Mutual entrainment effects in cold neutron-proton mixtures are studied in the framework of the self-consistent nuclear energy-density functional theory. Exact expressions for the mass currents, valid for both homogeneous and inhomogeneous systems, are directly derived from the time-dependent Hartree-Fock equations with no further approximation. The equivalence with the Fermi-liquid expression is also demonstrated. Focusing on neutron-star cores, a convenient and simple analytical formulation of the entrainment matrix in terms of the isovector effective mass is found, thus allowing to relate entrainment phenomena in neutron stars to isovector giant dipole resonances in finite nuclei. Results obtained with different functionals are presented. These include the Brussels-Montreal functionals, for which unified equations of state of neutron stars have been recently calculated.
The superfluid drag effect, in hydrodynamics of pulsating neutron stars, is conventionally described with the aid of the entrainment matrix relating the mass currents with the velocities of superfluid flows in the system. Equations for the entrainment matrix of a superfluid mixture of neutrons and protons are derived with allowance for the strong dependence of the energy gaps on the velocities of superfluid flows. The calculations are carried out in the frame of the Fermi-liquid theory. The equations obtained are highly nonlinear. Numerical solutions to the equations for some typical cases dem…
Mutual entrainment effects in hot neutron-proton superfluid mixtures are studied in the framework of the self-consistent nuclear energy-density functional theory. The local mass currents in homogeneous or inhomogeneous nuclear systems, which we derive from the time-dependent Hartree-Fock-Bogoliubov equations at finite temperatures, are shown to have the same formal expression as the ones we found earlier in the absence of pairing at zero temperature. Analytical expressions for the entrainment matrix are obtained for application to superfluid neutron-star cores. Results are compared to those ob…
We calculate the important quantity of superfluid hydrodynamics, the relativistic entrainment matrix for a nucleon-hyperon mixture at arbitrary temperature. In the nonrelativistic limit this matrix is also termed the Andreev-Bashkin or mass-density matrix. Our results can be useful for modeling the pulsations of massive neutron stars with superfluid nucleon-hyperon cores and for studies of the kinetic properties of superfluid baryon matter.
We investigate the slowly rotating general relativistic superfluid neutron stars including the entrainment effect in a two-fluid model, where one fluid represents the superfluid neutrons and the other is the charge-neutral fluid called the proton fluid, made of protons and electrons. The equation of state and the entrainment effect between the superfluid neutrons and the proton fluid are computed using a relativistic mean field (RMF) model where baryon-baryon interaction is mediated by the exchange of $σ$, $ω$, and $ρ$ mesons and scalar self interactions are also included. The equations govern…
We calculate the relativistic entrainment matrix Y_ik at zero temperature for nucleon-hyperon mixture composed of neutrons, protons, Lambda and Sigma^- hyperons, as well as of electrons and muons. This matrix is analogous to the entrainment matrix (also termed mass-density matrix or Andreev-Bashkin matrix) of non-relativistic theory. It is an important ingredient for modelling the pulsations of massive neutron stars with superfluid nucleon-hyperon cores. The calculation is done in the frame of the relativistic Landau Fermi-liquid theory generalized to the case of superfluid mixtures; the matri…
It is considered a closed system of non-linear equations for the entrainment matrix of a non-relativistic mixture of superfluid nucleons at arbitrary temperatures below the onset of neutron superfluidity, which takes into account the essential dependence of the superfluid energy gap in the nucleon spectra on the velocities of superfluid flows. It is assumed that the protons condense into the isotropic $^1$S$_0$ state, and the neutrons are paired into the spin-triplet $^3$P$_2$ state. It is derived an analytic solution to the non-linear equations for the entrainment matrix under temperatures ju…
We propose an experimental protocol to directly detect the Andreev-Bashkin effect (entrainment) in the bulk mixture of a bosonic and fermionic superfluid using a ring geometry. Our protocol involves the interferometric detection of the entrainment-induced phase gradient across a superfluid due to the flow of another in which it is immersed. The choice of ring geometry eliminates variations in the stronger mean-field interaction which can thwart the detection of entrainment in other geometries. A significant enhancement of the entrainment phase shift signal is possible, if the dimer-boson scatt…
