Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

TL;DR

This paper explores the unique finite-size scaling behavior in one-dimensional long-range Ising models above the upper critical dimension, revealing non-trivial correlation length scaling and modifications to classical relations.

Contribution

It demonstrates how long-range interactions lead to new scaling scenarios and non-trivial correlation length behavior above the upper critical dimension in Ising models.

Findings

Correlation length scales as a non-trivial power of system size

Disparity between correlation length and system size can be arbitrarily large

Modifications to Fisher relation due to new scaling scenario

Abstract

The correlation length plays a pivotal role in finite-size scaling and hyperscaling at continuous phase transitions. Below the upper critical dimension, where the correlation length is proportional to the system length, both finite-size scaling and hyperscaling take conventional forms. Above the upper critical dimension these forms break down and a new scaling scenario appears. Here we investigate this scaling behaviour in one-dimensional Ising ferromagnets with long-range interactions. We show that the correlation length scales as a non-trivial power of the linear system size and investigate the scaling forms. For interactions of sufficiently long range, the disparity between the correlation length and the system length can be made arbitrarily large, while maintaining the new scaling scenarios. We also investigate the behavior of the correlation function above the upper critical dimension and the modifications imposed by the new scaling scenario onto the associated Fisher relation.