Eur. Phys. J. B (2015) 88: 28
https://doi.org/10.1140/epjb/e2014-50683-1

Regular Article

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

¹ Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, UK
² Statistical Physics Group, Institut Jean Lamour, UMR CNRS 7198, Université de Lorraine, B.P. 70239, 54506 Vandœuvre lès Nancy Cedex, France

^a e-mail: floresse@uni.coventry.ac.uk

Received: 6 October 2014
Received in final form: 1 December 2014
Published online: 21 January 2015

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 by simulating 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.