The entrainment matrix (also termed the Andreev-Bashkin matrix or the mass-density matrix) for a neutron-proton mixture is derived at a finite temperature in a neutron star core. The calculation is performed in the frame of the Landau Fermi-liquid theory generalized to account for superfluidity of nucleons. It is shown, that the temperature dependence of the entrainment matrix is described by a universal function independent on an actual model of nucleon-nucleon interaction employed. The results are presented in the form convenient for their practical use. The entrainment matrix is important, …
Nuclear matter properties based on a relativistic approach suitable for the description of multi-component systems are calculated. We use a set of nuclear relativistic mean-field models that satisfy acceptable nuclear matter properties and neutron star observations. The effects of the density dependence of the symmetry energy and of the Landau quantization due to the presence of a strong external magnetic field are discussed. Properties such as the proton fraction, the Landau mass, Landau parameters and entrainment matrix, the adiabatic index and speed of sound are calculated for cold $β$-equi…
Hydrodynamical simulations of neutron star cores, based on a two fluid description in terms of a neutron-proton superfluid mixture, require the knowledge of the Andreev-Bashkin entrainment matrix which relates the momentum of one constituent to the currents of both constituents. This matrix is derived for arbitrary nuclear asymmetry at zero temperature and in the limits of small relative currents in the framework of the energy density functional theory. The Skyrme energy density functional is considered as a particular case. General analytic formulae for the entrainment parameters and various …
We calculate the finite-temperature r-mode spectrum of a superfluid neutron star accounting for both muons in the core and the entrainment between neutrons and protons. We show that the standard perturbation scheme, considering the rotation rate as an expansion parameter, breaks down in this case. We develop an original perturbation scheme which circumvents this problem by treating both the perturbations due to rotation and (weak) entrainment simultaneously. Applying this scheme, we propose a simple method for calculating the superfluid r-mode eigenfrequency in the limit of vanishing rotation …
I show that the usual model of the rotational response of a neutron star, which predicts rotation-induced neutronic vortices and no rotation-induced protonic vortices, does not hold (i) beyond a certain threshold of entrainment interaction strength nor (ii) in case of nonzero $Σ^-$ hyperon gap. I demonstrate that in both these cases the rotational response involves creation of phase windings in electrically charged condensate. Lattices of bound states of vortices which are caused these effects can (for a range of parameters) strongly reduce the interaction between rotation-induced vortices wit…
Background: In the solid crust of neutron stars, a variety of crystalline structure may exist. Recently the band theory of solids has been applied to the inner crust of neutron stars and significance of the entrainment between dripped neutrons and the solid crust was advocated. Since it influences interpretations of various phenomena of neutron stars, it has been desired to develop deeper understanding of the microphysics behind. Purpose: The purpose of the present article is to propose a fully self-consistent microscopic framework for describing time-dependent dynamics of neutron star matter,…
We present a numerical model for uniformly rotating superfluid neutron stars, for the first time with realistic microphysics including entrainment, in a fully general relativistic framework. We compute stationary and axisymmetric configurations of neutron stars composed of two fluids, namely superfluid neutrons and charged particles (protons and electrons), rotating with different rates around a common axis. Both fluids are coupled by entrainment, a non-dissipative interaction which in case of a non-vanishing relative velocity between the fluids, causes the fluid momenta being not aligned with…
A simple description of superfluid hydrodynamics in the inner crust of a neutron star is given. Particular attention is paid to the effect of the lattice of nuclei on the properties of the superfluid neutrons, and the effects of entrainment, the fact that some fraction of the neutrons are locked to the motion of the protons in nuclei.
