Infinite Cycles in Boson Lattice Models

TL;DR

This paper investigates the connection between long cycles and Bose-Einstein condensation in various boson lattice models, deriving new expressions and analyzing the conditions under which cycle densities relate to condensate densities.

Contribution

It introduces a new formulation for cycle densities in boson models and analyzes their relationship with BEC, including exact results for the hard-core case and conjectures for finite interactions.

Findings

Long cycle density coincides with BEC in known models.

In the hard-core case, only cycles of length one contribute.

Long cycle density and condensate density are generally not equal.

Abstract

We study the relationship between long cycles and Bose-Einstein condensation (BEC) in the case of several models. A convenient expression for the density of particles on cycles of length $q$ is obtained, in terms of $q$ unsymmetrised particles coupled with a boson field. Using this formulation we reproduce known results on the Ideal Bose Gas, Mean-Field and Perturbed Mean-Field Models, where the condensate density exactly equals the long cycle density. Then we consider the Infinite-Range-Hopping Bose-Hubbard model, with and without hard-cores. For the hard-core case, we find we can disregard the hopping contribution of the q unsymmetrised particles to obtain an exact expression for the density of particles on long cycles. It is shown that only the cycle of length one contributes to the cycle density. We conclude that while the existence of a non-zero long cycle density coincides with the occurrence of Bose-Einstein condensation, the respective densities are not necessarily equal. For the case of a finite on-site repulsion, we obtain an expression for the cycle density by neglecting the $q$ unsymmetrised hop. Inspired by the Approximating Hamiltonian method we conjecture a simplified expression for the short cycle density as a ratio of single-site partition functions. In the absence of condensation we prove that this simplification is exact and find that in this case the long-cycle density vanishes. In the presence of condensation we justify this simplification when a gauge-symmetry breaking term is introduced in the Hamiltonian. Assuming our conjecture is correct, we compare numerically the long-cycle density with the condensate and again find that though they coexist, in general they are not equal.