Local electronic properties of two-dimensional Penrose tilings: A renormalization-group approach.

Abstract

A real-space renormalization-group (RSRG) scheme is developed to study the local electronic properties of the two-dimensional Penrose tilings. The RSRG transformations are derived in terms of the inflation rule of the Penrose tilings. Two basic transformations ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$ and ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ are introduced, of which ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$ transforms a rhombus (R) Penrose tiling to a kite-dart (KD) Penrose tiling, while ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ transforms a KD Penrose tiling to a R Penrose tiling. Suitable combinations of them give rise to two transformations ${\mathit{T}}_{1}$ and ${\mathit{T}}_{2}$, of which ${\mathit{T}}_{1}$ or ${\mathit{T}}_{2}$ transforms a R or KD Penrose tiling to the same kind of quasiperiodic tiling that is scaled by a factor of ${\mathrm{\ensuremath{\tau}}}^{2}$, in which \ensuremath{\tau}=(\ensuremath{\surd}5 +1)/2 is the golden mean. By infinite iterations of ${\mathit{T}}_{1}$ or ${\mathit{T}}_{2}$, we obtain the local Green's functions and the local densities of states (LDOS) at the key sites of the R or KD Penrose tilings. The LDOS at the key sites of the R Penrose tilings are numerically calculated and are consistent with the results obtained by others.