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We address the problem of electronic transmission through a finite sequence of identical potential barriers in one dimension, which has embedded in it “correlated impurities” that is, smaller identical clusters of n-mers, each described by n identical impurity potentials. Analytic formulae are derived for the transmission coefficient, and the investigation of Dunlap et al. [Phys. Rev. Lett. 65 (1990) 88], which were carried out for the corresponding dimer problem in the tight-binding model, are re-examined and generalised. The dependence of the condition for perfect transmission is obtained as a function of the strength of the impurity potential, treated here as a control parameter, in the case of a sequence of δ-function potentials.

We re-examine the conventional idea of determining the nature of the electronic eigenfunctions (extended, critical or localised) of a Fibonacci lattice from a study of the invariant associated with the trace map. We demonstrate that this is insufficient, and a more detailed study of the renormalisation group transformation itself is required to ascertain the nature of the eigenfunctions. Suitable examples are provided.

Exact analytic formulae are obtained for the electronic structure of an ordered binary alloy in a Bethe lattice described by a tight-binding hamiltonian that includes nearest neighbour and second nearest neighbour interactions. Some interesting results are obtained and discussed in the text.

The Anderson hamiltonian has been studied for arbitrarily filled bands using the real space renormalization group (RSRG) method. The ground state energy is determined in one dimension for the scale factors ns = 3 and ns = 5. Direct application of RSRG gives, for a certain ns, the results for only ns number of discrete cases of band-filling. We then devise an interpolation scheme which spans the entire region of the band and gives the ground state energy for any general filling. The results, as provided by our scheme, satisfy the desired particle-hole symmetry exactly and are always an upper bound to the exact one. The non-linear dependence of the band energy on its filling is represented correctly, even for small ns. Finally, we discuss how the results improve for large ns.

We present here a new treatment of short-range order in a disordered binary alloy with Bethe lattice topology based on the single-site approximation. We show that our method yields the limits of segregation and binary sequence exactly, while it reduces to the CPA in the case of random order.

A new theory for incorporating short range order (SRO) in the cluster Bethe lattice method (CBLM) for disordered binary alloys is proposed. It is applied to determine the condition for the appearance of moments in the disordered phase of a narrow-band metal for general values of concentration and SRO parameter. The final result (i) yields all the known limiting conditions exactly and (ii) displays in a transparant form the SRO dependence, indicating the merit of the present approach.