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 $d\geq 3$, $d$ being the dimension of the lattice.
Categories: math-ph math.DS math.MP