skyrme forces which include the J 2 and J 2 q terms, nuclear matter is therefore unstable because of the tight correlations between the different coupling constants in the energy density. In order to illustrate the impact of the J 2 and J 2 q terms and their time-odd counterparts on the stability of nuclear matter, we have plotted in Fig. 2 the difference between the energy per particle in fully polarized NeuM and in unpolarized NeuM for the parametrizations SLy4 and BSk17. Both have been fitted to a realistic equation of state of NeuM, but BSk17 includes the J 2 and J 2 q terms while SLy4 does not. Removing all instabilities requires that we impose C s t ≥ 0 and C T t = 0. Since the first term in t 0 of the Skyrme force is generally associated with the long-range attractive part of the nucleon-nucleon interaction while the density-dependent term in t 3 is related to the strongly repulsive short-range part, the coupling constant C s 0 can be made positive for any density by choosing x 0 < 1/2 and x 3 > 0. With t 0 < 0 and t 3 > 0, the coefficient C s 1 will be positive, at least for not too high densities. Spin-and spin-isospin instabilities thus generally arise mainly from the coupling constants C T 0 and C T 1 , which in turn are generated by the momentum-dependent terms in t 1 and t 2 . Using Eqs. (60c) and (60d), the conditions C T t = 0 entail t 1 = t 2 and x 1 = −x 2 . Imposing these constraints would leave no degree of freedom for adjusting surface properties of nuclei, which also depend on the momentum dependent t 1 and t 2 terms through the coupling constants C ∆ρ t . This would also have an impact on the coupling constants C τ t which determine the nucleon effective masses, Eq. (18). There is little doubt that such a force would yield poor results when applied to nuclei. Since thermal effects on the spin polarization are rather small for temperatures found in protoneutron stars and supernova cores [17], one may be tempted to require the stability of cold nuclear matter only. But even in this case, it was shown in Refs. [7,8] that it is not possible to avoid spurious transitions to spin-ordered phases in nuclear matter above 2-3 times saturation density, and at the same time giving reasonable properties of SNM. We have found that the critical densities above which instabilities occur are even lower when more nuclear data are included in the fit of the effective interaction. In particular, conventional Skyrme forces fitted to essentially all experimental nuclear mass data predict a ferromagnetic transition in NeuM at a density slightly above saturation density [13] (see also Table V).
The stability of cold nuclear matter can only be restored by including additional components in the Skyrme interaction, thereby inducing new terms in the energy density. Two different extensions have been recently proposed. Margueron and Sagawa [12] considered extended Skyrme forces with two new t 3 like terms depending on the nucleon spin densities s q s q s q of the form In the energy density, Eqs. (2b) and (2c), these new terms modify the coefficients C s t . The additional parameters were adjusted so as to ensure the Landau stability conditions The nuclear mass model HFB-17 [40,41] was thus refitted with these new terms [55]. With this extended Skyrme force called BSk17st, it was possible to maintain the quality of the HFB-17 mass model, and at the same time the Landau parameters were adjusted so as to remove the spin and spin-isospin instabilities present in the original force BSk17. Unfortunately instabilities were still found for finite spin and spin-isospin polarizations [55]. The reason is that terms of the form given by Eq. (61), do not change the coefficients C T t and consequently, Eq. (44) is not guaranteed to remain positive for any spin and spin-isospin polarizations. Moreover, as noted in Ref. [55] the contributions of Eq. (61) to the energy density cancel in fully polarized NeuM so that BSk17st still predicts a ferromagnetic collapse of NeuM as BSk17 does. The extension of Ref. [12] does not affect the coefficients C T t hence also the effective masses of spin-up and spin-down nucleons are not affected, as can be seen from Eq. (17). This means that if the original Skyrme force violates the constraint (47), this will still be the case for the extended version of this force.
Alternatively, instabilities can be avoided by introducing into the force, density-dependent generalizations of the usual t 1 and t 2 terms of the form [13] 1 2 t 4 (1 + x 4 P σ ) 1 h 2 p 2 ij ρ(r r r) β δ(r r r ij ) + δ(r r r ij )ρ(r r r) β p 2 ij + t 5 (1 + x 5 P σ ) 1 h 2 p p p ij · ρ(r r r) γ δ(r r r ij ) p p p ij .(62) These new terms modify the coefficients C τ t , C T t , C ∆ρ t and C ∆s t thus providing more flexibility to remove instabilities without deteriorating the fit to nuclear data. We have constructed a new nuclear mass model, labeled HFB-18, with such a generalized Skyrme force [13]. The parameters t 5 , x 5 and γ were chosen in order to avoid a ferromagnetic collapse of neutronstar matter. For simplicity, the remaining parameters in Eq. (62) were fixed by the equations which ensure that the contributions of the new terms to the coefficients C τ t vanish identically. As a result, the t 4 and t 5 terms cancel exactly in unpolarized homogeneous nuclear matter.
This new model yields almost as good a mass fit as our previous model HFB-17, with the advantage that NeuM matter is now stable with respect to any degree of spin polarizations.
Even thouth this new force still predicts an isospin instability in SNM, this does not affect the interior of neutron stars which is now unpolarized. Moreover we have found that this isospin instability can be easily removed if the conditions (63a)-(63c) are released, without deteriorating the quality of the mass fit [56]. However we did not succeed in constructing a nuclear mass model that satisfies Eq. (47). As a consequence, nuclear matter could still become unstable at finite temperatures even though no phase transitions occur at zero temperature, as shown in Ref. [17].
