Periodic States in Systems with Competing Interactions

In this paper, we review some recent results about the existence of periodic states in Ising models with competing interactions, in two or more dimensions. The Hamiltonian of the system includes both the standard ferromagnetic nearest neighbor interaction, and an antiferromagnetic long range interaction, decaying polynomially, as (distance)^{-p}, with p>2d in dimension d. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the formation of domains of minus spins alternating with domains of opposite magnetization: this happens for values of J smaller than a critical value J_c(p), beyond which the ground state is homogeneous. In this paper we informally discuss a construction of the ground states of the model, for J in a left neighborhood of Jc(p). In particular, we show that the quasi-one-dimensional states consisting of infinite stripes, all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. We also discuss the main ideas involved in the proof, which combines localization bounds and reflection positivity.