Miniband and Gap Evolution in Gauss Chains

The Gauss chain is a one-dimensional quasiperiodic lattice with sites at zj=jnd, where j∈{0, 1, 2, …, N-1}, n∈{2, 3, 4, …}, 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 zj=j2d and a specific generalized Gauss chain, zj=(j2±12j)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.