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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.

The calculation of the static and dynamic structure factors of a dimerized Fibonacci chain are presented. This lattice has four kinds of bonds with an underlying quasiperiodicity, but does not preserve the scale invariance like the undistorted Fibonacci chain. This lattice is of interest from the point of view of possible Peierls' distortion in the presence of electron-phonon interactions. To calculate the static structure factor we use the ``cut and projection'' method, while a proposed numerical renormalization-group algorithm has been utilized to find the dynamic one. Our proposed algorithm is, in principle, exact, and can be implemented to any arbitrary degree of accuracy.

We present a detailed analysis of the nature of electronic eigenfunctions in one-dimensional quasiperiodic chains based on a clustering idea recently introduced by us [Sil et al., Phys. Rev. B 48, 4192 (1993)], within the framework of the real-space renormalization-group approach. It is shown that even in the absence of translational invariance, extended states arise in a class of such lattices if they possess a certain local correlation among the constituent atoms. We have applied these ideas to the quasiperiodic period-doubling chain, whose spectrum is found to exhibit a rich variety of behavior, including a crossover from critical to an extended regime, as a function of the Hamiltonian parameters. Contrary to prevailing ideas, the period-doubling lattice is shown to support an infinity of extended states, even though the polynomial invariant associated with the trace map is nonvanishing. Results are presented for different parameter regimes, yielding both periodic as well as nonperiodic eigenfunctions. We have also extended the present theory to a multiband model arising from a quasiperiodically arranged array of \ensuremath{\delta}-function potentials on the atomic sites. Finally, we present a multifractal analysis of these wave functions following the method of Godreche and Luck [J. Phys. A 23, 3769 (1990)] to confirm their extended character.

A new type of correlated disorder is shown to be responsible for the appearance of extended electronic states in one-dimensional aperiodic systems like the Thue-Morse lattice. Our analysis leads to an understanding of the underlying reason for the extended states in this system, for which only numerical evidence is available in the literature so far. The present work also sheds light on the restrictive conditions under which the extended states are supported by this lattice., Comment: 11 pages, LaTeX V2.09, 1 figure (available on request), to appear in Physical Review Letters

We discuss a very simple model of a 1-d disordered lattice, in which {\em all} the electronic eigenstates are extended. The nature of these states is examined from several viewpoints, and it is found that the eigenfunctions are not Bloch functions although they extend throughout the chain. Some typical wavefunctions are plotted. This problem originated in our earlier study of extended states in the quasiperiodic copper-mean lattice [ Sil, Karmakar, Moitra and Chakrabarti, Phys. Rev. B (1993) ]. In the present investigation extended states are found to arise from a different kind of correlation than that of the well-known dimer-type., Comment: 9 pages, 1 figure available on request, LaTex version 2.09, SINP-SSMP93-02

We present a real-space renormalization-group method for evaluating the exact dynamic structure factor S (q,ω) of a quasiperiodic Fibonacci chain. Contrary to earlier work that takes account only of the global aspects of the symmetry of the chain, our method additionally takes care of the local environmental aspects of the symmetry by separating the original lattice into a finite number of self-similar interpenetrating sublattices, followed by elimination of the coupling between them. Our method also yields correctly the positions of the Bragg peaks of the Fibonacci chain

The cluster-Bethe-lattice method is used to study the environment effects on the electronic density of states in narrow-band alloys. The intra-atomic Coulomb interaction, U, is treated in the statistical (0, 1) approximation and the coherent potential approximation is used for the alloy disorder. These approximations lead us to consider the alloy system as a quaternary alloy. A general approach for an n-component alloy in the Bethe lattice is presented using Yonezawa's approach. It is shown that the result obtained gives the result of Yonezawa in the limit z ∞, z being the coordination number of the alloy system. The quaternary alloy is then treated as a special case of an n-component alloy. A cluster consisting of (Z + 1) atoms is chosen to study environment effects. A comparison is made between the electronic density of states for the Bethe lattice and that for the cluster-Bethe lattice. Interesting features in the electronic density of states are observed due to clustering effects. Die Clustermethode von Bethe wird benutzt, um Einflusse der Umgebung auf die Elektronenzu-standsdichte in schmalbandigen Legierungen zu untersuchen. Die intraatomare Coulombwechsel-wirkung U wird in der statistischen (0.l)-Naherung behandelt und die koharente Potential-Approximation wird fur die Legierungsunordnung benutzt. Diese Naherungen fuhren dam, das Legierungssystem als quaternare Legierung zu betrachten. Ein allgemeines Verfahren fur eine n-komponentige Legierung im Bethe-Gitter wird mit der Methode von Yonezawa angegeben. Es wird gezeigt, das das erhaltene Ergebnis das Resultat von Yonezawa im Grenzfall z ∞ wieder-gibt, wobei z die Koordinationszahl des Legierungssystems ist. Die quaternare Legierung wird dann als Spezialfall einer n-komponentigen Legierung behandelt. Ein Cluster aus (Z + 1)-Atomen wird gewahlt, um die Umgebungseffekte zu untersuchen. Ein Vergleich zwischen der Elektro-nenzustandsdichte fur das Bethe-Gitter und das Cluster-Bethe-Gitter wird durchgefuhrt. Interes-sante Einzelheiten in der Elektronenzustandsdichte infolge der Clustereffekte werden beobachtet.

The authors propose a generalization of the 'cluster-Bethe lattice' approach of Falicov and Yndurain for the electronic density of states in binary alloys. In the present formulation it is possible to treat the Bethe lattice in any desired single-site approximation. Further, they obtain algebraic forms for the transfer matrices corresponding to the two types of atoms in the cluster, for which only interpolation formulae were previously given by Falicov and Yndurain (see Phys. Rev. B, vol.12, p.5664 (1975)).