A renormalization group analysis of extended electronic states in 1d quasiperiodic lattices

TL;DR

This paper uses a real-space renormalization group approach to analyze electronic states in 1D quasiperiodic lattices, revealing the existence of extended states due to local correlations even without translational symmetry.

Contribution

It introduces a clustering-based renormalization method to identify extended states in quasiperiodic chains, challenging previous beliefs about localization in such systems.

Findings

Extended states exist in certain quasiperiodic lattices due to local correlations.

The spectrum of the period-doubling chain shows a crossover from critical to extended regimes.

The theory is extended to multi-band models with quasiperiodic arrangements.

Abstract

We present a detailed analysis of the nature of electronic eigenfunctions in one-dimensional quasi-periodic chains based on a clustering idea recently introduced by us [Sil et al., Phys. Rev. {\bf 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 quasi-periodic period-doubling chain, whose spectrum is found to exhibit a rich variety of behaviour, including a cross-over 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 non-vanishing. Results are presented for different parameter regimes, yielding both periodic as well as non-periodic eigenfunctions. We have also extended the present theory to a multi-band model arising from a quasi-periodically arranged array of $\delta$-function potentials on the atomic sites. Finally, we present a multifractal analysis of these wavefunctions following the method of Godreche and Luck [ C. Godreche and J. M. Luck, J. Phys. A :Math. Gen. {\bf 23},