铁磁转变附近条纹和板片的形成
Formation of stripes and slabs near the ferromagnetic transition
摘要：
我们考虑d=2和d=3维的伊辛模型，其中最近邻的铁磁相互作用和长程反铁磁相互作用，后者在大的距离上随距离^（-p）衰减，p>2d.如果铁磁相互作用强度J大于临界值J_c，则基态是均匀的。当J小于J_c但接近J_c时，基态是周期性的条纹，条纹宽度为h=h（J），当J从下往上趋向J_c时，h趋向无穷大。(In d=3条纹表示板，而不是柱。）这里我们严格证明了，如果我们以均匀态的能量为零的方式归一化能量，那么当J从下往上趋向于J_c时，比值e_0（J）/e_S（J）趋向于1，其中e_S（J）是最佳周期条/板态的每格点能量，e_0（J）是系统的实际每格点基态能量。我们的证明给出了在小但有限的J_c-J处的差异e_0（J）-e_S（J）的明确界限，并且还表明在此参数范围内，基态在某种意义上是条纹状/板状的：即，如果观察一个随机选择的窗口，其大小为适当的l（与最佳条纹大小h（J）相比非常大），则以高概率找到条纹状/板状态。
英文摘要：
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.
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