On the absence of zero-temperature limit of equilibrium states for finite-range interactions on the lattice $\mathbb{Z}^2$
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 $(μ_β)_{β>0}$ in which $μ_β$ is an equilibrium state at inverse temperature $ β$ for this interaction, then $\lim_{β\to\infty}μ_β$ does not exist. This settles a question posed by the first author and Hochman who obtained such a non-convergence behavior when $d\geq 3$, $d$ being the dimension of the lattice.
math-ph↗