**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