  A relation between O(n) lattice spin models and Ising models defined on the
same lattice was recently put forward [L. Casetti, C. Nardini, and R.
Nerattini, Phys. Rev. Lett. 106, 057208 (2011)]. 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 the present 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 of the one-dimensional XY
model. We also discuss the possible consequences of this result for the general
case.
