Scaling theory for two-dimensional systems with competing interactions.

Abstract

We derive an analytic scaling theory for a two-dimensional system in which spontaneous patterns of stripes, bubbles, and intermediately shaped domains arise due to the competition of short-range attractions and long-range dipolar repulsions. The theory predicts temperature and domain-size scaling as a function of the relative repulsion strength eta, the ratio of the repulsive to the attractive coupling constant in the system's Hamiltonian. As eta decreases, the domain size explodes exponentially and the melting temperature for a system of ordered stripes increases. Our findings shed new light on the phase diagram and critical excitations for the dipolar Ising ferromagnet or lattice gas and their continuum analogs. We show that the features described by the scaling theory are insensitive to details like cutoffs for the dipolar interactions and, therefore, should be widely applicable. Our corresponding states analysis explains the experimentally observed stripe melting upon compression in a Langmuir monolayer. A phenomenological extension of the analytic scaling theory describes how the system's behavior is modified in the presence of magnetization or density fluctuations. Fluctuations are found to suppress domain size and the stripe melting temperature. In regimes where fluctuations are important, we predict that domain size will decrease with increasing temperature.