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

TL;DR

This paper demonstrates an exact relation between the density of states of certain O(n) spin models and Ising models, especially near phase transitions, with specific case studies on mean-field and 1D XY models.

Contribution

It establishes an exact form of the previously proposed relation between O(n) and Ising models' density of states, including special cases.

Findings

Exact relation holds for mean-field XY model.

Exact relation holds for one-dimensional XY model.

Implications for understanding phase transitions in spin models.

Abstract

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.