Emergence of limit-periodic order in tiling models

TL;DR

This paper investigates the emergence of limit-periodic order in 2D and 3D tiling models, revealing how specific interactions and quenches lead to complex ordered phases, including proofs of aperiodicity.

Contribution

It introduces new 3D models with only nearest-neighbor interactions that exhibit limit-periodic phases and provides exact diffraction calculations for the 2D Taylor-Socolar model.

Findings

Limit-periodic phases emerge during slow quenches.

First-order phase transitions observed in 3D models.

Proof of aperiodicity for certain simple tiles.

Abstract

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.