Phase transitions in the inner crust of neutron stars within the superfluid band theory: Competition between $^1\text{S}_0$ pairing and spin polarization under finite temperature and magnetic field

TL;DR

The paper develops a fully microscopic framework by extending the self-consistent superfluid band theory (based on Kohn–Sham DFT and SLDA) to finite temperature and magnetic fields, enabling a band-structure treatment of neutrons in the neutron-star inner crust under realistic conditions. Using a Skyrme-type EDF and a Bloch-boundary formulation, the authors compute pairing, density, and spin distributions for slab and uniform phases, incorporating finite-$T$ grand-canonical statistics and Landau-quantized electrons under strong $B$. They find two distinct thermal transitions—a neutron pairing transition at $T_c^{sf}$ and a crust-melting transition at $T_c^{melt}$—with $T_c^{sf}$ increasing and $T_c^{melt}$ decreasing with density, and a rich magnetic-field regime where protons polarize at modest fields while neutrons remain unpolarized overall but locally polarized, reflecting an intricate interplay between pairing and spin-dependent interactions. The results provide microscopic predictions for crust behavior in proto-neutron stars and magnetars, with implications for thermal conductivity, neutrino emission, and crustal dynamics, and lay the groundwork for future multi-dimensional extensions and exploration of exotic spin-ordered phases.

Abstract

Phase transitions of matter under changes of external environment such as temperature and magnetic field have attracted great interests to various quantum many-body systems. Several phase transitions must have occurred in neutron stars as well such as transitions from normal to superfluid/superconducting phases and crust formation. In this work, we extend the superfluid band theory, which has been formulated in our previous work [K. Yoshimura and K. Sekizawa, Phys. Rev. C 109, 065804 (2024)] based on the Kohn-Sham density functional theory (DFT) for superfluid systems, into the finite temperature and finite magnetic field systems. As a result of the finite temperature calculations, we find that the superfluidity of neutrons dissapears at around $k_\text{B}T=0.6$--$0.9\,$ MeV, and ``melting'' of nuclear slabs, that is, a structural change into the uniform matter, takes place at around $k_\text{B}T=2.5$--$4.5\,$ MeV. We also reveal that these transition temperatures exhibit a systematical dependence on the baryon densities. By turning on the magnetic field, we find that protons' spin gets polarized at around $B=10^{16}\,$G, whereas neutrons' spin is kept unpolarized on average up to around $B=10^{17}\,$G. Intriguingly, our microscopic calculations reveal that neutrons' spin is actually polarized locally inside and outside of the slab already at $B\sim10^{16}\,$G, while keeping the system unpolarized in total. As a conclusion, we have demonstrated validity and usefulness of the fully self-consistent superfluid nuclear band theory for describing neutron star matter under arbitrary temperature and magnetic field. Critical temperatures and magnetic fields have been predicted for 1) superfluid to normal transition, 2) crust formation, and 3) spin polarization, under conditions relevant to realistic neutron star environments.

Figures (7)

• Figure 1: (a) Density distributions and (b) pairing fields of neutrons and protons are shown as a function of $z$ coordinate at four representative temperatures, $k_\text{B}T = 0$, $1$, $3$, $4$, and $5\,{\mathrm{MeV}}$. In both panels, the upper green lines indicate the distribution of neutrons' quantities, while the red lines are that for protons' ones. In ascending order of temperatures, solid, dashed, dotted, dash-dotted, and long-dashed lines are used.
• Figure 2: (a) Total energy per nucleon, $E_{\mathrm{tot}}$, is shown as a function of temperature for a fixed baryon density, $n_\text{B}=0.04\,{\mathrm{fm}}^{-3}$. Two arrows indicates the position of kinks implying phase transitions. (b) Specific heat $C_V(T)$ is shown as a function of temperature, for different baryon number densities, $n_\text{B}=0.04$, 0.05, 0.06, and $0.07\,{\mathrm{fm}}^{-3}$. In ascending order of densities, solid, dashed, dotted, and dash-dotted lines are used.
• Figure 3: (a) The average pairing gap in the zero temperature $\Delta_0$, and (b) its ratio against the critical temperature $T_c$ plotted as a function of the baryon densities $n_B$. In both panels, results in the case of slab phases as well as uniform matter are demonstrated.
• Figure 4: (a) The number density $n_q$ fm$^{-3}$, (b) pairing field $\Delta_q$ MeV, and (c) the $z$ component of the spin density $s_{z,q}$ fm$^{-3}$ are shown as a function of the spatial coordinate $z$ in three cases with various magnetic field strengths $B_\star = 0$, $1000$, $5000$.
• Figure 5: (a) Average magnitude of the neutron pairing field, $\overline{\Delta}_n$, and (b) total spin polarization, $P_n$, of neutrons are shown as a function of the magnetic field strength, $B_\star$, at four representative temperatures, $k_\text{B}T=0$, $10$, $100$, and $1000\,\mathrm{keV}$. In ascending order of temperatures, circle, upward triangle, downward triangle, and rectangular symbols connected with solid, dashed, dotted, and dash-dotted lines are used, respectively.