Long cycles in the infinite-range-hopping Bose-Hubbard model

In this paper we study the relation between long cycles and Bose-Einstein condensation in the infinite-range Bose-Hubbard model. We obtain an expression for the cycle density involving the partition function for a Bose-Hubbard Hamiltonian with a single-site correction. 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 use it to show that in this case the long cycle density vanishes. In the presence of condensation we can 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 find that although they coexist, in general, they are not equal.