We calculate the frequencies of collective modes of neutrons, protons and electrons in the outer core of neutron stars. The neutrons and protons are treated in a hydrodynamic approximation and the electrons are regarded as collisionless. The coupling of the nucleons to the electrons leads to Landau damping of the collective modes and to significant dispersion of the low-lying modes. We investigate the sensitivity of the mode frequencies to the strength of entrainment between neutrons and protons, which is not well characterized. The contribution of collective modes to the thermal conductivity …
We develop a formalism to model slowly rotating superfluid Newtonian neutron stars. A two-fluid model is used to describe the matter, where one fluid is the superfluid neutrons that are believed to exist in the inner crust and core of mature neutron stars, while the other fluid is a charge neutral conglomerate of the remaining constituents (crust nuclei, core superconducting protons, electrons, etc). We include the entrainment effect, which is a non-dissipative interaction between the two fluids whereby a momentum induced in one of the fluids will cause part of the mass of the other fluid to b…
Superfluid extension of the self-consistent time-dependent band theory for neutron star matter: Anti-entrainment versus superfluid effects in the slab phase
A mixture of superconducting and superfluid nuclear liquids of protons coupled to the ultrarelativistic electron gas, and neutrons is considered. In the magnetohydrodynamic (MHD) approximation, the energy-momentum (stress) tensor is derived, and the entrainment contribution is found in the explicit form. It is shown that this contribution generates a force density, when a superfluid velocity lag and the magnetic field are simultaneously present. This force may be important in the nuclear "pasta" phase in neutron stars, if the proton and neutron Cooper pairing in the pasta phase is taken into a…
We investigate the tidal deformability of a superfluid neutron star. We calculate the equilibrium structure in the general relativistic two-fluid formalism with entrainment effect where we take neutron superfluid as one fluid and the other fluid is comprised of protons and electrons, making it a charge neutral fluid. We use a relativistic mean field model for the equation of state of matter where the interaction between baryons is mediated by the exchange $σ$, $ω$ and $ρ$ mesons. Then, we study the linear, static $l=2$ perturbation on the star to compute the electric-type Love number following…
We constrain the parameters of neutron superfluidity in the cores of neutron stars making use of the recently proposed effect of resonance stabilization of $r$-modes. To this end, we, for the first time, calculate the finite-temperature $r$-mode spectra for realistic models of rotating superfluid neutron stars, accounting for both muons and neutron-proton entrainment in their interiors. We find that the ordinary (normal) $r$-mode exhibits avoided crossings with superfluid $r$-modes at certain stellar temperatures and spin frequencies. Near the avoided crossings, the normal $r$-mode dissipates …
We study transport properties of a strongly interacting superfluid mixture of two Fermi-liquids. A typical example of such matter is the neutron-proton liquid in the cores of neutron stars. To describe the mixture, we employ the Landau theory of Fermi-liquids, generalized to allow for the effects of superfluidity. We formulate the kinetic equation and analyze linear response of the system to vector (e.g., electromagnetic) perturbation. In particular, we calculate the transverse and longitudinal polarization functions for both liquid components. We demonstrate, that they can be expressed throug…
We have performed calculations based on the Skyrme energy density functional (EDF) that includes arbitrary mixing between protons and neutrons. In this framework, single-particle states are generalized as mixtures of proton and neutron components. The model assumes that the Skyrme EDF is invariant under the rotation in isospin space and the Coulomb force is the only source of the isospin symmetry breaking. To control the isospin of the system, we employ the isocranking method, which is analogous to the standard cranking approach used for describing high-spin states. Here, we present results of…
We identify the possible ground states for a mixture of two superfluid condensates (one neutral, the other electrically charged) using a phenomenological Ginzburg-Landau model. While this framework is applicable to any interacting condensed-matter mixture of a charged and a neutral component, we focus on nuclear matter in neutron star cores, where proton and neutron condensates are coupled via non-dissipative entrainment. We employ the Skyrme interaction to determine the neutron star's equilibrium composition, and hence obtain realistic coefficients for our Ginzburg-Landau model at each depth …
We study the superfluid dynamics of the outer core of neutron stars by means of a hydrodynamic model made of a neutronic superfluid and a protonic superconductor, coupled by both the dynamic entrainment and the Skyrme SLy4 nucleon-nucleon interactions. The resulting nonlinear equations of motion are probed in the search for dynamical instabilities triggered by the relative motion of the superfluids that could be related to observed timing anomalies in pulsars. Through linear analysis, the origin and expected growth of the instabilities is explored for varying nuclear-matter density. Differentl…
