Exact results for an Ising system with long-range interactions on the Sierpiński-gasket lattice

Thermodynamic properties of an Ising model with two-point long-range interactions on the Sierpiński-gasket lattice are studied using an exact recursive methodfor calculating the partition function. The long-range interactions are introduced in a self-similar manner, reflecting the structure of the lattice, and are assumed to be of the form Ql = Q/2(l-1)σ, where Q ⩾ 0, σ ⩾ 0 are parameters, and l = 1, 2,...indicates that Ql couples pairs of spins on the distance 2l (measured in units of the lattice spacing). In the case of constant long-range interactions (σ = 0), exact analytic expression for the critical temperature and the critical exponent v are derived. It is also shown that the recursive method can be applied to analyze the problem of the existence of phase transitions at finite temperatures in the case of σ > 0, i.e. when the long-range coupling decrease with the distance.