Exact and approximate ensemble treatments of thermal pairing in a multilevel model

TL;DR

This paper compares exact and approximate methods for modeling thermal pairing in finite systems, showing that projected theories align closely with exact solutions and reveal that phase transitions are smoothed in small systems.

Contribution

It introduces a systematic comparison of various ensemble and approximate methods for thermal pairing, highlighting the effectiveness of Lipkin-Nogami projected theories.

Findings

FTLN1+SCQRPA aligns best with exact grand-canonical results.

All methods show phase transition smoothing in finite systems.

A new formula for empirical pairing gap extraction is proposed.

Abstract

A systematic comparison is conducted for pairing properties of finite systems at nonzero temperature as predicted by the exact solutions of the pairing problem embedded in three principal statistical ensembles, as well as the unprojected (FTBCS1+SCQRPA) and Lipkin-Nogami projected (FTLN1+SCQRPA) theories that include the quasiparticle number fluctuation and coupling to pair vibrations within the self-consistent quasiparticle random-phase approximation. The numerical calculations are performed for the pairing gap, total energy, heat capacity, entropy, and microcanonical temperature within the doubly-folded equidistant multilevel pairing model. The FTLN1+SCQRPA predictions agree best with the exact grand-canonical results. In general, all approaches clearly show that the superfluid-normal phase transition is smoothed out in finite systems. A novel formula is suggested for extracting the empirical pairing gap in reasonable agreement with the exact canonical results.