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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 J_{c}, then
the ground state is homogeneous. As J→J_{c}^{-}, it is
conjectured that the ground state is periodic and striped, with stripes of
constant width h=h(J), and h(J)→∞ 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 e_{0}(J)/e_{stripes}(J) tends to 1 as J→J_{c}^{-}, with e_{stripes}(J)
being the energy per site of the optimal periodic striped 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_{stripes}(J) at small but finite J_{c}-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.