Energy landscape methods make use of the stationary points of the energy function of a system to infer some of its collective properties. Recently this approach has been applied to equilibrium phase transitions, showing that a connection between some properties of the energy landscape and the occurrence of a phase transition exists at least for certain classes of models [1,2,3,4]. To better understand this connection we have studied the energy landscapes of classical \(O(n)\) models defined on regular lattices and with ferromagnetic interactions. Although a complete enumeration of all the stationary points is practically unfeasible when the number of degrees of freedom increases, we have been able to find at least one class of stationary configurations with interesting properties. In particular we found that a one-to-one relation between a class of stationary points of the energy landscape of O(n) models on a lattice and the configurations of an Ising model (\(n=1\)) defined on the same lattice and with the same interactions exists. This suggested an approximate expression for the microcanonical density of states of the \(O(n)\) models in terms of the microcanonical density of states of the Ising model [5]. If correct this would implies the equivalence of the critical values of the energy densities of a \(O(n)\) model with ferromagnetic interactions defined on a lattice and the \(n=1\) case, i.e., a system of Ising spins with the same interactions [5,6]. Both numerical analysis carried on the Ising model (\(n=1\)), the XY model (\(n=2\)), the Heisenberg model (\(n=3\)) and the \(O(4)\) model (\(n=4\)) in three dimensions and analytical calculations carried on the mean field XY model and on the one dimensional XY model with nearest neighbors interactions [6] showed the reasonableness of this approximation and gave us an empirical test of its range of validity. Generalizations to more general cases are still under investigations.
[1] R. Franzosi and M. Pettini, Phys. Rev. Lett., 92, 060601 (2004);