A relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of an Ising model defined on the same lattice suggests an approximate expression for the microcanonical density of states. Based on this approximation we conjecture that if a $O(n)$ model with ferromagnetic interactions on a lattice has a phase transition, its critical energy density is equal to that of the $n=1$ case, i.e., an Ising system with the same interactions. The conjecture holds true in the case of long-range interactions. For nearest-neighbor interactions, numerical results are consistent with the conjecture for $n=2$ and $n=3$ in three dimensions. For $n=2$ in two dimensions ($XY$ model) the conjecture yields a prediction for the critical energy of the Berežinskij-Kosterlitz-Thouless transition, which would be equal to that of the two-dimensional Ising model. We discuss available numerical data in this respect.

• Received 18 November 2010

DOI:https://doi.org/10.1103/PhysRevLett.106.057208