Metallic mean quasicrystals and their topological invariants

TL;DR

This paper derives the complete set of topological invariants for a family of one-dimensional metallic mean quasicrystals, linking their properties to quantum Hall systems and confirming the invariants through numerical edge state analysis.

Contribution

It introduces a gap labeling scheme for metallic mean quasicrystals and extends the understanding of their topological invariants in the quasiperiodic limit.

Findings

The scheme correctly predicts winding numbers of edge states.

Numerical results confirm the topological invariants across different quasicrystals.

Analysis of the Hofstadter butterfly reveals Landau level analogues.

Abstract

Topological invariants govern many important physical properties in condensed matter systems. In this work, we obtain the complete set of topological invariants for a family of one-dimensional quasicrystals. The first and best-studied member of the family is the Fibonacci chain, while the successive ones are known in the literature as silver, bronze... and collectively as the metallic mean chains. By considering rational approximants, and by making use of the relationship between these chains and two dimensional Quantum Hall problems, we write down a gap labeling scheme for finite systems, and extend it to the quasiperiodic limit. We show, by numerical computations on open chains, that the proposed scheme correctly yields the winding numbers of edge states in each of the gaps, in all of the quasicrystals. In the strict 1D limit, we discuss properties of a simplified Hofstadter ``butterfly" diagram, with the analogues of Landau levels appearing in the asymptotic limit.