We consider the hydrodynamics of the outer core of a neutron star under conditions when both neutrons and protons are superfluid. Starting from the equation of motion for the phases of the wave functions of the condensates of neutron pairs and proton pairs we derive the generalization of the Euler equation for a onecomponent fluid. These equations are supplemented by the conditions for conservation of neutron number and proton number. Of particular interest is the effect of entrainment, the fact that the current of one nucleon species depends on the momenta per nucleon of both condensates. We find that the nonlinear terms in the Euler-like equation contain contributions that have not always been taken into account in previous applications of superfluid hydrodynamics. We apply the formalism to determine the frequency of oscillations about a state with stationary condensates and states with a spatially uniform counterflow of neutrons and protons. The velocities of the coupled sound-like modes of neutrons and protons are calculated from properties of uniform neutron star matter evaluated on the basis of chiral effective field theory. We also derive the condition for the two-stream instability to occur.
--In this article we study the hydrodynamic spectrum of a superfluid confined in a cylindrical trap. We show that the dispersion relation $ω$(q) of the phonon branch scales like $\sqrt$ q at large q, leading to a vanishingly small superfluid critical velocity. In practice the critical velocity is set by the breakdown of the hydrodynamic approximation. For a broad class of superfluids, this entails a reduction of the critical velocity by a factor ($ω$ $\perp$ /ï1/2c) 1/3 with respect to the free-space prediction (here $ω$ $\perp$ is the trapping frequency and ï1/2c the chemical potential of the…
I study vortex ring oscillations in a superfluid, trapped in an elongated trap, under the conditions of the Local Density Approximation. On the basis of the Hamiltonian formalism I develop a hydrodynamic theory, which is valid for an arbitrary superfluid and depends only on the equation of state. The problem is reduced to an ordinary differential equation for the ring radius. The cases of the dilute BEC and the Fermi gas at unitarity are investigated in detail. Simple analytical equations for the periods of small oscillations are obtained and the equations of non-linear dynamics are solved in …
From flow without dissipation of energy to the formation of vortices when placed within a rotating container, the superfluid state of matter has proven to be a very interesting physical phenomenon. Here we present the key mechanisms behind superfluidity in fermionic systems and apply our understanding to an exotic system found deep within the universe -- the superfluid found deep within a neutron star. A defining trait of a superfluid is the pairing gap, which the cooling curves of neutron stars depend on. The extreme conditions surrounding a neutron star prevent us from directly probing the s…
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…
We present a study of the hydrodynamics of compressible superfluids in confined geometries. We use a perturbative procedure in terms of the dimensionless expansion parameter $(v/v_s)^2$ where $v$ is the typical speed of the flow and $v_s$ the speed of sound. A zero value of this parameter corresponds to the incompressible limit. We apply the procedure to two specific problems: the case of a trapped superfluid with a gaussian profile of the local density, and that of a superfluid confined in a rotating obstructed cylinder. We find that the corrections due to finite compressibility which are, as…
We clarify a relation between the Gross-Pitaevskii energy functional for spin-2 spinor Bose-Einstein condensates and the Ginzburg-Landau theory for neutron $^3P_2$ superfluidis (spin-triplet $P$-wave pairing with total angular momentum two). We then classify all uniform states with nontrivial unbroken symmetries, with the help of geometric invariant theory.
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…
Superfluid neutron matter is a key ingredient in the composition of neutron stars. The physics of the inner crust is largely dependent on that of its $S$-wave neutron superfluid which has made its presence known through pulsar glitches and modifications on the neutron star cooling. Moreover, with recent gravitational-wave observations of neutron star mergers, the need for an equation of state for the matter of these compact stars is further accentuated and a model-independent treatment of neutron superfluidity is important. \textit{Ab initio} techniques developed for finite systems can be guid…
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 report results of fully non-perturbative calculations, based on Auxiliary Field Quantum Monte Carlo (AFQMC) approach, for the dilute neutron matter at the density $ρ=0.003\fm^{-3}$. Fundamental quantities which characterize the superfluid state: the single particle energy gap $Δ(T)$, and the critical temperature $\Tc$ have been determined. The large value of $Δ(0)/\Tc\approx 3.2$ indicates that the system is not a BCS-type superfluid at low temperatures.
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…
We show that spinor Bose gases subject to a quadratic Zeeman effect exhibit coexisting superfluidity and spin superfluidity, and study the interplay between these two distinct types of superfluidity. To illustrate that the basic principles governing these two types of superfluidity are the same, we describe the magnetization and particle-density dynamics in a single hydrodynamic framework. In this description spin and mass supercurrents are driven by their respective chemical potential gradients. As an application, we propose an experimentally accessible stationary state, where the two types o…
We determine the hydrodynamic modes of the superfluid analog of a smectic-A phase in liquid crystals, i.e., a state in which both gauge invariance and translational invariance along a single direction are spontaneously broken. Such a superfluid smectic provides an idealized description of the incommensurate supersolid state realized in Bose-Einstein condensates with strong dipolar interactions as well as of the stripe phase in Bose gases with spin-orbit coupling. We show that the presence of a finite normal fluid density in the ground state of these systems gives rise to a well-defined second-…
Extending the quantum effective approach of Son and Nicolis and incorporating dissipation, we develop a MIS formalism for describing a superfluid out of equilibrium by including the Goldstone boson and the condensate together with the hydrodynamic modes as the effective degrees of freedom. We find that the evolution of the superfluid undergoing Bjorken flow is governed by the conventional hydrodynamic attractor with unbroken symmetry and an even number of novel non-dissipative fixed points with broken symmetry. If the initial temperature is super-critical, then the condensate becomes exponenti…
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…
Superfluid extension of the self-consistent time-dependent band theory for neutron star matter: Anti-entrainment versus superfluid effects in the slab phase
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…
