Trace map and eigenstates of a Thue-Morse chain in a general model

By the standard method proposed by Kolar and Nori [Phys. Rev. B 42, 1062 (1990)], a rigorous eight-dimensional (8D) trace map for a general model of Thue-Morse (TM) sequences is obtained. Using this trace map, the characteristics of electronic eigenstates in TM lattices are explored in a very broad way. Simultaneously, a constraint condition for energy parameters, under which the complex 8D trace map can be simplified into the ordinary form, is found. It is also proved analytically that all eigenstates of TM lattices are extended when this constraint condition is fulfilled. Furthermore, the properties of eigenstates beyond this constraint are investigated and some wave functions with critical features are discovered by the multifractal analysis. Our results support the previous viewpoint that a TM lattice is an intermediate stage between periodic and Fibonacci structures.