In neutron star matter, there exist $^{1}S_{0}$ superfluids in lower density in the crust while $^{3}P_{2}$ superfluids are believed to exist at higher density deep inside the core. In the latter, depending on the temperature and magnetic field, either the uniaxial nematic phase, the D$_{2}$-biaxial nematic phase, or the D$_{4}$-biaxial nematic phase appears. In this paper, we discuss a mixture of the $^{1}S_{0}$ and $^{3}P_{2}$ superfluids and find their coexistence. Adopting the loop expansion and the weak-coupling approximation for the interaction between two neutrons, we obtain the Ginzburg-Landau (GL) free energy in which both of the $^{1}S_{0}$ and $^{3}P_{2}$ condensates are taken into account by including the coupling terms between them. We analyze the GL free energy and obtain the phase diagram for the temperature and magnetic field. We find that the $^{1}S_{0}$ superfluid excludes the $^{3}P_{2}$ superfluid completely in the absence of magnetic field, they can coexist for weak magnetic fields, and the $^{1}S_{0}$ superfluid is expelled by the $^{3}P_{2}$ superfluid at strong magnetic fields, thereby proving the robustness of $^{3}P_{2}$ superfluid against the magnetic field. We further show that the D$_{4}$-BN phase covers the whole region of the $^{3}P_{2}$ superfluidity as a result of the coupling term, in contrast to the case of a pure $^{3}P_{2}$ superfluid studied before in which the D$_{4}$-BN phase is realized only under strong magnetic fields. Thus, the D$_{4}$-BN phase is topologically the most interesting phase, e.g., admitting half-quantized non-Abelian vortices relevant not only in magnetars but also in ordinary neutron stars.
Nonequilibrium conditions imposed by neutrino cooling through the liquid-solid transition lead to disorder in the solid crust of neutron stars. Disorder reduces the superfluid fraction, $ρ_s/ρ$, at densities above that of neutron drip, $ρ_d \approx 4\times 10^{11}\,g/cm^3$. For an amorphous solid crust the suppression of $ρ_s$ is small, except in the highest density regions of the crust. In contrast to the strong reduction in neutron conduction predicted for coherent Bragg scattering in a crystalline crust, the disordered solid crust supports sufficient neutron superfluid density to account fo…
The neutron superfluidity in the inner crust of a neutron star has been traditionally studied considering either homogeneous neutron matter or only a small number of nucleons confined inside the spherical Wigner-Seitz cell. Drawing analogies with the recently discovered multi-band superconductors, we have solved the anisotropic multi-band BCS gap equations with Bloch boundary conditions, thus providing a unified description taking consistently into account both the free neutrons and the nuclear clusters. Calculations have been carried out using the effective interaction underlying our recent H…
Axial anomalous contributions into neutrino PBF losses due to triplet pairing of neutrons are still ignored in modeling the evolution of neutron stars. In this paper, the influence of the anomalous axial contributions onto the rate of neutron stars cooling is estimated.
We discuss collective excitations (both fundamental and solitonic excitations) of quantized superfluid vortices in neutron $^3P_2$ superfluids, which likely exist in high density neutron matter such as neutron stars. Besides the well-known Kelvin modes (translational zero modes), we find a gapfull mode whose low-energy description takes the simple form of a double sine-Gordon model. The associated kink solution and its effects on spontaneous magnetization inside the vortex core are analyzed in detail.
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.
