Wednesday, April 24, 2013

1304.6344 (Alessandro Giuliani et al.)

Stripe formation at the ferromagnetic transition    [PDF]

Alessandro Giuliani, Elliott H. Lieb, Robert Seiringer
We consider a two-dimensional Ising model with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)^(-p), p>4, 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 tends to J_c from the left, it is conjectured that 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. 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) to e_s(J) tends to 1 as J tends to J_c, with e_s(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_s(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^*), we see a striped state with high probability.
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