BCS Pairing Gap in the Infrared Limit of the Similarity Renormalization Group

TL;DR

This paper investigates how phase-shift equivalent interactions evolved via the Similarity Renormalization Group affect the BCS pairing gap in nuclear matter, revealing that the gap vanishes in the continuum limit but can be computed from phase-shifts in finite systems.

Contribution

It demonstrates that the momentum grid encodes physical information and that finite size effects are crucial for understanding the pairing gap in SRG-evolved interactions.

Findings

Pairing gap vanishes in the on-shell and continuum limits.

Finite size systems allow pairing gap computation from phase-shifts.

Momentum grid encodes finite size effects in pairing calculations.

Abstract

Effective interactions have been used to compute the pairing gap for nuclear and neutron matter in several schemes. In this work we analyze the impact of phase-shift equivalent interactions within the BCS theory on the $^1S_0$-channel pairing gap for a translational invariant many-fermion system such as nuclear and neutron matter. We solve the BCS pairing gap equation on a finite momentum grid for a toy model separable Gaussian potential in the $^1S_0$-channel explicitly evolved through the Similarity Renormalization Group (SRG) transformation and show that in the on-shell and continuum limits the pairing gap vanishes. For finite size systems the momentum is quantized and the on-shell limit is realized for SRG cutoffs comparable to the momentum resolution. In this case the pairing gap can be computed directly from the scattering phase-shifts by an energy-shift formula. While the momentum grid is usually used as an auxiliary way of solving the BCS pairing gap equation, we show that it actually encodes some relevant physical information, suggesting that in fact finite grids may represent the finite size of the system.