One might be tempted to enforce the stability conditions C T t = 0 by adding a zerorange tensor force to the conventional Skyrme interaction (5) with suitable adjustments of the parameters, like the parametrization T22 of Ref. [44]. Unfortunately, a tensor force introduces new terms in the functional which also affect the stability of nuclear matter [57].
The stability of 41 different Skyrme interactions having a tensor component has been recently studied in Ref. [58]. In particular, the recent Skyrme forces from the Saclay-Lyon group [44] which include tensor forces and which were fitted following the same protocol as the older SLy family [35], still predict various spin and spin-isospin instabilities. This is notably the case for the force T22 for which C T t = 0.

V. CONCLUSION
Nuclear energy density functional theory has been traditionally restricted to very specific phenomenological semi-local functionals of the form given by Eqs. (2a)-(2c), based on effective forces [1,6]. However the use of effective forces introduces tight correlations between different terms of the functional, which can generate various kinds of instabilities. In particular, the time-odd terms s t s t s t · T t T t T t induced by the momentum-dependent part of Skyrme forces (which contribute also to the coupling constants C τ t , C ∆ρ t and C ∆s t ) are responsible for spurious spin and spin-isospin instabilities in infinite homogeneous nuclear matter at densities encountered in the interior of neutron stars. In some cases, instabilities arise in symmetric nuclear matter below saturation densities and could thus also contaminate calculations in finite nuclei (Note that the coupling constants C ∆ρ t alone were found to drive finite-size instabilities [59]). These correlations between different parts of the nuclear energy density functional hamper the development of more accurate functionals since adding one term in the effective force can induce several new terms in the functional. Moreover, the coupling constants of the time-odd terms are generally not directly fitted to experimental data but are calculated a posteriori using the parameters of Skyrme force. However, there is no guarantee that the effects associated with the time-odd terms will be correctly described in this way. As shown in Refs. [7,8] it is not possible to avoid spurious transitions to spin-ordered phases in nuclear matter above 2-3 times saturation density. The critical densities above which these instabilities occur, decrease when more nuclear data are included in the fit of the parameters of the Skyrme force [13]. For instance, for our nuclear mass model HFB-17 [40,41], the ground state of neutron matter becomes ferromagnetic above 0.17 fm −3 . These instabilities can be (at least partially) removed by suitable extensions of the Skyrme force, as proposed for instance in Refs. [12,13]. However an unphysical spin-ordering could still occur at finite temperatures thus spoiling the application of Skyrme forces to the hot nuclear matter found in protoneutron stars and supernova cores. Alternatively the terms s t s t s t · T t T t T t that are responsible for spin and spin-isospin instabilities could be canceled by suitable adjustments of an additional tensor component to the Skyrme force [44]. Unfortunately a tensor force would also generate new terms in the energy density which still lead to instabilities [58].
On the other hand, the concept of effective forces leads to formal inconsistencies as recently discussed in Ref. [60]. Lots of efforts are now devoted to the construction of non-empirical functionals from realistic interactions directly without resorting to effective forces [32]. If one adopts the point of view that the nuclear functional is more fundamental than effective forces, the different terms appearing in Eqs.(2b) and (2c) can be treated independently (apart from the requirements of gauge invariance and cancellation of selfinteractions). It is therefore perfectly legitimate to set C J t = −C T t ≡ 0. Actually the J 2 and J 2 q terms are dropped in most Skyrme forces, not only because of simplicity but also because it seems to be favored by global fits to nuclear data and basic nuclear matter properties [61].
Moreover, the J 2 and J 2 q terms might even lead to instabilities in the single-particle spectra of finite nuclei, as discussed for instance in Ref. [44]. However in all previous studies of spin and spin-isospin instabilities in nuclear matter [7-14, 17, 18], the associated time-odd terms s s s · T T T and (s n s n s n − s p s p s p ) · (T n T n T n − T p T p T p ) have been included in the residual interaction thus violating gauge symmetry. We have therefore reexamined the stability of nuclear matter by setting C T t ≡ 0 for those Skyrme parametrizations which were fitted without the J 2 and J 2 q terms. We have found that this simple prescription not only improves the values of the Landau parameters G 0 , G ′ 0 and G NeuM 0 . But this also generally removes all kinds of instabilities in asymmetric nuclear matter both at zero and finite temperatures. Nevertheless this prescription yields unrealistic values of the Landau parameters G 1 , G ′ 1 and G NeuM 1 , hence also of the effective masses M * qσ in polarized matter. Further improvements thus require extensions of existing Skyrme functionals. in neutron matter (at saturation density) for selected Skyrme forces which were fitted without the J 2 and J 2 q terms.
Values in parenthesis were obtained by setting C T t = 0. The last line shows the Landau parameters predicted by microscopic calculations using realistic interactions: Ref. [48] for symmetric nuclear matter and Ref. [49] for neutron matter. criterion for selected Skyrme forces which were fitted without the J 2 and J 2 q terms. The first two column are for symmetric nuclear matter, while the last column is for pure neutron matter. The densities indicated in parenthesis were obtained by setting C T t = 0.  terms. The last line shows the Landau parameters predicted by microscopic calculations using realistic interactions: Ref. [48] for symmetric nuclear matter and Ref. [49] for neutron matter.