Formation of stripes and slabs near the ferromagnetic transition

TL;DR

This paper rigorously proves that near the ferromagnetic transition, the ground state of certain Ising models with competing interactions is predominantly striped or slabbed, with energies close to the optimal periodic configurations.

Contribution

It provides a rigorous proof that the ground state energy approaches that of the optimal striped/slabbed state near the transition, with explicit bounds and probabilistic structure.

Findings

Ground state energy ratio approaches 1 near transition

Ground states are striped/slabbed with high probability in large windows

Explicit bounds on energy differences near critical point

Abstract

We consider Ising models in d=2 and d=3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)^(-p), p>2d, at large distances. If the strength J of the ferromagnetic interaction is larger than a critical value J_c, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to J_c the ground state is periodic and striped, with stripes of constant width h=h(J), and h tends to infinity as J tends to J_c from below. (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 e_0(J)/e_S(J) tends to 1 as J tends to J_c from below, with e_S(J) being the energy per site of the optimal periodic striped/slabbed state and e_0(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e_0(J)-e_S(J) at small but finite J_c-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 l (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability.