Discrete Scale Invariance in Topological Semimetals

TL;DR

This paper reveals discrete scale invariance in topological semimetals, leading to fractal-like bound state spectra and ln B periodic oscillations in magnetoresistivity, observed in ZrTe$_{5}$, TaAs, and Bi.

Contribution

It identifies a novel physical effect of discrete scale invariance in Weyl and Dirac semimetals due to their unique dispersion relations.

Findings

Observation of ln B periodic oscillations in magnetoresistivity.

Detection of discrete scale invariance in three topological semimetals.

Bound state spectra exhibit fractal-like repeating patterns.

Abstract

The discovery of Weyl and Dirac semimetals has produced a number of dramatic physical effects, including the chiral anomaly and topological Fermi arc surface states. We point out that a very different but no less dramatic physical effect is also to be found in these materials: discrete scale invariance. This invariance leads to bound state spectra for Coulomb impurities that repeat when the binding energy is changed by a fixed factor, reminiscent of fractal behavior. We show that this effect follows from the peculiar dispersion relation in Weyl and Dirac semimetals. It is observed when such a material is placed in very strong magnetic field B: there are oscillations in the magnetoresistivity somewhat similar to Shubnikov-de Haas oscillations but with a periodicity in ln B rather than 1/B. These oscillations should be present in other thermodynamic and transport properties. The oscillations have now been seen in three topological semimetals: ZrTe$_{5}$, TaAs, and Bi.