Phase transitions and tricritical points: An exactly soluble model for magnetic or distortive systems

We present and solve exactly a lattice model for magnetic or for structural phase transitions. The model proposed here can be seen as an extension of the spherical model. We obtain the following results: (i) The free energy of the system is calculated rigorously in a general case for lattices of any dimensionality $d$ or structure. (ii) The critical properties are worked out explicitly for a special case of the interaction on a cubic $d$-dimensional lattice ($d$ may be a fractional number). If $d>\mathrm{}2\mathrm{}$, second-order phase transitions may occur. The critical exponents are those of the sperical model. (iii) For a special choice of the interaction and if $d\ge 3$, the existence of a line of tricritical points can be demonstrated. The tricritical exponents are computed explicitly; they are identical to the exponents of the "Gaussian" model. (iv) Finally, first-order phase transitions are shown to exist in one and two dimensions.