A 2D lattice model defined on a triangular lattice with nearest- and
next-nearest-neighbor interactions based on the Taylor-Socolar monotile is
known to have a limit-periodic ground state. The system reaches that state
during a slow quench through an infinite sequence of phase transitions. We
study the model as a function of the strength of the next-nearest-neighbor
interactions, and introduce closely related 3D models with only
nearest-neighbor interactions that exhibit limit-periodic phases. For models
with no next-nearest-neighbor interactions of the Taylor-Socolar type, there is
a large degenerate classes of ground states, including crystalline patterns and
limit-periodic ones, but a slow quench still yields the limit-periodic state.
For the Taylor-Socolar lattice model, we present calculations of the
diffraction pattern for a particular decoration of the tile that permits exact
expressions for the amplitudes, and identify domain walls that slow the
relaxation times in the ordered phases. For one of the 3D models, we show that
the phase transitions are first order, with equilibrium structures that can be
more complex than in the 2D case, and we include a proof of aperiodicity for a
geometrically simple tile with only nearest-neighbor matching rules.Comment: 25 pages, 28 figures; To appear in Physical Review
