Splitting rules for the electronic energy spectra of two-dimensional Thue-Morse lattices with three kinds of atom and one bond length

In this paper we study the splitting rules of the electronic energy spectra for two-dimensional Thue-Morse lattices with three kinds of atom and one bond length by means of a decomposition-decimation method. It is found that the symmetries of systems, boundary conditions, and then the branching types of energy spectra for the second and third hierarchies are all related to generations. For example, under the second-order approximation, the energy spectra of four-atom A molecules split in the proportion of one-to-five for even generations and one-to-three for odd ones. The analytical results are confirmed by numerical simulations.