We study the effect of a strong magnetic field on the properties of neutron stars with a quarkhadron phase transition. It is shown that the magnetic field prevents the appearance of a quark phase, enhances the leptonic fraction, decreases the baryonic density extension of the mixed phase and stiffens the total equation of state, including both the stellar matter and the magnetic field contributions. Two parametrisations of a density dependent static magnetic field, increasing, respectively, fast and slowly with the density and reaching 2 − 4 × 10 18 G in the center of the star, are considered. The compact stars with strong magnetic fields have maximum mass configurations with larger masses and radius and smaller quark fractions. The parametrisation of the magnetic field with density has a strong influence on the star properties.

The effect of the density dependence of the nucleonic equation of state and the hyperon meson couplings on the star properties, including strangeness content, mass and radius, are studied within a relativistic mean field formalism. It is shown that there is still lacking information on the nucleonic equation of state at supra-saturation densities and on the hyperon interactions in nuclear matter that will allow a clear answer to the question whether the mass of the pulsar J1614-2230 could rule out exotic degrees of freedom from the interior of compact stars. We show that some star properties are affected in a similar way by the density dependence of the symmetry energy and the hyperon content of the star. To disentangle these two effects it is essential to have a good knowledge of the equation of state at supra-saturation densities. A linear correlation between the radius and the strangeness content of a star with a fixed mass is obtained.

The effect of strong magnetic fields on the equation of state (EoS) for compact stars described with density dependent relativistic hadronic models is studied. A comparison with other mean-field relativistic models is done. It is shown that the largest differences between models occur for low densities and that the magnetic field affects the crust properties of star, namely its extension. PACS numbers: 26.60.-c 26.60.Kp 97.60.Jd 24.10.Jv I. INTRODUCTIONThe study of very asymmetric nuclear matter is presently an important issue due to the radioactive beams which will be operating in the near future and which will allow the investigation of a region of the nuclear matter phase space unaccessible till recently. Asymmetric nuclear matter is of particular interest for the description of stellar matter of compact stars.Compact star properties depend a lot on the model used to describe the hadronic equation of state (EoS). In particular relativistic nuclear mean-field models [1,2] are very popular to describe stellar matter because causality will always be satisfied. The imposition of constraints, both coming from measured star properties or from relativistic heavy ion collisions in the laboratory, is essential to test the different models [3].Magnetars are neutron stars which may have surface magnetic fields larger that 10 15 G [4-6] and which were discovered at the x-ray and γ-ray energies (for a review refer [7]). They are identified with the anomalous x-ray pulsars (AXP) and soft γ-ray repeaters. Taking as reference the critical

In the present study we analyse the effect of the density dependence of the symmetry energy on the hyperonic content of neutron stars within a relativistic mean field description of stellar matter. For the Λ-hyperon, we consider parametrizations calibrated to Λ-hypernuclei. For the Σ and Ξ-hyperons uncertainties that reflect the present lack of experimental information on Σ and Ξ-hypernuclei are taken into account. We perform our study considering nuclear equations of state that predict two solar mass stars, and satisfy other well settled nuclear matter properties. The effect of the presence of hyperons on the radius, the direct Urca processes, and the cooling of accreting neutron stars are discussed. We show that some star properties are affected in a similar way by the density dependence of the symmetry energy and the hyperon content of the star. To disentangle these two effects it is essential to have a good knowledge of the equation of state at supra-saturation densities. The density dependence of the symmetry energy affects the order of appearance of the different hyperons, which may have direct implications on the neutron star cooling as different hyperonic neutrino processes processes may operate at the center of massive stars. For models which allow for the direct Urca process to operate, hyperonic and purely nucleonic ones are shown to have a similar luminosity when hyperons are included in agreement with modern experimental data. It is shown that for a density dependent hadronic model constrained by experimental, theoretical and observational data, the low-luminosity of SAX J1808.4 − 3658 can only be modelled for a hyperonic NS, suggesting that hyperons could be present in its core.

