An Ultimate Frustration in Classical Lattice-Gas Models

We compare tiling systems with square-like tiles and classical lattice-gas models with translation-invariant, finite-range interactions between particles. For a given tiling, there is a natural construction of a corresponding lattice-gas model. With oneto-one correspondence between particles and tiles, we simply assign a positive energy to pairs of nearest-neighbor particles which do not match as tiles; otherwise the energy of interaction is zero. Such models of interacting particles are called nonfrustrated all interactions can attain their minima simultaneously. Ground-state configurations of these models correspond to tilings; they have the minimal energy density equal to zero. There are frustrated lattice-gas models; antiferromagnetic Ising model on the triangular lattice is a standard example. However, in all such models known so far, one could always find a nonfrustrated interaction having the same ground-state configurations. Here we constructed an uncountable family of classical lattice-gas models with unique ground-state measures which are not uniquely ergodic measures of any tiling system, or more generally, of any system of finite type. Therefore, we have shown that the family of structures which are unique ground states of some translation-invariant, finite-range interactions is larger than the family of tilings which form single isomorphism classes. Such ground-state measures cannot be ground-state measures of any translation-invariant, finite-range, nonfrustrated potential. Our ground-state configurations are two-dimensional analogs of one-dimensional, most homogeneous ground-state configurations of infinite-range, convex, repulsive interactions in models with devil’s staircases.