Binary tiling quasicrystals and matching rules

Abstract

A general theory on the transfer of perfect matching rules for a quasiperiodic tiling to perfect matching rules for an atomic decoration of the tiling is presented. General conditions on the possibility of such a transfer are discussed, and an upper bound on the range of the matching rules for the atomic structure is derived. This range is identical to the range of interactions needed to stabilize such a quasiperiodic ground state. The main tool in this analysis is the concept of mutual local derivability. The general principles are then applied to two examples of binary tiling quasicrystals. The first one, based on the Tübingen triangle tiling, needs matching rules of a rather long range, whereas the second example, which is a decoration of the Penrose rhombus tiling, has matching rules of reasonable range. Finally, the concepts put forward in this paper are set into a broader context, and we compare them with other theories for the propagation of quasiperiodic order.