Scaling hypothesis for modulated systems

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 $\ensuremath{\alpha}$ 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 $L$, ${L}^{\ensuremath{\Delta}(\ensuremath{\alpha},d)}$. The scaling dimensions $\ensuremath{\Delta}(\ensuremath{\alpha},d)$ only depend on the dimensionality of the system $d$ and the exponent $\ensuremath{\alpha}$. Scaling predictions are in agreement with experiments on ultrathin ferromagnetic films and computational results. Finally, our scaling hypothesis implies that, for some range of values $\ensuremath{\alpha}>d$, inverse-symmetry-breaking transitions may appear systematically in the considered class of frustrated systems.