Jacek Miekisz
An Ultimate Frustration in Classical-Lattice Gas Models
(43K, Latex)
ABSTRACT. 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 one-to-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.