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arXiv:cond-mat/0204430 · Journal of Physics A Mathematical and General

Percolation transition in the Bose gas: II

András Sütö

Published Aug 7, 2002• 68 citations
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Abstract

In an earlier paper (Sütö A 1993 J. Phys. A: Math. Gen. 26 4689) we introduced the notion of cycle percolation in the Bose gas and conjectured that it occurs if and only if there is Bose–Einstein condensation. Here we give a complete proof of this statement for the perfect and the imperfect (mean-field) Bose gas and also show that in the condensate there is an infinite number of macroscopic cycles.

DOI
10.1088/0305-4470/35/33/303

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In an earlier paper (Sütö A 1993 J. Phys. A: Math. Gen. 26 4689) we introduced the notion of cycle percolation in the Bose gas and conjectured that it occurs if and only if there is Bose–Einstein condensation. Here we give a complete proof of this statement for the perfect and the imperfect (mean-field) Bose gas and also show that in the condensate there is an infinite number of macroscopic cycles.
The canonical partition function of a Bose gas gives rise to a probability distribution over the permutations of N particles. The author studies the probability and mean value of the cycle lengths in the cyclic permutations, their relation to physical quantities like pair correlations, and their thermodynamic limit. He shows that in the ground state of most interacting boson gases the mean cycle length diverges in the bulk limit and the particles form macroscopic cycles. In the free Bose gas Bose-Einstein condensation is accompanied by a percolation transition: the appearance of infinite cycle with nonvanishing probability.
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 qqq is obtained, in terms of qqq 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 qqq 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.