Miniband and Gap Evolution in Gauss Chains

The Gauss chain is a one-dimensional quasiperiodic lattice with sites at $z_j=j^{n}d$, where $j\in \{0,1,2,\dots ,N-1\}$, $n\in \{2,3,4,\dots \}$, and d is the underlying lattice constant. We numerically study the formation of a hierarchy of minibands and gaps as N increases using a Kronig–Penney model. Increasing n empirically results in a more fragmented miniband and gap structure due to the rapid increase in the number of minibands and gaps as n increases, in agreement with previous studies. We show that the Gauss chain $z_j=j^{2}d$ and a specific generalized Gauss chain, $z_j=(j^2\pm \frac{1}{2}j)d$, are treatable by a real-space renormalization group approach. These appear to be the only Gauss chains treatable by this approach, suggesting a hidden symmetry for the quadratic cases.