We consider free bosons hopping on a network (infinite graph). The condition for Bose–Einstein condensation is given in terms of the random walk on a graph. In case of periodic lattices, we also consider boson moving in an external periodic potential and obatin the criterion for Bose–Einstein condensation.
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.
Abstract We investigate Bose–Einstein condensation of a noninteracting gas of Bose particles moving in the background of a periodic lattice of delta functions. In the one-dimensional case, where one has no condensation in the free case, this property persists also in the presence of the lattice for all examples which are considered in the present paper and we could only formulate some conditions which are necessary for condensation. We also considered the three-dimensional case and showed that the lattice does not destroy condensation. We calculated, for small coupling, the change in the critical temperature, which is lowered by the lattice. Finally, we took another, more general view on the problem using heat kernel expansion, and discuss BEC for Casimir effect related configurations.
We study the filling of states in a pure hopping boson model on the comb lattice, a low-dimensional discrete structure where geometrical inhomogeneity induces Bose-Einstein condensation (BEC) at finite temperature. By a careful analysis of the thermodynamic limit on combs we show that, unlike the standard lattice case, BEC is characterized by a macroscopic occupation of a finite number of states with energy belonging to a small neighborhood of the ground state energy. Such a remarkable feature gives rise to an anomalous behavior in the large distance two-point correlation functions. Finally, we prove a general theorem providing the conditions for the pure hopping model to exhibit the standard behavior, i.e. to present a macroscopic occupation of the ground state only.
In this work, we provide exact solutions for the ideal boson lattice gas on the infinitely ramified star and wheel graphs. Within a tight-binding description, we show that Bose-Einstein condensation (BEC) takes place at a finite temperature after a proper rescaling of the hoping integral ɛ connecting a central site to the peripheral ones. Analytical expressions for the transition temperature, the condensed gas fraction, and the specific heat are given for the star graph as a function of the density of particles n. In particular, the specific heat has a mean-field character, being null in the high-temperature noncondensed phase with a discontinuity at T(c). In the wheel graph, on which the peripheral sites form a closed chain with hopping integral t, BEC takes place only above a critical value of the ratio ɛ/t for which a gap ΔE appears between the ground state and a one-dimensional band. A detailed analysis of the BEC characteristics as a function of n and ΔE is provided. The specific heat in the high-temperature phase of the wheel graph remains finite due to correlations among the peripheral sites.