Hidden dimers and the matrix maps: Fibonacci chains revisited

Abstract

The existence of cycles of the matrix maps in Fibonacci class of lattices is well established. We show that such cycles are intimately connected with the presence of interesting positional correlations among the constituent ``atoms'' in a one-dimensional quasiperiodic lattice. We particularly address the transfer model of the classic golden mean Fibonacci chain where a six cycle of the full matrix map exists at the center of the spectrum [Kohmoto et al., Phys. Rev. B 35, 1020 (1987)]. In addition, we show that our prescription leads to a determination of other energy values for a mixed model of the Fibonacci chain, for which the full matrix map may have similar cyclic behavior. Apart from the standard transfer model of a golden mean Fibonacci chain, we address a variant of it and the silver mean lattice, where four cycles of the matrix map is already known to exist. The underlying positional correlations for all such cases are discussed in detail.