Magnetic Manifestation of Discrete Scaling Symmetry in Dirac Semimetals

TL;DR

This paper investigates how magnetic fields influence the discrete scaling symmetry of quasibound states in Dirac semimetals, revealing that magnetic fields can modify, shift, or destroy these states and their symmetry, with implications for experimental observations.

Contribution

It demonstrates the magnetic field's effect on quasibound states and their discrete scaling symmetry in Dirac semimetals, extending understanding of these phenomena in different dimensions.

Findings

Magnetic fields can shift or destroy quasibound states in 2D Dirac systems.

The discrete scaling symmetry persists in the quantum limit of 3D Dirac semimetals.

Magnetic oscillations reflect the same discrete scaling symmetry as in zero field.

Abstract

Two-dimensional and three-dimensional massless Dirac fermions can form a sequence of quasibound states with an attractive charged impurity. These quasibound states exhibit a discrete scaling symmetry, i.e., the energy ratio between two successive states is a constant. Through the calculation of the energy spectrum directly, we find that in two dimension an applied magnetic field can shift or even destroy the quasibound states around the Dirac point and their discrete scaling symmetry disappears. However, as the magnetic field increases, the remaining quasibound states are pushed up to the Dirac point. When one quasibound state is close to the Dirac point, the spectrum is modified significantly, due to the resonant scattering. The magnetic oscillation of the spectrum displays the same discrete scaling symmetry as the quasibound state does in zero magnetic field. This phenomenon also occurs in the quantum limit of three-dimensional Dirac semimetals, where the system becomes quasi one-dimensional essentially. Our theoretical analysis are in good agreement with the recent experimental observations.