Local and Nonlocal Continuum Limits of Ising-Type Energies for Spin Systems

We study, through a !-convergence procedure, the discrete to continuum limit of Ising-type energies of the form F"(u) = −Pi,j c" i,juiuj , where u is a spin variable defined on a portion of a cubic lattice "Zd 8 ⌦, ⌦ being a regular bounded open set, and valued in {−1, 1}. If the constants c" i,j are nonnegative and satisfy suitable coercivity and decay assumptions, we show that all possible !-limits of surface scalings of the functionals F" are finite on BV (⌦; {±1}) and of the formR Su'(x, ⌫u) dH11 d−1. If such decay assumptions are violated, we show that we may approximate nonlocal functionals of the form R Su '(⌫u) dHd−1+ K(x, y)g(u(x), u(y)) dxdy. We focus on the approximation of two relevant examples: fractional perimeters and Ohta–Kawasaki-type energies. Eventually, we provide a general criterion for a ferromagnetic behavior of the energies F" even when the constants c" i,j change sign. If such a criterion is satisfied, the ground states of F" are still the uniform states 1 and −1 and the continuum limit of the scaled energies is an integral surface energy of the form above.