Bose-Einstein condensation in the infinitely ramified star and wheel graphs

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.

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