Rachele	Nerattini - Università di Firenze #
On a microcanonical relation between continuous and discrete spin models #
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. We considered classical spin models and found that a relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of a Ising model defined on the same lattice exists and suggests an approximate expression for the microcanonical density of states in terms of the energy density of the Ising model. Assuming this approximation is correct close to the phase transition implies that the critical energy density of a O(n) model with ferromagnetic interactions on a lattice is equal to that of the n = 1 case, i.e., a system of Ising spins with the same interactions. This holds true in the case of long-range interactions, and at least in the special case of the mean-field XY model the expression of the density of states in terms of the Ising one can be exactly derived. For nearest-neighbor interactions, numerical results are consistent with the equality of critical energy densities for n = 2 and n = 3 in three dimensions. According to the approximation, also the critical energy of the Berezinskij-Kosterlitz-Thouless (BKT) transition for n = 2 in two dimensions (XY model) should be equal to that of the two-dimensional Ising model. However, numerical results show that the critical energies of these two models are different, although close, the difference being around 2%. The transition energies may be really equal for all cases but the BKT one, due to very different nature of the BKT phase transitions with respect to the ferromagnetic one; otherwise, transition energies might be equal only for the long-range case and different in all the other cases, with a difference that is very small and not masked by numerical errors only in the BKT case. Numerical investigations will hopefully help to clarify this point. Reference: PRL 106, 057208 (2011)