Local electronic properties of one-dimensional quasiperiodic systems.

Abstract

An exact real-space renormalization-group approach is developed to calculate the local Green's function and the local density of states at any site in an infinite Fibonacci chain, in which three basic renormalization transformations ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathit{T}}_{\ensuremath{\gamma}}$ are introduced. Transformations ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ and ${\mathit{T}}_{\ensuremath{\gamma}}$ divide the Fibonacci chains of different generations into two distinct subclasses with key sites ${\mathit{S}}_{\mathrm{\ensuremath{\beta}}}$ and ${\mathit{S}}_{\ensuremath{\gamma}}$, respectively. The local Green's function and the local density of states at the key site ${\mathit{S}}_{\mathrm{\ensuremath{\beta}}}$ or ${\mathit{S}}_{\ensuremath{\gamma}}$ can be obtained by successive iterations of the transformation ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ or ${\mathit{T}}_{\ensuremath{\gamma}}$. Any other site can be transferred to the key site of a renormalized Fibonacci chain by suitable combinations of the transformations ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathit{T}}_{\ensuremath{\gamma}}$. In this approach, the relevant scale concerning the key site is ${\mathrm{\ensuremath{\tau}}}^{2}$, where \ensuremath{\tau} is the golden mean.