The interior of a neutron star is expected to be occupied by a neutron $^3P_2$ superfluid, which is the condensate of spin-triplet $p$-wave Cooper pairs of neutrons with total angular momentum $J=2$. Here we investigate the thermodynamic stability of $^3P_2$ superfluids in a neutron-star interior under a strong magnetic field. Using the theory incorporating the finite size correction of neutron Fermi surface, we show that the spin-polarized phases of $^3P_2$ superfluids, the magnetized biaxial nematic phase and the ferromagnetic phase, appear in high temperatures and high magnetic fields. Thes…
Despite the absence of viscous drag, the neutron superfluid permeating the inner crust of a neutron star can still be strongly coupled to nuclei due to non-dissipative entrainment effects. Neutron superfluidity and entrainment have been systematically studied in all regions of the inner crust of a cold non-accreting neutron star in the framework of the band theory of solids. It is shown that in the intermediate layers of the inner crust a large fraction of "free" neutrons are actually entrained by the crust. The results suggest that a revision of the interpretation of many observable astrophys…
We study surface effects of neutron $^{3}P_{2}$ superfluids in neutron stars. $^{3}P_{2}$ superfluids are in uniaxial nematic (UN), D$_{2}$ biaxial nematic (BN), or D$_{4}$ BN phase, depending on the strength of magnetic fields from small to large. We suppose a neutron $^{3}P_{2}$ superfluid in a ball with a spherical boundary. Adopting a suitable boundary condition for $^{3}P_{2}$ condensates, we solve the Ginzburg-Landau equation to find several surface properties for the neutron $^{3}P_{2}$ superfluid. First, the phase on the surface can be different from that of the bulk, and symmetry rest…
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…
In the Ginzburg-Landau limit, a possible texture without singular cores in a rotating neutron spin-triplet superfluid is studied. It is constructed the lattice of non-singular vortices with a vorticity diffusely distributed over the entire unit cell. The upper limit of the free energy associated with this structure is estimated, and the result shows that non-singular vortices are more preferable in the core of rotating neutron stars than ordinary linear singular vortices. The order parameter of the studied neutron system belongs to the unitary class. This implies that the superfluid under cons…
We propose that the observed cooling of the neutron star in Cassiopeia A is due to enhanced neutrino emission from the recent onset of the breaking and formation of neutron Cooper pairs in the 3P2 channel. We find that the critical temperature for this superfluid transition is ~0.5x10^9 K. The observed rapidity of the cooling implies that protons were already in a superconducting state with a larger critical temperature. Our prediction that this cooling will continue for several decades at the present rate can be tested by continuous monitoring of this neutron star.
We consider how superfluidity of dripped neutrons in the crust of a neutron star affects the frequencies of the crust's fundamental torsional oscillations. A nonnegligible superfluid part of dripped neutrons, which do not comove with nuclei, act to reduce the enthalpy density and thus enhance the oscillation frequencies. By assuming that the quasi-periodic oscillations observed in giant flares of soft gamma repeaters arise from the fundamental torsional oscillations and that the mass and radius of the neutron star is in the range of 1.4 < M/M_SUN < 1.8 and 10 km < R < 14 km, we constrain the d…
This review focuses on applications of the ideas of superfluidity and superconductivity in neutron stars in a broader context, ranging from the microphysics of pairing in nucleonic superfluids to macroscopic manifestations of superfluidity in pulsars. The exposition of the basics of pairing, vorticity and mutual friction can serve as an introduction to the subject. We also review some topics of recent interest, including the various types of pinning of vortices, glitches, and oscillations in neutron stars containing superfluid phases of baryonic matter.
In spite of the absence of viscous drag, the neutron superfluid permeating the inner crust of a neutron star cannot flow freely, and is entrained by the nuclear lattice similarly to laboratory superfluid atomic gases in optical lattices. The role of entrainment on the neutron superfluid dynamics is reviewed. For this purpose, a minimal hydrodynamical model of superfluidity in neutron-star crusts is presented. This model relies on a fully four-dimensionally covariant action principle. The equivalence of this formulation with the more traditional approach is demonstrated. In addition, the differ…
Calculations of the effects of band structure on the neutron superfluid density in the crust of neutron stars made under the assumption that the effects of pairing are small [N. Chamel, Phys. Rev. C 85, 035801 (2012)] lead to moments of inertia of superfluid neutrons so small that the crust alone is insufficient to account for the magnitude of neutron star glitches. Inspired by earlier work on ultracold atomic gases in an optical lattice, we investigate fermions with attractive interactions in a periodic lattice in the mean-field approximation. The effects of band structure are suppressed when…
We show that suppression of the baryon energy gaps, caused by the relative motion of superfluid and normal liquid components, can substantially influence dynamical properties and evolution of neutron stars. This effect has been previously ignored in the neutron-star literature.
