### Abstract

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.

Original language | English |
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Pages (from-to) | 761-781 |

Number of pages | 21 |

Journal | Journal of Statistical Physics |

Volume | 63 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - May 1991 |

### Keywords

- Bose-Einstein condensation
- mean-field theories
- quantum lattice gas
- XY model