Formation of Stripes and Slabs Near the Ferromagnetic Transition

We consider Ising models in d = 2 and d = 3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic in-teractions, the latter decaying as (distance)−p, p> 2d, at large dis-tances. If the strength J of the ferromagnetic interaction is larger than a critical value Jc, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to Jc the ground state is periodic and striped, with stripes of constant width h = h(J), and h → ∞ as J → J−c. (In d = 3 stripes mean slabs, not columns.) Here we rigorously prove that, if we normalize the energy in such a way that the energy of the homogeneous state is zero, then the ratio e0(J)/eS(J) tends to 1 as J → J−c, with eS(J) being the energy per site of the optimal periodic striped/slabbed state and e0(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e0(J) − eS(J) at small but fi-nite Jc − J, and also shows that in this parameter range the ground state is striped/slabbed in a certain sense: namely, if one looks at a randomly chosen window, of suitable size ` (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability. c © 2013 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. 1 1 Introduction an