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Exact local Green function for phonons in a Fibonacci chain: a new real-space renormalisation group approach

An exact real-space renormalisation group (RSRG) approach for the local Green function (LGF) for a quasi-periodic Fibonacci chain has been developed. The method has been used to study the local density of states (LDOS) for phonons, both in the one-site and transfer models. The scale factor in this work is tau 3, tau being the golden ratio. Apart from mapping a Fibonacci chain onto itself, the transformation accounts for the additional constraining symmetry, namely, that the local environment around the site at which the LGF is being found should remain invariant. The Cantor set-like behaviour of the energy spectrum results naturally from the alternative formulation.