Metallic mean quasicrystals and their topological invariants

TL;DR

The paper addresses topological invariants in a family of 1D metallic mean quasicrystals by mapping finite periodic approximants to 2D Quantum Hall problems and establishing a gap-labeling framework. It introduces the metallic mean chain family defined by $\omega_n$ solving $\omega^2=n\omega+1$, with substitution-based approximants and a cut-and-project construction, to derive a unified topological description. The main result is a complete set of gap labels $I_n(j)=p_j+q_j\frac{P^{(k)}_n}{Q^{(k)}_n}$, with a large-$n$ asymptotic form $I_n(q)=\mathrm{Mod}[q \frac{\omega_n}{1+\omega_n},1]$, and numerical verification via edge-state winding in open chains. This work provides a global topological framework for the metallic mean family, enabling robust edge transport features and predicting Landau-like levels at the Hofstadter butterfly corners in the strict 1D limit.

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.

Figures (3)

• Figure 1: The energy spectra $E$ (in units of $t$) of the first three metallic mean chains, for small k (values of $n$ and $k$ as indicated in each figure). Bands are shown in grey, gaps in white. The absolute values of the gap index are indicated in blue within each of the gaps. For reasons of symmetry only the lower half of the spectrum is shown.
• Figure 2: A Hofstadter "butterfly" diagram for the 1D hopping models of the metallic mean chains Eq.\ref{['eq:fhham']}. The energy spectra for several metallic mean chains (between $n=1$ and $n=40$) are plotted against $\Omega_n$ (see text for definition). Each of the individual spectra (series of vertical dots) are Cantor sets in the limit $k\rightarrow \infty$, but for this figure, they were numerically computed for large approximants, with $t_A=1, t_B=0.25$, under periodic boundary conditions. Band edges are outlined in grey, and the four main gaps $q=\pm 1$ are outlined in red. At the corners, levels group into levels which evolve linearly as a function of flux, like the Landau levels of the Hofstadter butterfly.
• Figure 3: Plots of the lower half of the energy spectrum of an open 1D silver mean approximant system of $N=17$ sites, showing the evolution of energies as the phason angle $\theta$ takes 17 discrete values between 0 and $2\pi$. Finite system of 8 unit cells was considered, with $t_A=0.7t,t_B=t$. The energies E is given in units of $t$. The lines are a only a guide to the eye, to help see the distinction between bulk states and edge states. The number of crossings of the energy of edge states in each gap can be seen to correspond to the gap label $q$ for this $n=2, k=4$ approximant chain.