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.