Renormalization transformation of periodic and aperiodic lattices

In this work we introduce a similarity transformation acting on transfer matrices describing the propagation of elementary excitations through either periodic or Fibonacci lattices. The proposed transformation can act at two different scale lengths. At the atomic scale the transformation allows one to express the systems' global transfer matrix in terms of an equivalent on-site model one. Correlation effects among different hopping terms are described by a series of local phase factors in that case. When acting on larger scale lengths, corresponding to short segments of the original lattice, the similarity transformation can be properly regarded as describing an effective renormalization of the chain. The nature of the resulting renormalized lattice significantly depends on the kind of order (i.e., periodic or quasiperiodic) of the original lattice, expressing a delicate balance between chemical complexity and topological order as a consequence of the renormalization process