Proton Superconductor Research Articles

Magnetic coupling through flux branching of adjacent type-I and -II superconductors in a neutron star

Abstract The inner and outer cores of neutron stars are believed to contain type-I and -II proton superconductors, respectively. The type-I superconductor exists in an intermediate state, comprising macroscopic flux-free and flux-containing regions, while the type-II superconductor is flux-free, except for microscopic, quantized flux tubes. Here, we show that the inner and outer cores are coupled magnetically, when the macroscopic flux tubes subdivide dendritically into quantized flux tubes, a phenomenon called flux branching. An important implication is that up to ∼1012(r1/106 cm) erg of energy are required to separate a quantized flux tube from its progenitor macroscopic flux tube, where r1 is the length of the macroscopic flux tube. Approximating the normal-superconducting boundary as sharp, we calculate the magnetic coupling energy between a quantized and macroscopic flux tube due to flux branching as a function of, f1, the radius of the type-I inner core divided by the radius of the type-II outer core. Strong coupling delays magnetic field decay in the type-II superconductor. For an idealised inner core containing only a type-I proton superconductor and poloidal flux, and in the absence of ambipolar diffusion and diamagnetic screening, the low magnetic moments (≲ 1027 G cm3) of recycled pulsars imply f1 ≲ 10−1.5.

Nambu Jona-Lasinio model of relativistic superconductivity

Abstract We propose a Nambu Jona-Lasinio (NJL) effective model of relativistic superconductivity. In this framework, we discuss possible electromagnetic (EM) behaviors of (specifically) type-II superconductivity in line with the nonrelativistic Ginzburg–Landau (GL) theory. We comment on possible solitonic solutions of this model. Our investigation could be of relevance to describe type-II proton superconductivity in neutron-star crusts.

Thermal properties of the core of magnetar

During very early age of neutron stars, the core cools down faster compared to the crust creating a large thermal gradient in the interior of the star. During 10−100 years, a cooling wave propagates from the core to the crust causing the interior of the star to thermalize. During this duration thermal properties of the core material is of great importance to understand the dynamics of the interior of the star. The heat capacity and thermal conductivity of the core depends on the behaviour of matter inside the core. We investigate these two properties in case of magnetars. Due to presence of large magnetic field, the proton superconductivity is quenched partially inside the magnetars depending upon the comparative values of upper critical field and the strength of the magnetic field present. This produces non-uniformity in the behaviour of matter throughout the star. Moreover, such non-uniformity arises from the variation of nature of the pairing and values of the pairing gap energy. We find that the heat capacity is substantially reduced due to the presence of superfluidity. On the other hand, the thermal conductivity of neutron is enhanced due to proton superconductivity and gets reduced due to neutron superfluidity. Hence, the variation of the thermal properties due to superfluidity in presence of magnetic field is different at different radius inside the star. However, in all the cases the maximum variation is of the order one. This affects the thermal relaxation time of the star and eventually its the thermal evolution.

Long-lasting accretion-powered chemical heating of millisecond pulsars

ABSTRACT We analyse the effect of magnetic field in superconducting neutron star (NS) cores on the chemical heating of millisecond pulsars (MSPs). We argue that the magnetic field destroys proton superconductivity in some volume fraction of the stellar core, thus allowing for unsuppressed non-equilibrium reactions of particle mutual transformations there. The reactions transform the chemical energy, accumulated by an NS core during the low-mass X-ray binary stage, into heat. This heating may keep an NS warm at the MSP stage (with the surface temperature $\sim 10^5\, \rm K$) for more than a billion of years after ceasing of accretion, without appealing to the rotochemical heating mechanism.

Roton instabilities in the superfluid outer core of neutron stars

We study the superfluid dynamics of the outer core of neutron stars by means of a generalized 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 of potential relevance to observational features of neutron stars. Through linear analysis, the origin and expected growth of the instabilities is explored for varying nuclear-matter density. Differently from previous findings, the dispersion of linear excitations in our model shows rotonic structures below the pair-breaking energy threshold, which lies at the origin of the dynamical instabilities and could eventually lead to emergent vorticity along with modulations of the superfluid density.

Proton superconductivity in pasta phases in neutron star crusts

In the so-called pasta phases predicted to occur in neutron star crusts, protons are able to move easily over large distances because the nuclear matter regions are extended in space. Consequently, electrical currents can be carried by protons, an effect not possible in conventional crystalline matter with isolated nuclei. With emphasis on the so-called lasagna phase, which has sheet-like nuclei, we describe the magnetic properties of the pasta phases allowing for proton superconductivity. We predict that these phases will be Type II superconductors and we calculate the energy per unit length of a flux line, which is shown to be strongly anisotropic. If, as seems likely, the pasta structure is imperfect, flux lines will be pinned and matter will behave as a good electrical conductor and flux decay times will be long. We describe some possible astrophysical manifestations of our results.

