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 (∼${10}^{20}$ tile edges) to determine whether the interface is rough. A blocking scheme suggests a roughening temperature ${\mathit{T}}_{\mathit{R}}$=0.28±0.01 (all temperatures given in units of J/${\mathit{k}}_{\mathit{B}}$). If a finite nonzero ${\mathit{T}}_{\mathit{R}}$ does exist, our results show that the roughness exponent must be discontinuous at ${\mathit{T}}_{\mathit{R}}$. We also study the model using an approximate mapping to a one-dimensional Schrödinger equation. In this approximation, for T≲0.45, 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 T≳0.45, and we cannot say if there exists a finite ${\mathit{T}}_{\mathit{R}}$. We compare the two approaches.

Close

PrevNext

View figure in article