On a microcanonical relation between continuous and discrete spin models

TL;DR

This paper explores a proposed relation between continuous and discrete spin models, suggesting that their critical energies are connected, with implications for understanding phase transitions in various lattice systems.

Contribution

It introduces an approximate relation between continuous and discrete spin models' energy landscapes and conjectures a universal critical energy density for models with phase transitions.

Findings

The conjecture holds for long-range interactions.

Numerical results support the conjecture for n=2,3 in 3D.

Predictions for the XY model's BKT transition energy are discussed.

Abstract

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 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., a system of Ising spins 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 Berezinskij-Kosterlitz-Thouless transition, which would be equal to that of the two-dimensional Ising model. We discuss available numerical data in this respect.