Proton pairing in neutron stars from chiral effective field theory

We study the $^{1}S_{0}$ proton pairing gap in $\ensuremath{\beta}$-equilibrated neutron star matter within the framework of chiral effective field theory. We focus on the role of three-body forces, which strongly modify the effective proton-proton spin-singlet interaction in dense matter. We find that three-body forces generically reduce both the size of the pairing gap and the maximum density at which proton pairing may occur. The pairing gap is computed within Bardeen-Cooper-Schrieffer theory using a single-particle dispersion relation calculated up to second order in perturbation theory. Model uncertainties are estimated by varying the nuclear potential (its order in the chiral expansion and high-momentum cutoff) and the choice of single-particle spectrum in the gap equation. We find that a second-order perturbative treatment of the single-particle spectrum suppresses the proton $^{1}S_{0}$ pairing gap relative to the use of a free spectrum. We estimate the critical temperature for the onset of proton superconductivity to be ${T}_{c}=(3.2\text{--}5.1)\ifmmode\times\else\texttimes\fi{}{10}^{9}$ K, which is consistent with previous theoretical results in the literature and marginally within the range deduced from a recent Bayesian analysis of neutron star cooling observations.

Neutron Star Cooling Within the Equation of State With Induced Surface Tension

We study the thermal evolution of neutron stars described within the equation of state with induced surface tension (IST) that reproduces properties of normal nuclear matter, fulfills the proton flow constraint, provides a high-quality description of hadron multiplicities created during the nuclear-nuclear collision experiments, and it is equally compatible with the constraints from astrophysical observations and the GW170817 event. The model features strong direct Urca processes for the stars above 1.91M⊙. The IST equation of state shows very good agreement with the available cooling data, even without introducing nuclear pairing. We also analysed the effect of the singlet proton/neutron and triplet neutron pairing on the cooling of neutron stars of different mass. We show that the description of the compact object in the center of the Cassiopeia A does not necessarily require an inclusion of neutron superfluidity and/or proton superconductivity. Our results indicate that data of Cassiopeia A can be adequately well reproduced by a 1.66M⊙ star with an atmosphere of light elements. Moreover, the IST EoS reproduces each of the observational datasets for the surface temperature of Cassiopeia A either by a rapidly cooling ∼1.955M⊙ star with paired and unpaired matter or by a 1.91M⊙ star with the inclusion of neutron and proton pairings in the singlet channel.

Surface energy of magnetized superconducting matter in neutron star cores

In this paper, an effective field theory for proton superconductor (SC) interacting with neutron superfluid (SF), both with scalar order parameters, is developed and applied to the surface energy (SE) of a magnetized SC body in neutron stars (NS). Essentially, the SE studied here differs from the nuclear SE: here, the proton SF density decays to zero while the total proton density is constant across the surface. Interactions between the condensates are parameterized phenomenologically and their effects determined from calculations of a planar SE as the ranges of parameters are varied. The critical Ginzburg-Landau (GL) parameter $\kappa_c$ which renders the SE equal to zero is found analytically by noting that in a system with vanishing SE the thermodynamic critical MF is equivalent to the upper critical MF. In the case of weak coupling, $\kappa_c$ is shown to be a linear function of SF-SF density-density coupling, in agreement with the earlier results based on asymptotic intervortex interactions. Numerical simulations corroborate our analytical predictions. Coupling due to the mixed term arising from a scalar product of gradients of the SF densities, which had been considered in the earlier literature, is seen to have practically no effect on the superconductivity type. However, this coupling does produce a frozen wave packet of the SF neutron density localized at the surface. It is shown that the leading contribution from the gradient coupling arises from a novel mixed quantum pressure term, but still does not affect the planar SE. The present calculations provide an initial map of superconductivity types in the phenomenological effective field theory and will serve as a landmark for future studies, which require microscopic calculations of the coupling parameters introduced here phenomenologically.

Two-fluid simulations of the magnetic field evolution in neutron star cores in the weak-coupling regime

ABSTRACT In a previous paper, we reported simulations of the evolution of the magnetic field in neutron star (NS) cores through ambipolar diffusion, taking the neutrons as a motionless uniform background. However, in real NSs, neutrons are free to move, and a strong composition gradient leads to stable stratification (stability against convective motions) both of which might impact on the time-scales of evolution. Here, we address these issues by providing the first long-term two-fluid simulations of the evolution of an axially symmetric magnetic field in a neutron star core composed of neutrons, protons, and electrons with density and composition gradients. Again, we find that the magnetic field evolves towards barotropic ‘Grad–Shafranov equillibria’, in which the magnetic force is balanced by the degeneracy pressure gradient and gravitational force of the charged particles. However, the evolution is found to be faster than in the case of motionless neutrons, as the movement of charged particles (which are coupled to the magnetic field, but are also limited by the collisional drag forces exerted by neutrons) is less constrained, since neutrons are now allowed to move. The possible impact of non-axisymmetric instabilities on these equilibria, as well as beta decays, proton superconductivity, and neutron superfluidity, are left for future work.

