More surprises in the general theory of lattice systems

I use Israel's methods to prove new theorems of "ubiquitous pathology" for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ρ{variant} be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φ_k} of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ρ{variant}_k, such that {ρ{variant}_k} converges to ρ{variant}. In certain situations the perturbations Φ_k-Φ can be chosen to lie in a cone of "antiferromagnetic pair interactions." I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r² interaction (Thouless effect).