On the absence of zero-temperature limit of equilibrium states for finite-range interactions on the lattice ${\mathbb{Z}}^2$

Abstract

We construct finite-range interactions on ${\mathcal{S}}^{{\mathbb{Z}}^2}$, where $\mathcal{S}$ is a finite set, for which the associated equilibrium states (i.e., the shift-invariant Gibbs states) fail to converge as temperature goes to zero. More precisely, if we pick any one-parameter family $({\mu }_{\beta }{)}_{\beta >0}$ in which ${\mu }_{\beta }$ is an equilibrium state at inverse temperature $\beta$ for this interaction, then ${\lim }_{\beta \to \infty }{\mu }_{\beta }$ does not exist. This settles a question posed by the first author and Hochman who obtained such a non-convergence behavior when $d\ge 3$, $d$ being the dimension of the lattice.

25 pages, 8 figures