The inner crust of neutron stars is supposed to be inhomogeneous and composed of dense structures (clusters) that are immersed in a dilute gas of unbound neutrons. Here we consider spherical clusters forming a BCC crystal and cylindrical rods arranged in a hexagonal lattice. We study the relative motion of these dense structures and the neutron gas using superfluid hydrodynamics. Within this approach, which relies on the assumption that Cooper pairs are small compared to the crystalline structures, we find that the entrainment of neutrons by the clusters is very weak since neutrons of the gas …
The supernova remnant Cassiopeia A contains the youngest known neutron star which is also the first one for which real time cooling has ever been observed. In order to explain the rapid cooling of this neutron star, we first present the fundamental properties of neutron stars that control their thermal evolution with emphasis on the neutrino emission processes and neutron/proton superfluidity/superconductivity. Equipped with these results, we present a scenario in which the observed cooling of the neutron star in Cassiopeia A is interpreted as being due to the recent onset of neutron superflui…
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 …
I show that the standard picture of the neutron star core containing coexisting neutron and proton superfluids, with the proton component forming a type II superconductor threaded by flux tubes, is inconsistent with observations of long-period (~1 yr) precession in isolated pulsars. I conclude that either the two superfluids coexist nowhere in the stellar core, or the core is a type I superconductor rather than type II. Either possibility would have interesting implications for neutron star cooling and theories of spin jumps (glitches).
We investigate the effect of a microscopic three-body force on the proton and neutron superfluidity in the $^1S_0$ channel in $β$-stable neutron star matter. It is found that the three-body force has only a small effect on the neutron $^1S_0$ pairing gap, but it suppresses strongly the proton $^1S_0$ superfluidity in $β$-stable neutron star matter.
The possible $^{1}S_{0}$ protonic superfluidity is investigated in neutron star matter, and the corresponding energy gap as a function of baryonic density is calculated on the basis of BCS gap equation. We have discussed particularly the influence of hyperon degrees of freedom on $^{1}S_{0}$ protonic superfluidity. It is found that the appearance of hyperons leads to a slight decrease of $^{1}S_{0}$ protonic pairing energy gap in most density range of existing $^{1}S_{0}$ protonic superfluidity. However, when the baryonic density $ρ_{B}>$0.377 (or 0.409) fm${^{-3}}$ for TM1 (or TMA) parameter …
We review recent progress in the theory of neutron matter with particular emphasis on its superfluid properties. Results of quantum Monte Carlo calculations of simple and realistic models of uniform superfluid neutron gas are discussed along with those of neutrons interacting in a potential well chosen to approximate neutron-rich oxygen isotopes. The properties of dilute superfluid Fermi gases that may be produced in atom traps, and their relations with neutron matter, are illustrated. The density dependence of the effective interaction between neutrons, used to describe neutron-rich systems w…
We have performed tilted-axis-cranked Hartree-Fock-Bogoliubov calculations for a neutron-rich hafnium isotope ($^{182}$Hf) whose proton and neutron numbers are both in the upper shell region. We study whether the shell effects play a role in producing high-$K$ isomers or highly gamma-deformed states at high spin. In particular, the possibility of shape coexistence and the effect of wobbling motion are discussed.
