Phase diagram of a random tiling quasicrystal

We study the phase diagram of a two-dimensional random tiling model for quasicrystals. At proper concentrations the model has 8-fold rotational symmetry. Landau theory correctly gives most of the qualitative features of the phase diagram, which is in turn studied in detail numerically using a transfer matrix approach. We find that the system can enter the quasicrystal phase from many other crystalline and incommensurate phases through first-order or continuous transitions. Exact solutions are given in all phases except for the quasicrystal phase, and for the phase boundaries between them. We calculate numerically the phason elastic constants and entropy density, and confirm that the entropy density reaches its maximum at the point where phason strains are zero and the system possesses 8-fold rotational symmetry. In addition to the obvious application to quasicrystals, this study generalizes certain surface roughening models to two-dimensional surfaces in four dimensions.