Simultaneous fitting of neutron star structure and cooling data

Using a model for the equation of state and composition of dense matter and the magnitude of singlet proton superconductivity and triplet neutron superfluidity, we perform the first simultaneous fit of neutron star masses and radii determined from observations of quiescent low-mass x-ray binaries and luminosities and ages determined from observations of isolated neutron stars. We find that the Vela pulsar strongly determines the values inferred for the superfluid/superconducting gaps and the neutron star radius. We find, regardless of whether or not the Vela pulsar is included in the analysis, that the threshold density for the direct Urca process lies between the central density of 1.7 and 2 solar mass neutron stars. We also find that two solar mass stars are unlikely to cool principally by the direct Urca process because of the suppression by neutron triplet superfluidity.

Magnetic Field Generation in Hybrid Stars

The mechanism for magnetic field generation in hybrid neutron stars (containing “npe,” hadron, “2SC” and “CFL” quark phases) is discussed. It is assumed that the rotational vortices in “npe” and “CFL” phases with a quantum of circulation h/2m also continue in the “2SC” phase. Since the superconducting components in the “npe” and “2SC” phases are charged, entrainment currents develop around the vortices and generate a magnetic field. The average magnetic field in the quark phase is on the order of 5·1015 G and exceeds the field in the “npe” phase by 2-3 orders of magnitude. The magnetic field penetrates into the “CFL” phase by means of magnetic vortices with a flux 2Φ0 and it can partially destroy the proton superconductivity in the “npe” phase. On the star’s surface, the magnetic field reaches 5·1014 G, a level comparable to the magnetic field of magnetars. Magnetars may, therefore, contain quark matter.

Stability of interlinked neutron vortex and proton flux-tube arrays in a neutron star – II. Far-from-equilibrium dynamics

The equilibrium configurations of neutron superfluid vortices interacting with proton superconductor flux tubes in a rotating, harmonic trap are non-trivial in general, when the magnetorotational symmetry is broken. A non-zero angle $\theta$ between the magnetic and rotation axes leads to tangled vorticity due to competition between vortex-vortex repulsion and vortex-flux-tube pinning. Here we investigate the far-from-equilibrium behaviour of the vortices, as the trap decelerates, by solving the time-dependent, stochastic, Gross-Pitaevskii equation numerically in three dimensions. The numerical simulations reveal new vortex behaviours. Key geometrical attributes of the evolving vortex tangle are characterised, as is the degree to which pinning impedes the deceleration of the neutron condensate as a function of $\eta$, the pinning strength, and $\theta$. The simulated system is a partial analogue of the outer core of a decelerating neutron star, albeit in a very different parameter regime.

Stability of interlinked neutron vortex and proton flux tube arrays in a neutron star: equilibrium configurations

Three-dimensional, Gross-Pitaevskii equation (GPE) simulations are presented of the interaction between neutron superfluid vortices and proton superconductor flux tubes in a rotating, harmonic trap, representing an idealised model of the outer core of a neutron star. Low-energy states of the neutron condensate are calculated by evolving the GPE in imaginary time in the presence of a prescribed, static, rectilinear flux tube array. The calculations are carried out as a function of the angle between the global magnetic and rotation axes, and the amplitude and sign of the current-current and density couplings between the neutron and proton condensates. It is found that the system is frustrated by the competition between vortex-vortex repulsion and vortex-flux-tube attraction (pinning), leading to the formation of vortex tangles and "glassy" behaviour characterized by multiple metastable states spaced closely in energy. The dimensionless parameters in the simulations are ordered as one expects in a neutron star, but the dynamic range is many orders of magnitude smaller than in reality, so caution must be exercised when assessing the astrophysical implications. Nevertheless the results suggest that tangled vorticity may be endemic in neutron star outer cores.

Flux-Vortex Pinning and Neutron Star Evolution

G. Srinivasan et al. (1990) proposed a simple and elegant explanation for the reduction of the neutron star magnetic dipole moment during binary evolution leading to low mass X-ray binaries and eventually to millisecond pulsars: Quantized vortex lines in the neutron star core superfluid will pin against the quantized flux lines of the proton superconductor. As the neutron star spins down in the wind accretion phase of binary evolution, outward motion of vortex lines will reduce the dipole magnetic moment in proportion to the rotation rate. The presence of a toroidal array of flux lines makes this mechanism inevitable and independent of the angle between the rotation and magnetic axes. The incompressibility of the flux-line array (Abrikosov lattice) determines the epoch when the mechanism will be effective throughout the neutron star. Flux vortex pinning will not be effective during the initial young radio pulsar phase. It will, however, be effective and reduce the dipole moment in proportion with the rotation rate during the epoch of spindown by wind accretion as proposed by Srinivasan et al. The mechanism operates also in the presence of vortex creep.