Several phenomena occurring in neutron stars are affected by the elementary excitations that characterize the stellar matter. In particular, low-energy excitations can play a major role in the emission and propagation of neutrinos, neutron star cooling and transport processes. In this paper, we consider the elementary modes in the star region where both proton and neutron components are superfluid. We study the overall spectral functions of protons, neutrons and electrons on the basis of the Coulomb and nuclear interactions. This study is performed in the framework of the Random Phase Approxim…
Nuclear matter and finite nuclei exhibit the property of superfluidity by forming Cooper pairs. We review the microscopic theories and methods that are being employed to understand the basic properties of superfluid nuclear systems, with emphasis on the spatially extended matter encountered in neutron stars, supernova envelopes, and nuclear collisions. Our survey of quantum many-body methods includes techniques that employ Green functions, correlated basis functions, and Monte Carlo sampling of quantum states. With respect to empirical realizations of nucleonic and hadronic superfluids, this r…
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…
The linear response of a neutron spin-triplet superfluid onto external weak axial-vector field is studied for the case of $^{3}P_{2}$ pairing with a projection of the total angular momentum $m_{j}=0$. The problem is considered in the BCS approximation discarding Fermi-liquid effects. The anomalous axial-vector vertices of neutron quasiparticles possess singularities at some frequencies which specify existence of undamped spin-density waves in the Cooper condensate. The spin waves are of a low excitation energy and are kinematically able to decay into neutrino pairs through neutral weak current…
We review the effective field theory associated with the superfluid phonons that we use for the study of transport properties in the core of superfluid neutrons stars in their low temperature regime. We then discuss the shear and bulk viscosities together with the thermal conductivity coming from the collisions of superfluid phonons in neutron stars. With regard to shear, bulk, and thermal transport coefficients, the phonon collisional processes are obtained in terms of the equation of state and the superfluid gap. We compare the shear coefficient due to the interaction among superfluid phonon…
We present a quantitative analysis of superfluidity and superconductivity in dense matter from observations of isolated neutron stars in the context of the minimal cooling model. Our new approach produces the best fit neutron triplet superfluid critical temperature, the best fit proton singlet superconducting critical temperature, and their associated statistical uncertainties. We find that the neutron triplet critical temperature is likely $2.09^{+4.37}_{-1.41} \times 10^{8}$ K and that the proton singlet critical temperature is $7.59^{+2.48}_{-5.81} \times 10^{9}$ K. However, we also show th…
We investigate ground states properties of Pb isotopes located between neutron and proton drip-lines estimated by two-neutron (two-proton) separation energies and Fermi energies within the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc). First, we report some candidates of nuclear shape coexistence in the isotope chain. They are accessed by calculating total energy as a function of the deformation parameter $β_2$, and for the coexistence candidates we take a couple of the deformation region bringing about minima of the energy within energy difference $ΔE < $ 1 MeV. Second,…
We discuss quantized vortices in neutron $^3P_2$ superfluids, which are believed to realize in high density neutron matter such as neutron stars. By using the Ginzburg-Landau free energy for $^3P_2$ superfluids, we determine the ground state in the absence and presence of the external magnetic field, and numerically construct $^3P_2$ quantized vortices in the absence and presence of the external magnetic field along the vortex axis (poloidal) or angular direction (toroidal). We find in certain situations the spontaneous magnetization of the vortex core, whose typical magnitude is about $10^{7-…
We compare two approaches to describe the inner crust of neutron stars: on the one hand, the simple coexistence of a liquid (clusters) and a gas phase, and on the other hand, the energy minimization with respect to the density profile, including Coulomb and surface effects. We find that the phase-coexistence model gives a reasonable description of the densities in the clusters and in the gas, but the precision is not high enough to obtain the correct proton fraction at low baryon densities. We also discuss the surface tension and neutron skin obtained within the energy minimization.
The neutron-star inner crust is assumed to be superfluid at relevant temperatures. The contribution of neutron quasiparticles to thermodynamic and transport properties of the crust is therefore strongly suppressed by the pairing gap. Nevertheless, the neutron gas still has low-energy excitations, namely long-wavelength collective modes. We summarize different approaches to describe the collective modes in the crystalline phases of the inner crust and present an improved model for the description of the collective modes in the pasta phases within superfluid hydrodynamics.
