Density of states of continuous and discrete spin models: a case study

A relation between O(n) lattice spin models and Ising models defined on the same lattice was recently put forward (Casetti et al 2011 Phys. Rev. Lett. 106 057208). Such a relation, inspired by an energy landscape analysis, implies that the density of states of an O(n) spin model on a lattice can be effectively approximated, at least close to the phase transition, in terms of the density of states of an Ising model defined on the same lattice and with the same interactions. In this paper we show that such a relation exactly holds, albeit in a slightly modified form, in the special cases of the mean-field XY model and the one-dimensional XY model. We also discuss the possible consequences of this result for the general case.