Coexistence of neutron-neutron, proton-proton, and neutron-proton pairings in asymmetric nuclear matter with medium-corrected interactions

In homogeneous asymmetric nuclear matter, the coupled BCS gap equations for n-n and p-p pairs (isovector $I=1$, $^1S_0$) and n-p pairs (isoscalar $I=0$, $^3S_1$-$^3D_1$) admit coexistence solutions only when the n-p pairing interaction is stronger than the like-nucleon ones, and the n-n/p-p condensates either enhance n-p pairing (Fermi-surface dispersion) or reduce it (pair competition), depending on the relative channel strengths (Yan, Shang, Dong, and Zuo, 2021). Those conclusions rest on a contact pairing interaction calibrated to Hartree-Fock-level gaps, on condensates with zero center-of-mass momentum, on $T=0$, and on infinite matter. The problem asks for the quantitative, robust version of the coexistence prediction: effective pairing interactions in both channels that include medium-polarization (induced-interaction) and self-energy corrections; whether the coexistence conditions and the mutual enhancement/suppression survive those corrections; the stability of the coexistence solutions against inhomogeneous finite-momentum (FFLO-type) pairing; the finite-temperature behavior of the coupled system; and, as a further prospect noted in the source, the connection to finite nuclei near $N\approx Z$.

Background

Nuclear matter — an idealized infinite system of neutrons and protons interacting only through the strong force — is the reference medium for both neutron-star interiors and the bulk part of nuclei. Its composition is measured by the isospin asymmetry $\beta=(\rho_n-\rho_p)/\rho$, where $\rho_n$ and $\rho_p$ are the neutron and proton densities; $\beta=0$ is symmetric matter and $\beta=1$ pure neutron matter. Near the Fermi surface (the boundary between occupied and empty single-particle states at zero temperature), the attractive part of the nucleon-nucleon interaction drives superfluid pairing, described at mean-field level by BCS theory (the Bardeen-Cooper-Schrieffer theory of superconductivity). Two low-energy channels compete: the isovector channel with total isospin $I=1$, spin $S=0$, and orbital angular momentum $L=0$ (the $^1S_0$ channel), which pairs like nucleons into neutron-neutron (n-n) and proton-proton (p-p) Cooper pairs, and the isoscalar channel with $I=0$, $S=1$ (the $^3S_1$-$^3D_1$ channel, that of the deuteron), which pairs neutrons with protons (n-p). In asymmetric matter the neutron and proton Fermi momenta differ, and the resulting mismatch $\delta\mu=(\mu_n-\mu_p)/2$ of the chemical potentials suppresses n-p pairing — the nuclear analog of density-imbalance physics in cold atomic Fermi gases, where it produces breached-pair states (gapless superfluids that coexist with unpaired excess majority fermions) and LOFF/FFLO phases: inhomogeneous superfluids, named after Larkin, Ovchinnikov, Fulde, and Ferrell, in which Cooper pairs carry nonzero center-of-mass momentum (He, Jin, and Zhuang, 2006).

Microscopic work over several decades also shows that pairing in a nuclear medium is strongly renormalized beyond the bare two-body force: self-energy corrections (the energy dependence of the in-medium single-particle spectrum) and polarization or vertex corrections (the induced interaction generated by exchange of density, spin, and spin-isospin collective modes, computed for example in the random-phase approximation, a standard linear-response summation of particle-hole ring diagrams) can quench pairing gaps by factors of order two (Wambach, Ainsworth, and Pines, 1993). For the isoscalar channel, Guo, Lombardo, and Schuck (2019) found that self-energy effects confine the $^3SD_1$ n-p superfluid to very low density and strongly quench its gap, while the induced interaction is attractive and partly compensating; Fan et al. (2019) computed self-energy corrections to the n-p gap up to third order in an extended Brueckner-Hartree-Fock (a microscopic scheme built on in-medium two-body reaction matrices) plus BCS framework. Finite-temperature and FFLO phase structures are likewise known for the n-p channel in isolation, from dilute matter (Stein, Huang, Sedrakian, and Clark, 2012) to the recent phase diagrams of Duan, Zhang, and Shang (2025) and Duan and Shang (2026). Medium corrections to pairing have been computed only channel-by-channel, and FFLO and finite-temperature phase diagrams exist only for the n-p channel in isolation.