We investigate effects of thermal and quantum fluctuations of the superfluid order parameter in $^1S_{0}$ superfluids in neutron stars. We construct a separable potential to reproduce the $^1S_{0}$ phase shift reconstructed by using the partial wave analysis from nucleon scattering data. We include superfluid fluctuations within a strong-coupling approximation developed by Nozières and Schmitt-Rink and determine self-consistently the superfluid order parameter as well as the chemical potential. We show that the quantum depletion, which gives a fraction of noncondensed neutrons at zero temperat…
Superfluid extension of the self-consistent time-dependent band theory for neutron star matter: Anti-entrainment versus superfluid effects in the slab phase
We consider sound waves in superfluid nucleon-hyperon matter of massive neutron-star cores. We calculate and analyze the speeds of sound modes and their damping times due to the shear viscosity and non-equilibrium weak processes of particle transformations. For that, we employ the dissipative relativistic hydrodynamics of a superfluid nucleon-hyperon mixture, formulated recently [M.E. Gusakov and E.M. Kantor, Phys. Rev. D78, 083006 (2008)]. We demonstrate that the damping times of sound modes calculated using this hydrodynamics and the ordinary (nonsuperfluid) one, can differ from each other b…
The properties of the low-lying states, especially the relevant shape coexistence in $^{80}$Ge close to one of most neutron-rich doubly magic nuclei at $N=50$ and $Z=28$ have been investigated within the framework of the proton-neutron interacting model (IBM-2). Based on the fact that the relative energy of the $d$ neutron boson is different from proton bosons', the calculated energy levels of low-lying states, $E2$ transition strengths can reproduce the experimental data very well. Particularly, the first excited state 0$^+_2$ is reproduced quite nicely, which is intimately related with the s…
In this work we present results for pairing gaps in $β$--stable neutron star matter with electrons and muons using a relativistic Dirac--Brueckner--Hartree--Fock approach, starting with modern meson--exchange models for the nucleon--nucleon interaction. Results are given for superconducting $^1S_0$ protons and $^3P_2$ and $^1D_2$ neutron superfluids. A comparison is made with recent non--relativistic calculations and the implications for neutron star cooling are discussed.
A microscopic, quantum mechanical model for neutron vortices in the crust of a neutron star is presented. After a brief introduction to the Bogoliubov- de Gennes equations, which form the basis for our calculations, we present results for density distributions, vortex core sizes and vortex energies, both for an isolated neutron vortex and for the case when the vortex core overlaps with a cylindrical nucleus. Earlier results on the vortex core size are confirmed, indicating a much less dramatic variation of the vortex core size with density than predicted by the BCS formula.
The coherence length of superfluid neutron matter is calculated from the microscopic BCS wavefunction of a Cooper pair in momentum space making use of the Bonn meson-exchange potential. We find that the coherence length is proportional to the Fermi momentum-to pairing gap ratio, in good agreement with simple estimates used in the literature, and we establish the appropriate fitting constants using our numerical data. Our calculations can be applied to the problem of inhomogeneous superfluidity of hadronic matter in the crust of a neutron star.
The observed rapid cooling of the Cassiopeia A neutron star can be interpreted as being caused by neutron and proton transitions from normal to superfluid and superconducting states in the stellar core. Here we present two new Chandra ACIS-S Graded observations of this neutron star and measurements of the neutron star mass M and radius R found from consistent fitting of both the X-ray spectra and cooling behavior. This comparison is only possible for individual nuclear equations of state. We test phenomenological superfluid and superconducting gap models which mimic many of the known theoretic…
In a simple model it is demonstrated that the neutron star surface temperature evolution is sensitive to the phase state of the triplet superfluid condensate. A multicomponent triplet pairing of superfluid neutrons in the core of a neutron star with participation of several magnetic quantum numbers leads to neutrino energy losses exceeding the losses from the unicomponent pairing. A phase transition of the neutron condensate into the multicomponent state triggers more rapid cooling of superfluid core in neutron stars. This makes it possible to simulate an anomalously rapid cooling of neutron s…
We study shape coexistence and low-energy excitation spectra in neutron-deficient Pb isotopes using configuration mixing of angular-momentum and particle-number projected self-consistent mean-field states. The same Skyrme interaction SLy6 is used everywhere in connection with a density-dependent zero-range pairing force.
