Properties of the Thue-Morse Chain

Abstract

In recent years there has been a great progress in the theory of elementary excitations in non-periodic systems. In particular, electronic states in quasicrystals are peculiar because they seem to be neither localized, nor extended, but critical. That is the case of the Fibonacci chain. Of course, localization of states due to disorder in one dimension is ill-defined, since a chain is infinitely unstable facing disorder. However, there are other non-periodic one-dimensional systems that are constructed by inflation rules that are not quasiperiodic, and definitely neither disordered. A good example of these interesting systems is the Thue-Morse chain. The electronic tight-binding spectra of finite Thue-Morse chains show bands of perfectly extended states, together with a fractal distribution of gaps, whose edges present truly localized states. In this work we examine this spectrum exactly, in the limit of infinite chains, where there are dramatic changes to this picture.