Renormalization group analysis of graphene with a supercritical Coulomb impurity

TL;DR

This paper develops a field-theoretic renormalization group approach to analyze supercritical Coulomb impurities in graphene, revealing discrete scale invariance and log-periodic behavior in scattering and induced charge densities.

Contribution

It introduces a novel RG analysis with an Aharonov-Bohm solenoid to study supercritical Coulomb impurities in graphene, demonstrating limit cycles and discrete scale invariance.

Findings

Discovery of RG limit cycle behavior in scattering amplitudes

Power-law tails in induced charge and current densities with log-periodic coefficients

Consistency with previous Dirac equation solutions and experimental feasibility

Abstract

We develop a field-theoretic approach to massless Dirac fermions in a supercritical Coulomb potential. By introducing an Aharonov--Bohm solenoid at the potential center, the critical Coulomb charge can be made arbitrarily small for one partial-wave sector, where a perturbative renormalization group analysis becomes possible. We show that a scattering amplitude for reflection of particle at the potential center exhibits the renormalization group limit cycle, i.e., log-periodic revolutions as a function of the scattering energy, revealing the emergence of discrete scale invariance. This outcome is further incorporated in computing the induced charge and current densities, which turn out to have power-law tails with coefficients log-periodic with respect to the distance from the potential center. Our findings are consistent with the previous prediction obtained by directly solving the Dirac equation and can in principle be realized by graphene experiments with charged impurities.