Isospectral But Physically Distinct: Modular Symmetries and their Implications for Carbon Nanotori

TL;DR

This paper reveals that many physically different carbon nanotori share identical electronic spectra due to modular symmetries, significantly reducing the number of spectrally distinct nanotori and impacting their predicted electronic properties.

Contribution

The study uncovers modular symmetries in carbon nanotori that cause spectral equivalences, challenging previous assumptions and extending to cases with magnetic flux.

Findings

Many distinct nanotori have identical band structures due to modular symmetries.

The number of spectrally distinct nanotori is much smaller than the total number of physical configurations.

The fraction of metallic nanotori is approximately three times higher than previously estimated.

Abstract

Recently there has been considerable interest in the properties of carbon nanotori. Such nanotori can be parametrized according to their radii, their chiralities, and the twists that occur upon joining opposite ends of the nanotubes from which they are derived. In this paper, however, we demonstrate that many physically distinct nanotori with wildly different parameters nevertheless share identical band structures, energy spectra, and electrical conductivities. This occurs as a result of certain geometric symmetries known as modular symmetries which are direct consequences of the properties of the compactified graphene sheet. Using these symmetries, we show that there is a dramatic reduction in the number of spectrally distinct carbon nanotori compared with the number of physically distinct carbon nanotori. The existence of these modular symmetries also allows us to demonstrate that many statements in the literature concerning the electronic properties of nanotori are incomplete because they fail to respect the spectral equivalences that follow from these symmetries. We also find that as a result of these modular symmetries, the fraction of spectrally distinct nanotori which are metallic is approximately three times greater than would naively be expected on the basis of standard results in the literature. Finally, we demonstrate that these modular symmetries also extend to cases in which our carbon nanotori enclose non-zero magnetic fluxes.