The seniority scheme has been shown to be extremely useful for the classification of nuclear states in semi-magic nuclei. The neutron-proton ($np$) correlation breaks the seniority symmetry in a major way. As a result, the corresponding wave function is a mixture of many components with different seniority quantum numbers. In this contribution we show that the $np$ interaction may favor a new kind of coupling in $N=Z$ nuclei, i.e., the so-called isoscalar spin-aligned $np$ pair mode. Shell model calculations reveal that the ground and low-lying yrast states of the $N = Z$ nuclei $^{92}$Pd and …
We apply a novel equation of state (EoS) that includes the surface tension contribution induced by interparticle interaction and asymmetry between neutrons and protons, to the study of neutron star (NS) properties. This elaborated EoS is obtained from the virial expansion applied to multicomponent particle mixtures with hard core repulsion. The considered model is in full concordance with all the known properties of normal nuclear matter, provides a high-quality description of the proton flow constraints, hadron multiplicities created during the nuclear-nuclear collision experiments, and equal…
The mass, radius and crustal fraction of moment of inertia in neutron stars are calculated using $β$-equilibrated nuclear matter obtained from Skyrme effective interaction. The transition density, pressure and proton fraction at the inner edge separating the liquid core from the solid crust of the neutron stars are determined from the thermodynamic stability conditions using the KDE0v1 set. The neutron star masses obtained by solving the Tolman-Oppenheimer-Volkoff equations using neutron star matter obtained from this set is able to describe highly massive compact stars $\sim$2$M_{\odot}$. The…
We explore novel features of a nonrelativistic fermionic superfluid in which the pairing interaction includes a contribution from the exchange of a dynamical bosonic mode. We show that the dynamical boson exchange (DBE), which causes a retarded pairing interaction and thus violates the Galilean invariance of the fermion sector, generically leads to a quantum reduction of the superfluid density and hence a nonzero normal fraction even at zero temperature. For spin-singlet pairing, the DBE also leads to a nonvanishing spin susceptibility at zero temperature, providing a mechanism for the coexist…
In a recent paper by Link, it was pointed out that the standard picture of the neutron star core composed of a mixture of a neutron superfluid and a proton type-II superconductor is inconsistent with observations of a long period precession in isolated pulsars. In the following we will show that an appropriate treatment of the interacting two-component superfluid (made of neutron and proton Cooper pairs), when the structure of proton vortices is strongly modified, may dramatically change the standard picture, resulting in a type-I superconductor. In this case the magnetic field is expelled fro…
We discuss experimental evidence for a nuclear phase transition driven by the different concentration of neutrons to protons. Different ratios of the neutron to proton concentrations lead to different critical points for the phase transition. This is analogous to the phase transitions occurring in 4He-3He liquid mixtures. We present experimental results which reveal the N/A (or Z/A) dependence of the phase transition and discuss possible implications of these observations in terms of the Landau Free Energy description of critical phenomena.
A nucleus is a quantum many body system made of strongly interacting Fermions, protons and neutrons (nucleons). This produces a rich Nuclear Equation of State whose knowledge is crucial to our understanding of the composition and evolution of celestial objects. The nuclear equation of state displays many different features; first neutrons and protons might be treated as identical particles or nucleons, but when the differences between protons and neutrons are spelled out, we can have completely different scenarios, just by changing slightly their interactions. At zero temperature and for neutr…
The Borromean nucleus $^{17}$Ne ($^{15}$O$ + p + p$) is investigated by using the hyperspheric adiabatic expansion for a a three-body system. The measured size of $^{15}$O and the low-lying resonances of $^{16}$F ($^{15}$O$ + p$) are first used as constraints to determine both central and spin-dependent two-body interactions. Then, the ground state structure of $^{17}$Ne is found to be an almost equal mixture of $s^2$ and $d^2$ proton-$^{15}$O relative states, the two lowest excited states have about 80% of $sd$-mixed components, and for the next two excited three-body states the proton-$^{15}…
In a recent paper by Link, it was pointed out that the the standard picture of the neutron star core, composed of a mixture of a neutron superfluid and a proton type-II superconductor, is inconsistent with observations of long period precession in isolated pulsars. In the following we will show that intervortex force between the magnetic flux tubes may be attractive resulting in a type-I (rather than type-II) superconductor. In this case the magnetic field cannot exist in the form of magnetic flux tubes, supporting Link's observation. This behavior of the system is due to the strong interactio…
We study a global nuclear structure in the framework of experimental observables. With the aid of large nuclear structure data at the national nuclear data center, NNDC, we present the distinctive systematic patterns emerged in the first 2+ excited energies, E(2+) and their energy ratios to the first 4+ levels, R = E(4+)/E(2+), in the even-even nuclei, over 50 < Z < 82 for protons, and 50 < N < 126 for neutrons. We introduce the so-called pseudo-shell configurations from the subshells mixture in order to explain a semi-double shell closure, a shape phase transition, and a reinforced deformatio…
