Criteria of off-diagonal long-range order in Bose and Fermi systems based on the Lee-Yang cluster expansion method

TL;DR

This paper extends the Lee-Yang cluster expansion method to analyze off-diagonal long-range order in Bose and Fermi systems, providing criteria for Bose-Einstein condensation and Fermi pairing through graph summations.

Contribution

It introduces a systematic cluster expansion framework applicable to both uniform and trapped systems without local-density approximation, linking Lee-Yang graphs to ODLRO criteria.

Findings

Infinite ladder-type Lee-Yang 0-graphs lead to Bose-Einstein condensation of dimers.

Series of Lee-Yang 1-graphs determine ODLRO in Bose systems.

Series of Lee-Yang 2-graphs determine ODLRO in Fermi systems.

Abstract

The quantum-statistical cluster expansion method of Lee and Yang is extended to investigate off-diagonal long-range order (ODLRO) in one- and multi-component mixtures of bosons or fermions. Our formulation is applicable to both a uniform system and a trapped system without local-density approximation and allows systematic expansions of one- and multi-particle reduced density matrices in terms of cluster functions which are defined for the same system with Boltzmann statistics. Each term in this expansion can be associated with a Lee-Yang graph. We elucidate a physical meaning of each Lee-Yang graph; in particular, for a mixture of ultracold atoms and bound dimers, an infinite sum of the ladder-type Lee-Yang 0-graphs is shown to lead to Bose-Einstein condensation of dimers below the critical temperature. In the case of Bose statistics, an infinite series of Lee-Yang 1-graphs is shown to converge and gives the criteria of ODLRO at the one-particle level. Applications to a dilute Bose system of hard spheres are also made. In the case of Fermi statistics, an infinite series of Lee-Yang 2-graphs is shown to converge and gives the criteria of ODLRO at the two-particle level. Applications to a two-component Fermi gas in the tightly bound limit are also made.