Perron-Frobenius theorem on the superfluid transition of an ultracold Fermi gas

TL;DR

This paper applies the Perron-Frobenius theorem to analyze the superfluid transition in an ultracold Fermi gas, revealing a unique phase transition point and implications for Bose-Einstein condensates of dimers.

Contribution

It introduces a novel application of the Perron-Frobenius theorem to identify the superfluid transition via Lee-Yang contracted graphs in a Fermi gas.

Findings

Existence of a unique fugacity at the phase transition point

No fragmentation of Bose-Einstein condensates at the ladder-approximation level

Application to strongly bound dimer Bose-Einstein condensates

Abstract

The Perron-Frobenius theorem is applied to identify the superfluid transition of a two-component Fermi gas with a zero-range s-wave interaction. According to the quantum cluster expansion method of Lee and Yang, the grand partition function is expressed by the Lee-Yang contracted 0-graphs. A singularity of an infinite series of ladder-type Lee-Yang contracted 0-graphs is analyzed. We point out that the singularity is governed by the Perron-Frobenius eigenvalue of a certain primitive matrix which is defined in terms of the two-body cluster functions and the Fermi distribution functions. As a consequence, it is found that there exists a unique fugacity at the phase transition point, which implies that there is no fragmentation of Bose-Einstein condensates of dimers and Cooper pairs at the ladder-approximation level of Lee-Yang contracted 0-graphs. An application to a Bose-Einstein condensate of strongly bounded dimers is also made.