Magnetic field evolution and equilibrium configurations in neutron star cores: the effect of ambipolar diffusion

As another step towards understanding the long-term evolution of the magnetic field in neutron stars, we provide the first simulations of ambipolar diffusion in a spherical star. Restricting ourselves to axial symmetry, we consider a charged-particle fluid of protons and electrons carrying the magnetic flux through a motionless, uniform background of neutrons that exerts a collisional drag force on the former. We also ignore the possible impact of beta decays, proton superconductivity, and neutron superfluidity. All initial magnetic field configurations considered are found to evolve on the analytically expected time-scales towards "barotropic equilibria" satisfying the "Grad-Shafranov equation", in which the magnetic force is balanced by the degeneracy pressure gradient, so ambipolar diffusion is choked. These equilibria are so-called "twisted torus" configurations, which include poloidal and toroidal components, the latter restricted to the toroidal volumes in which the poloidal field lines close inside the star. In axial symmetry, they appear to be stable, although they are likely to undergo non-axially symmetric instabilities.

Mixing of charged and neutral Bose condensates at nonzero temperature and magnetic field

It is expected that in the interior of compact stars a proton superconductor coexists with and couples to a neutron superfluid. Starting from a field-theoretical model for two complex scalar fields - one of which is electrically charged - we derive a Ginzburg-Landau potential which includes entrainment between the two fluids and temperature effects from thermal excitations of the two scalar fields and the gauge field. The Ginzburg-Landau description is then used for an analysis of the phase structure in the presence of an external magnetic field. In particular, we study the effect of the superfluid on the flux tube phase by computing the various critical magnetic fields and deriving an approximation for the flux tube interaction. As a result, we point out differences to the naive expectations from an isolated superconductor, for instance the existence of a first-order flux tube onset, resulting in a more complicated phase structure in the region between type-I and type-II superconductivity.

Relativistic dynamics of superfluid-superconducting mixtures in the presence of topological defects and an electromagnetic field with application to neutron stars

The relativistic dynamic equations are derived for a superfluid-superconducting mixture coupled to the electromagnetic field. For definiteness, and bearing in mind possible applications of our results to neutron stars, it is assumed that the mixture is composed of superfluid neutrons, superconducting protons, and normal electrons. Proton superconductivity of both I and II types is analysed, and possible presence of neutron and proton vortices (or magnetic domains in the case of type-I proton superconductivity) is allowed for. The derived equations neglect all dissipative effects except for the mutual friction dissipation and are valid for arbitrary temperatures (i.e. they do not imply that all nucleons are paired), which is especially important for magnetar conditions. It is demonstrated that these general equations can be substantially simplified for typical neutron stars, for which a kind of magnetohydrodynamic approximation is justified. Our results are compared to the nonrelativistic formulations existing in the literature and a number of discrepancies are found. In particular, it is shown that, generally, the electric displacement ${\pmb D}$ does not coincide with the electric field ${\pmb E}$, contrary to what is stated in the previous works. The relativistic framework developed here is easily extendable to account for more sophisticated microphysics models and it provides the necessary basis for realistic modelling of neutron stars.

Rapid rotational crust-core relaxation in magnetars

If a magnetar interior $B$-field exceeds $10^{15}$~G, it will unpair the proton superconductor in the stellar core by inducing diamagnetic currents that destroy the Cooper pair coherence. Then, the $P$-wave neutron superfluid in these non-superconducting regions will couple to the stellar plasma by scattering of protons off the quasiparticles that are confined in the cores of neutron vortices by the strong (nuclear) force. The dynamical timescales associated with this interaction span from several minutes at the crust-core interface to a few seconds in the deep core. We show that (a) the rapid crust-core coupling is incompatible with oscillation models of magnetars that completely decouple the core superfluid from the crust and (b) magnetar precession is damped by the coupling of normal fluids to the superfluid core and, if observed, needs to be forced or continuously excited by seismic activity.

Upper critical field and (non)-superconductivity of magnetars

We construct equilibrium models of compact stars using a realistic equation of state and obtain the density range occupied by the proton superconductor in strong B-fields. We do so by combining the density profiles of our models with microscopic calculations of proton pairing gaps and the critical unpairing field H c2 above which the proton type-II superconductivity is destroyed. We find that magnetars with interior homogeneous field within the range 0.1 ≤ B 16 ≤ 2, where B 16 = B/1016 G, are partially superconducting, whereas those with B 16 > 2 are void of superconductivity. We briefly discuss the neutrino emissivity and superfluid dynamics of magnetars in the light of their (non)-superconductivity.