Roughening of two-dimensional quasicrystals: A study on the Penrose tiling

We explore the roughening transition of a two-dimensional quasicrystal using a transfer-matrix approach. We study a solid-on-solid model on a Penrose tiling, thus retaining full two-dimensional quasiperiodicity. We find that the interface is smooth at low temperatures. It is necessary to compute the transfer matrix over extremely long distances (1020 tile edges) to determine whether the interface is rough. A blocking scheme suggests a roughening temperature TR=0.280.01 (all temperatures given in units of J/kB). If a finite nonzero TR does exist, our results show that the roughness exponent must be discontinuous at TR. We also study the model using an approximate mapping to a one-dimensional Schrödinger equation. In this approximation, for T0.45, the interface appears rough with oscillatory corrections to scaling for a large x range, and crosses over to a smooth one as x. The crossover region becomes very large for T0.45, and we cannot say if there exists a finite TR. We compare the two approaches.