Within this landscape, Yan, Shang, Dong, and Zuo (2021) solved the three BCS gap equations for the gaps $\Delta_{nn}$, $\Delta_{pp}$, and $\Delta_{np}$ simultaneously in homogeneous asymmetric nuclear matter at $T=0$, using density-dependent contact pairing interactions with parameters calibrated to Hartree-Fock-level microscopic gaps. They found that the three pairings never coexist in symmetric matter; that in asymmetric matter they coexist only when the n-p pairing interaction is stronger than the n-n and p-p ones; and that the n-n and p-p condensates influence n-p pairing through two competing mechanisms — dispersion of the neutron and proton Fermi surfaces, which enlarges the n-p phase-space overlap and enhances n-p pairing, versus pair competition, in which forming two n-p pairs requires breaking an n-n and a p-p pair and reduces the n-p pairing probability. The authors themselves state the limitations that define the open problem: the interaction omits medium polarization and correlation (self-energy) effects; only homogeneous condensates with zero center-of-mass momentum were considered, so the solutions are thermodynamically but not dynamically stable; the calculations are at zero temperature; and the results concern infinite matter only, with the extension to finite nuclei near $N\approx Z$ left as a prospect where spin-orbit splitting and shell structure enter.

Problem Statement

The problem is to replace the model-level coexistence results of Yan et al. (2021) by quantitative, robust predictions for the coupled multi-channel pairing system in asymmetric nuclear matter. Its components are linked by this single objective.

(1) Effective interactions. Construct effective pairing interactions for the $I=1$ (n-n, p-p) and $I=0$ (n-p) channels in asymmetric nuclear matter that include the medium-polarization (induced-interaction or vertex) corrections and the self-energy (correlation) corrections known to quench pairing, treated consistently across the coupled channels, and derive or calibrate them so that quantitative pairing gaps can be predicted rather than fitted to Hartree-Fock-level gaps.

(2) Coexistence with medium corrections. With these interactions, solve the coupled n-n, p-p, and n-p gap equations self-consistently — with the number constraints and the free-energy comparison that selects the stable phase — over the density-asymmetry plane, and determine whether the coexistence conditions found in 2021 (coexistence only when the n-p pairing strength exceeds the like-nucleon strength) and the two competing effects of the n-n and p-p condensates on n-p pairing (enhancement by Fermi-surface dispersion versus reduction by pair competition) persist, are weakened, or change sign once medium corrections are included.

(3) Dynamical stability and finite temperature. Test the stability of the coexistence solutions against inhomogeneous pairing states with nonzero Cooper-pair momentum (FFLO-type states), since the 2021 solutions are thermodynamically but not dynamically stable; and extend the phase structure of the coupled system to finite temperature, examining how thermal smearing of the split neutron and proton Fermi surfaces modifies the enhancement/suppression balance, the critical asymmetry at which n-p pairing disappears, and the coexistence region in the temperature-asymmetry plane.

(4) Finite nuclei (prospect). As an outlook identified in the source, extend the infinite-matter coexistence picture to finite nuclei near $N\approx Z$, where spin-orbit splitting, shell structure, and deformation enter, by extending or validating the coexistence findings with more realistic interactions. This component is a further prospect rather than a condition for settling the coexistence question in the coupled system.

The problem is complete when the coexistence question for the coupled system is settled with medium-corrected interactions, including the stability of its solutions against inhomogeneous pairing and their finite-temperature behavior; the connection to finite nuclei near $N\approx Z$ is a further prospect discussed in the source, not a completion condition. Studies of a single pairing channel in isolation, or of coupled channels with uncorrected phenomenological interactions, do not answer it.

References

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