Formation of stripes and slabs near the ferromagnetic transition

FM 7-13, arXiv:1304.6344

Authors Alessandro Giuliani, Elliott H. Lieb, Robert Seiringer

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 Jc, then the ground state is homogeneous. As J→Jc-, it is conjectured that the ground state is periodic and striped, with stripes of constant width h=h(J), and h(J)→∞ as J→Jc-. (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)/estripes(J) tends to 1 as J→Jc-, with estripes(J) being the energy per site of the optimal periodic striped 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)-estripes(J) at small but finite Jc-J, and also shows that in this parameter range the ground state is striped in a certain sense: namely, if we look at a randomly chosen window, of suitable size l (very large compared to the optimal stripe size h(J)), we see a striped state with high probability.