Emergence of quasiperiodic Bloch wave functions in quasicrystals

TL;DR

This paper investigates how quasiperiodic Bloch wave functions can emerge in quasicrystals, revealing that superpositions of eigenfunctions form extended states and that disorder influences their eigenfunction status, with implications for quasicrystal physics.

Contribution

It introduces a novel understanding of quasiperiodic Bloch functions as superpositions of eigenstates and explores their behavior under disorder in quasicrystals.

Findings

Superpositions of eigenfunctions form quasiperiodic Bloch functions.

An effective crystal momentum characterizes these functions.

Weak disorder can cause quasiperiodic Bloch functions to become eigenfunctions.

Abstract

We study the emergence of quasiperiodic Bloch wave functions in quasicrystals, employing the one-dimensional Fibonacci model as a test case. We find that despite the fact that Bloch functions are not eigenfunctions themselves, superpositions of relatively small numbers of nearly degenerate eigenfunctions give rise to extended quasiperiodic Bloch functions. These functions possess the structure of earlier ancestors of the underlying Fibonacci potential, and it is often possible to obtain different ancestors as different superpositions around the same energy. There exists an effective crystal momentum that characterizes these ancestors, which is determined by the mean energy of the superimposed eigenfunctions, giving rise to an effective dispersion curve. We also find that quasiperiodic Bloch functions do emerge as eigenfunctions when weak disorder is introduced into the otherwise perfect quasiperiodic potential. These theoretical results may explain a number of experimental observations, and may have practical consequences on emerging theories of band topology and correlated electrons in quasicrystals.