Renormalization-group methods for the spectra of disordered chains

A family of real-space renormalization techniques for calculating the Green's functions of disordered chains is developed and explored. The techniques are based on a recently proposed renormalization method which is rederived here and shown to be equivalent to a virtual-crystal approximation on a renormalized Hamiltonian. The derivation suggests how other conventional alloy methods can be coupled to the renormalization concept. Various examples are discussed. Short-range order in the occupation of alloy sites and very general disorder in the Hamiltonian—diagonal, off-diagonal, and environmental—are readily incorporated. The techniques are exact in the limits of high and low concentration and of complete short-range order and for the Lloyd model. All states are found to be localized, in agreement with exact treatments. Results for the alloy density of states are presented for various cases and compared to numerical simulations on long chains (${10}^5$ atoms).