Discrete approximations to vector spin models

TL;DR

This paper investigates how discretising continuous-spin models affects their interactions, showing effective interactions exist at moderate temperatures but not at very low temperatures, and compares two discretisation methods.

Contribution

It strengthens previous results on effective interactions in discretised models and compares two discretisation methods, highlighting temperature-dependent behaviors in two and higher dimensions.

Findings

Effective interactions exist at moderate temperatures.

Discretisation methods behave similarly except at very low temperatures.

Behavior in higher dimensions remains an open discussion.

Abstract

We strengthen a result of two of us on the existence of effective interactions for discretised continuous-spin models. We also point out that such an interaction cannot exist at very low temperatures. Moreover, we compare two ways of discretising continuous-spin models, and show that, except for very low temperatures, they behave similarly in two dimensions. We also discuss some possibilities in higher dimensions.