Splitting rules for the 2nd hierarchy structure of the electronic spectra of 2D FC(n) quasicrystals

On the basis of our former work and by means of the decomposition-decimation method, we study the splitting rules for the second hierarchy of the electronic energy spectra for two-dimensional Fibonacci-class quasicrystals with one kind of atom and two bond lengths. It is found that every line of the sub-spectra for n x n and (n + 1) x (n + 1) clusters of FC(n) <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$(n \geq 2)$</EquationSource> </InlineEquation> splits according to the type Y‘<Subscript>(n-1)-2-1</Subscript> and type Y <Subscript> n-2-1</Subscript> respectively. The one for n x (n + 1) clusters of FC(n) consists of three sub-subbands when <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$n \le 2$</EquationSource> </InlineEquation>, and five sub-subbranches when <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$n \ge 3$</EquationSource> </InlineEquation>. The general formulae of the number of energy levels for the spectra of the second hierarchy are obtained. The analytical results are confirmed by numerical simulations. Copyright Springer-Verlag Berlin/Heidelberg 2004