A Scaling Hypothesis for Modulated Systems

TL;DR

This paper introduces a scaling hypothesis for pattern-forming systems with competing short- and long-range interactions, predicting how thermodynamic quantities scale with system size and explaining experimental and computational observations.

Contribution

It proposes a novel scaling framework linking system size, interaction range, and pattern formation in modulated systems, with implications for phase transitions.

Findings

Scaling predictions match experimental data on ferromagnetic films.

Thermodynamic quantities scale as powers of system size L.

Inverse-symmetry-breaking transitions may occur for certain interaction exponents.

Abstract

We propose a scaling hypothesis for pattern-forming systems in which modulation of the order parameter results from the competition between a short-ranged interaction and a long-ranged interaction decaying with some power $\alpha$ of the inverse distance. With L being a spatial length characterizing the modulated phase, all thermodynamic quantities are predicted to scale like some power of L. The scaling dimensions with respect to L only depend on the dimensionality of the system d and the exponent \alpha. Scaling predictions are in agreement with experiments on ultra-thin ferromagnetic films and computational results. Finally, our scaling hypothesis implies that, for some range of values \alpha>d, Inverse-Symmetry-Breaking transitions may appear systematically in the considered class of frustrated systems.