Bose-Einstein condensation in an exactly soluble system of interacting particles

The model investigated recently by Tóth, a lattice gas of bosons with hard-core repulsion on a complete graph, is studied here by diagonalizing the Hamiltonian. The thermodynamic free energy per site is shown to be f, where {Mathematical expression} where ß is the inverse temperature and ??[0, 1] is the number of particles per site. This formula is equivalent to the one obtained by Tóth. There is a phase transition at ß +ß^* (?) = (1-2 ?)^-1 log[(1-?)/?]. If ß=ß^*(?), Bose-Einstein condensation is shown to be present, the condensate density (number of condensed particles per site) in the thermodynamic limit being [?-x^*][l-?-x^*], where x^* is the minimizing value of x, satisfying ß^*(x^*)=ß. © 1991 Plenum Publishing Corporation.