Renormalization-group method for exact Green’s functions of self-similar lattices: Application to generalized Fibonacci chains

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Abstract
We present a real-space renormalization-group method for calculating the exact local and average Green’s functions for generalized Fibonacci lattices. The fundamental requirement of the present method is that one should be able to split the original lattice into a finite number of sublattices, all of which need not be simple scaled versions of the parrent lattice. In this method we determine exactly the renormalized local environment, up to infinite order, of any arbitrarily chosen site in the original lattice. We have also been able to classify the generalized Fibonacci chains into two groups, depending on their spectral properties. For one of the groups (silver-mean class) eigenstates are shown to be critical, whereas, the other group (copper-mean class) has some states extended. This method is readily applicable to any self-similar lattice in one and higher dimensions.