A novel way of recasting the Bardeen-Cooper-Schrieffer gap equations

TL;DR

This work tackles the longstanding challenge of solving the BCS gap equations by recasting them as a nonlinear algebraic system for the kernel $F(k)=\Delta(k)/E(k)$ on a collocation grid. By expanding $F(k)$ as $F_n(k)=\sum_i g_i\phi_i(k)$ and enforcing the recast equation at collocation points, the problem becomes a Newton-solvable system with precomputed integrals $\psi_{ij}$. The approach leverages Legendre grids, barycentric interpolation, and a tail-mapping joint to stabilize long-range potentials, achieving rapid convergence across several nuclear interactions and showing robustness near phase transitions. It promises a universal solver for gap equations, extensible to coupled channels and added constraints, with demonstrated applicability to neutron matter and potential impact on mean-field descriptions of exotic nuclear superfluidity.

Abstract

The gap equations lie at the core of the Bardeen-Cooper-Schrieffer (BCS) theory, a standard tool in the description of superfluidity. As a set of non-linear integral equations, the gap equations' inherent difficulties oftentimes hinder even the crudest descriptions of superfluid states. Hard-core potentials, high-density superfluids, and coupled-channel pairing are all reasons that have historically required one to provide special treatment to the gap equations to get a solution. In this paper we present a new method for solving the gap equations that holds the promise of being an efficient universal solver that requires the minimum amount of \textit{a priori} knowledge of the targeted solutions. With theoretical evidence of exotic nuclear superfluidity posing new questions to our understanding of this fundamental property of nuclear systems, the presented method can be a valuable tool when exploring new pairing states, finite-temperature properties, or the development of sophisticated descriptions of nuclear superfludity.

Figures (8)

• Figure 1: The weights for the barycentric formula in Eq. (\ref{['eq:bary']}) for 100 Legendre nodes in the interval $[0,10]$.
• Figure 2: Solutions of the gap equations with various potentials: Pöschl-Teller potential (grey solid line), $\textrm{N}^{2}\textrm{LO}$ potential with $R_0=0.9~\textrm{fm},~1.0~\textrm{fm},~1.1~\textrm{fm},~1.2~\textrm{fm}$ (solid light blue, dashed-dotted navy blue, dashed red, dotted dark red), and $\textrm{N}^{3}\textrm{LO}$ (450 MeV) potential (green dashed). Solutions from iterative methods are included for the $\textrm{N}^{2}\textrm{LO}$ potential (solid circles).
• Figure 3: Convergence of Newton's method in $\delta g$ for the PT potential (thick grey solid line), the $\textrm{N}^{2}\textrm{LO}$ potentials with $R_0=0.9~\textrm{fm}$ (blue solid line), $R_0=1.0~\textrm{fm}$ (dark blue dashed dotted line), $R_0=1.1~\textrm{fm}$ (dashed red line), $R_0=1.2~\textrm{fm}$ (dotted red line), and the $\textrm{N}^{3}\textrm{LO}$ potential (green dashed line). Three regimes are separated by the rate of convergence and analyzed in Fig. \ref{['fig:regimes']}.
• Figure 4: Convergence of Newton's method in $\textrm{max}_if_i$ for the PT potential (thick grey solid line), the $\textrm{N}^{2}\textrm{LO}$ potentials with $R_0=0.9~\textrm{fm}$ (blue solid line), $R_0=1.0~\textrm{fm}$ (dark blue dashed dotted line), $R_0=1.1~\textrm{fm}$ (dashed red line), $R_0=1.2~\textrm{fm}$ (dotted red line), and the $\textrm{N}^{3}\textrm{LO}$ potential (green dashed line).
• Figure 5: The three convergence regimes of $F(k)$ for the $\textrm{N}^{3}\textrm{LO}$ potential: a qualitative regime ($0<\textrm{step}\lessapprox5$, top panel), a quantitative regime ($5<\textrm{step}\lessapprox10$, middle panel), and a final regime ($10<\textrm{step}$, bottom panel). The green squares in the top and bottom panels mark the position of discontinuity created at the node of $F(k)$ while the green curve in the bottom panel highlights the effect of correcting it in the way described in sec \ref{['sec:smooth']}.