Simultaneous occurrence of off-diagonal long-range order and infinite permutation cycles in systems of interacting atoms

TL;DR

This paper establishes a precise link between off-diagonal long-range order and infinite permutation cycles in interacting bosonic systems, extending known results from ideal gases to more general interactions.

Contribution

It proves that in interacting boson systems, off-diagonal long-range order occurs if and only if a nonzero fraction of particles form infinite permutation cycles, generalizing previous ideal gas results.

Findings

Off-diagonal long-range order corresponds to infinite permutation cycles.

Bose-Einstein condensation occurs when cycle lengths grow at least as fast as N^{2/d} in d≥3.

Extension of ideal Bose gas results to interacting systems.

Abstract

Based on the paper "Fourier formula for quantum partition functions", arXiv:2106.10032 [math-ph], we show that in an infinite system of identical bosons interacting via a positive-type pair potential there is off-diagonal long-range order if and only if a nonzero fraction of the particles form infinite permutation cycles. In particular, there is Bose-Einstein condensation if and only if the diverging cycle lengths increase at least as fast with $N$, the number of particles, as $N^{2/d}$ in $d\geq 3$ dimensions. This extends a similar result known for the ideal Bose gas.