FM 16-08; to appear in the AIP conference proceedings of the 10th Granada Seminar on Computational Physics, Sept. 15-19, 2008.

Authors : Alessandro Giuliani, Joel L. Lebowitz, Elliott H. Lieb

Title: Pattern formation in systems with competing interactions

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.

Keywords: Striped order, periodic ground state, Ising model, reflection positivity.

Alessandro Giuliani,
Dipartimento di Matematica,
Universita' di Roma Tre,
L.go S. Leonardo Murialdo 1, 00146 Roma, Italy

Joel L. Lebowitz,
Department of Mathematics and Physics,
Rutgers University, Rutgers University, Piscataway, NJ 08854 USA

Elliott H. Lieb,
Department of Mathematics and Physics,
Princeton University, Princeton 08544 NJ, USA