Topology of the Superconducting Heart of Neutron Stars: Effects of Microphysics and Gravitational-Wave Signatures

TL;DR

This work develops a general-relativistic framework to map proton superconductivity in magnetized neutron-star cores by solving the Einstein–Maxwell equations with the XNS code for toroidal, poloidal, and twisted-torus field geometries, incorporating microscopically derived proton pairing gaps from Brueckner–Hartree–Fock theory across two representative EoS. It demonstrates that proton $S$-wave superconductivity is confined to intermediate densities and exhibits rich three-dimensional topologies (e.g., doughnut-shaped toroidal regions and prolate outer cores), with the superconducting domain size sensitive to EoS stiffness and three-body forces that suppress the gap. The study then assesses continuous gravitational-wave signatures from millisecond pulsars, showing that type-II superconductivity can enhance magnetic deformations and GW strains, potentially detectable by next-generation detectors, thus offering a rare observational window into internal magnetic fields and dense-matter microphysics. Taken together, the results underscore the critical role of microphysical inputs and EoS in shaping superconducting topology and GW observables, and point to multi-messenger observations as a path to constrain neutron-star interior physics.

Abstract

We present a general-relativistic study of the distribution of proton superconductivity in strongly magnetized neutron stars (NSs), using the XNS code to solve the coupled Einstein-Maxwell equations. We investigate equilibrium configurations with both toroidal and poloidal magnetic field geometries and incorporate complex many-body effects through microscopically derived proton pairing gaps. The models employ equations of state (EoS) obtained from microscopic many-body theory - including realistic two- and three-body nuclear interactions - as well as from relativistic mean-field approaches. We compare superconducting topologies across our collection of EoS and explore the influences of magnetic field geometry in stellar models parameterized by central density. Our models confirm the absence of $S$-wave superconductivity in the inner core and, importantly, reveal that non-superconducting regions exhibit complex three-dimensional geometries: doughnut-shaped for toroidal fields and prolate-shaped for poloidal fields -- spatial structures that are inherently absent in one-dimensional analyses. We also compute magnetic deformations and ellipticities for several millisecond pulsars (MSPs), estimating their continuous gravitational wave strain. While these MSPs remain undetectable by current detectors, next-generation instruments such as the Einstein Telescope and Cosmic Explorer may detect their signals, opening an observational window into internal superconductivity and internal magnetic field of NSs, as well as the fundamental microphysics of dense matter.

Figures (14)

• Figure 1: Left panels: The magnitude of the stellar magnetic field is shown as a function of the radial coordinate for fixed values of the polar angle $\theta$. Each model is specified by the stellar mass and the maximum field strength, as indicated in the corresponding panel. The upper and middle panels present the purely toroidal and purely poloidal configurations, respectively, while the lower panel corresponds to the twisted-torus configuration and displays separately the toroidal and poloidal components, and the total magnetic field, as labeled. Right panels: Color maps illustrating the spatial distribution of the stellar magnetic field in the $x$–$y$ plane of Cartesian coordinates, with the magnetic axis aligned along the $y$-direction. The top, middle, and bottom panels correspond to the toroidal, poloidal, and twisted-torus configurations, respectively.
• Figure 2: (a) The fractions of constituents $Y_{q}=n_q/n$, $q\in n,\, p,\, e,\, \mu$, as functions of normalized density $n/n_s$Sharma2015. (b) Dependence of proton pairing gap $\Delta$ on normalized density at different temperatures, computed at zero temperature using Eqs. \ref{['eq:D_2B']} or \ref{['eq:D_3B']} and extended to finite temperature using Eq. \ref{['eq:gapt']}.
• Figure 3: $\lambda$, $\xi$ and $\kappa$ for a $1.4M_\odot$ NS as functions of equatorial radial coordinate, showing the possible superconducting region of type-I ($\kappa<1/\sqrt{2}$) and type-II ($\kappa>1/\sqrt{2}$).
• Figure 4: Panel (a) shows the critical temperature $T_c$; panels (b) and (c) show the critical fields $H_{c1}$, $H_{c2}$, and $H_{cm}$ for 2B and 2B+3B force models with the gaps defined via Eqs. \ref{['eq:D_2B']} and \ref{['eq:D_3B']} respectively, along with the $H$-field derived from the Einstein-Maxwell solutions, as a function of equatorial-plane radial coordinate (plane perpendicular to the external field). The NS model has $M = 1.4\,M_\odot$, $R=11$ km, $T = 10^8$ K, and maximal toroidal magnetic field $B_{S\text{max}} = 10^{16}$ G and was constructed using the BBB EoS. Two vertical lines indicate the transition points: one at $\kappa = 1/\sqrt{2}$ marking the boundary between type-I and type-II superconductivity, and another where the superconducting gap vanishes ($\Delta \to 0$), indicating a transition to the normal (unpaired) state. Shaded regions in this and the following similar figures highlight superconducting phases. For type-II superconductors, these include the Meissner state ($H < H_{c1}$) and the flux-tube array state ($H_{c1} < H < H_{c2}$); for type-I superconductors, they correspond to the Meissner or layered-domain state with $H < H_{cm}$ (see text). In the present figure, only flux-tube phases occur for the given magnetic-field ($H$) profile.
• Figure 5: Color maps of the distribution of the toroidal $H$-field in the $x$–$y$ plane, where the magnetic axis is aligned along the $y$-direction in Cartesian coordinates. Superconducting regions for NS models using the BBB EoS with $M = 1.4\,M_\odot$, $T = 10^8$ K, $R=11$ km, and toroidal $B_{S\text{max}} = 10^{16}$ G are shown for 2B and 2B+3B nucleonic force models in panels (a) and (b), respectively. The superconducting phases are identified as follows: red crosses mark the type-II region where $\kappa > 1/\sqrt{2}$ and $H_{c1} < H < H_{c2}$, indicating the presence of a flux tube lattice; black dots represent the Meissner state within a type-II region, characterized by $\kappa > 1/\sqrt{2}$ and $H < H_{c1}$; and magenta vertical hatching denotes the Meissner or layered-domain state in the type-I region where $\kappa < 1/\sqrt{2}$ and $H < H_{cm}$. Note that, if Meissner expulsion of the flux is fast, the magnetic induction vanishes in type-II regions when $H\le H_{c1}$. The same applies to type-I regions when $H\le H_{cm}$. However, the slow field-expulsion timescales Baym1969 imply that flux tubes in type-II regions and normal domains in type-I regions may still exist. Unshaded regions indicate nonsuperconducting zones where $H > H_{cm}$ in the type-I case or $H > H_{c2}$ in the type-II case.