Weinberg Eigenvalues and Pairing with Low-Momentum Potentials

TL;DR

This paper extends Weinberg eigenvalue analysis to in-medium conditions near the Fermi surface, providing a diagnostic for pairing instabilities and estimating the pairing gap in nucleon-nucleon interactions at low momentum.

Contribution

It introduces a method to analyze pairing in nuclear matter using Weinberg eigenvalues with Nambu-Gorkov Green's functions, linking eigenvalues to the pairing gap.

Findings

Eigenvalues increase near the Fermi surface, indicating pairing instability.

Largest attractive eigenvalue approaches unity, signaling bound Cooper pairs.

Eigenvector analysis yields the self-consistent pairing gap.

Abstract

The nonperturbative nature of nucleon-nucleon interactions evolved to low momentum has recently been investigated in free space and at finite density using Weinberg eigenvalues as a diagnostic. This analysis is extended here to the in-medium eigenvalues near the Fermi surface to study pairing. For a fixed value of density and cutoff Lambda, the eigenvalues increase arbitrarily in magnitude close to the Fermi surface, signaling the pairing instability. When using normal-phase propagators, the Weinberg analysis with complex energies becomes a form of stability analysis and the pairing gap can be estimated from the largest attractive eigenvalue. With Nambu-Gorkov Green's functions, the largest attractive eigenvalue goes to unity close to the Fermi surface, indicating the presence of bound states (Cooper pairs), and the corresponding eigenvector leads to the self-consistent gap function.