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

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 $d\ge 3$ dimensions. This extends a similar result known for the ideal Bose gas.

27 pages, minor changes

References in corpus (4)

• Spatial random permutations and infinite cycles
• A variational formula for the free energy of an interacting many-particle system
• The relation between Feynman cycles and off-diagonal long-range order
• Fourier formula for quantum partition functions