Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals

Abstract

Random tilings that comprise squares and equilateral triangles can model quasicrystals with twelvefold symmetry. A (phason) elastic theory for such tilings is constructed, whose order parameter is the phason field, and whose entropy density includes terms up to third order in the phason strain. Due to an unusual constraint, the phason field of any square-triangle tiling is irrotational and, as a result, the form of the entropy density is simpler than the general form that is required by twelvefold symmetry alone. Using an update move, which rearranges a closed, nonlocal, one-dimensional chain of squares and triangles, the unknown parameters of the elastic theory are estimated via Monte Carlo simulations: (i) One of the two second-order elastic constants and the third-order elastic constant are found by measuring phason fluctuations; athermal systems (maximally random ensembles) with the same background phason strain but different sizes of unit cell are simulated to distinguish the effects of a finite background phason strain from the effects of finite unit-cell size. (ii) The entropy per unit area at zero phason strain and the other second-order elastic constant are found from the entropies that thermal systems (canonical ensembles) gain between zero and infinite temperature, which are estimated using Ferrenberg and Swendsen’s histogram method. A way to set up transfer-matrix calculations for random square-triangle tilings is also presented.