Evolution with Magnetic Field of Discrete Scale Invariant Supercritical States in Graphene

TL;DR

This paper studies how magnetic fields influence supercritical Coulomb impurity states in graphene, revealing signatures of discrete scale invariance that can be experimentally observed through scanning tunneling microscopy.

Contribution

It demonstrates the magnetic field-induced transition of supercritical states to subcritical states and identifies observable signatures of discrete scale invariance in graphene.

Findings

Supercritical states become subcritical with increasing magnetic field.

Local density of states shows signatures of discrete scale invariance.

Experimental detection is feasible with current STM technology.

Abstract

We investigate the quasi-bound states of a Coulomb impurity in graphene in the presence of a magnetic field. \ These states exhibit the dramatic and rather rare property of discrete scale invariance when the Coulomb potential is supercritical. \ We show using both Wentzel-Kramers-Brillouin (WKB) approximation and numerical studies that the supercritical states are converted to subcritical states as the field is increased. \ The local density of states is calculated and it shows direct signatures of discrete scale invariance. \ In a magnetic field, these signatures are gradually destroyed in a systematic way. \ Hence the effect that we propose can be detected via scanning tunneling microscope experiments. \ The range of magnetic field and energy resolution required are compatible with existing experimental setups. \ These experiments can be performed in a single sample by changing the field; they do not involve changing the nuclear charge.