The self-consistent random phase approximation (SCRPA) is applied to the exactly solvable model with fermion-boson coupling proposed by Schütte and Da Providencia. Very encouraging results in comparison with the exact solution of the model for various observables are obtained. The transition from the normal phase to the phase with a spontaneously broken symmetry is carefully investigated. The strong reduction of the variance in SCRPA vs Hartree-Fock is pointed out. PACS number(s): 21.60. Jz, 24.10.Cn During the last decade the so-called self-consistent version of the random phase approximation (SCRPA) has seen very encouraging successes in a number of nontrivial model cases (see, for example, Ref.[1] for a detailed description of the method and Ref.[2] for the application of SCRPA to the many-level pairing model). In spite of these performances of the theory, there are remaining problems. In first place this concerns situations with spontaneously broken symmetries. Such situations were treated in Refs. [1,3,4]. Whereas in the Lipkin model [3] the symmetry broken ("deformed") phase caused no problem because the broken symmetry is discrete (parity), in the other two cases [1,4], with a continuous broken symmetry, problems appeared with the low-lying mode known to be exactly at zero energy in the standard HF-RPA approach (the spurious or Goldstone mode), where HF stands for Hartree-Fock. In the two cases cited [1,4] the low-lying mode does not appear at zero energy in SCRPA because the RPA operator does not contain the symmetry operator as a limit case. Indeed, e.g., the number operator in quasiparticle (BCS) representation contains a purely Hermitian piece ␣ k † ␣ k which cannot be incorporated in the RPA operator which by definition is non-Hermitian. The same situation is present in the Schütte-Da-Providenica boson-fermion model [5] where the symmetry operator contains the boson and fermion number operators. The violation of the Goldstone theorem signifies that the Ward identities and conservation laws are not respected. Though this violation seems relatively mild and to go away in macroscopic systems (the Hermitian pieces becoming of zero weight), the situation remains annoying for finite systems.In this paper which can be considered as a sequel of Ref.[4] we want again to investigate the Schütte-Da Providenica model:where the a † , a are fermion operators. In analogy to the work in Ref. ͓4͔ we will introduce the more general RPA operatorThe operators T ± , T 0 are obtained from ± , 0 , by writing the latter ones in the deformed basisThe bosons operators B † and B are obtained from the original ones by a shift transformation B → b − , where is a c number characterizing the appearance of the Bose condensate. The introduction of the boson pair operators ␤ † ␤ † is motivated by the fact that otherwise there exists a certain dissymmetry between fermions and bosons, the fermions being in any case bilinear whereas the bosons are otherwise only contained to linear order in Eq. ͑2͒. Also the symmetry operator P = b † ...

We investigate the effects of strong magnetic fields on the equation of state of warm stellar matter as it may occur in a protoneutron star. Both neutrino free and neutrino trapped matter at a fixed entropy per baryon are analyzed. A relativistic meanfield nuclear model, including the possibility of hyperon formation, is considered. A density dependent magnetic field with the magnitude 10 15 G at the surface and not more than 3 × 10 18 G at the center is considered. The magnetic field gives rise to a neutrino suppression, mainly at low densities, in matter with trapped neutrinos. It is shown that an hybrid protoneutron star will not evolve to a low mass blackhole if the magnetic field is strong enough and the magnetic field does not decay. However, the decay of the magnetic field after cooling may give rise to the formation of a low mass blackhole.

We investigate the effects of strong magnetic fields on the equation of state of dense stellar neutrino-free and neutrino-trapped matter. Relativistic nuclear models both with constant couplings (NLW) and with density dependent parameters (DDRH) and including hyperons are considered . It is shown that at low densities neutrinos are suppressed in the presence of the magnetic field. The magnetic field reduces the strangeness fraction of neutrino-free matter and increases the strangeness fraction of neutrino-trapped matter. The mass-radius relation of stars described by these equations of state are determined. The magnetic field makes the overall equation of state stiffer and the stronger the field the larger the mass of maximum mass star and the smaller the baryon density at the center of the star. As a consequence in the presence of strong magnetic fields the possibility that a protoneutron star evolves to a blackhole is smaller.

Self-consistent quasiparticle random phase approximation ͑SCQRPA͒ is for the first time applied to a more level pairing case. Various filling situations and values for the coupling constant are considered. Very encouraging results in comparison with the exact solution of the model are obtained. The nature of the low-lying mode in SCQRPA is identified. The strong reduction of the number fluctuation in SCQRPA vs BCS is pointed out. The transition from superfluidity to the normal fluid case is carefully investigated.