Pattern formation in systems with competing interactions

TL;DR

This paper reviews recent theoretical work demonstrating the existence of periodic structures in systems with competing short-range ferromagnetic and long-range dipole-dipole interactions, relevant to thin film phenomena.

Contribution

It proves the existence of periodic structures in lattice and continuum models with competing interactions, advancing understanding of pattern formation in such systems.

Findings

Existence of periodic structures in lattice models.

Existence of periodic structures in continuum models.

Relevance to thin film and micromagnet systems.

Abstract

There is a growing interest, inspired by advances in technology, in the low temperature physics of thin films. These quasi-2D systems show a wide range of ordering effects including formation of striped states, reorientation transitions, bubble formation in strong magnetic fields, etc. The origins of these phenomena are, in many cases, traced to competition between short ranged exchange ferromagnetic interactions, favoring a homogeneous ordered state, and the long ranged dipole-dipole interaction, which opposes such ordering on the scale of the whole sample. The present theoretical understanding of these phenomena is based on a combination of variational methods and a variety of approximations, e.g., mean-field and spin-wave theory. The comparison between the predictions of these approximate methods and the results of MonteCarlo simulations are often difficult because of the slow relaxation dynamics associated with the long-range nature of the dipole-dipole interactions. In this note we will review recent work where we prove existence of periodic structures in some lattice and continuum model systems with competing interactions. The continuum models have also been used to describe micromagnets, diblock polymers, etc.