Authors: Alessandro Giuliani, Elliott H. Lieb, Robert Seiringer

Title: Realization of stripes and slabs in two and three dimensions
 
Abstract: We consider Ising models in two and three dimensions with nearest neighbor ferromagnetic interactions and long range, power law decaying, antiferromagnetic interactions. If the strength of the ferromagnetic coupling J is larger than a critical value J_c, then the ground state is homogeneous and ferromagnetic. As the critical value is approached from smaller values of J, it is believed that the ground state consists of a periodic array of stripes (d=2) or slabs (d=3), all of the same size and alternating magnetization. Here we prove rigorously that the ground state energy per site converges to that of the optimal periodic striped/slabbed state, in the limit that J tends to the ferromagnetic transition point. While this theorem does not prove rigorously that the ground state is precisely striped/slabbed, it does prove that in any suitably large box the ground state is striped/slabbed with high probability.

Keywords: Competing interactions, long range interactions, periodic patterns, striped state

A. Giuliani
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre
L.go S. Leonardo Murialdo 1, 00146, Rome, Italy
E-mail address: giuliani AT mat DOT uniroma3 DOT it

E. H. Lieb
Department of Mathematics and Physics, Princeton University
Jadwin Hall, Princeton, NJ 08542-0708
E-mail address: lieb AT princeton DOT edu

R. Seiringer
IST Austria
Am Campus 1, 3400 Klosterneuburg, Austria
E-mail address: robert DOT seiringer AT ist DOT ac DOT at