The Coulomb impurity problem in graphene

TL;DR

This paper investigates the effects of an unscreened Coulomb impurity in graphene by comparing exact lattice solutions with continuum Dirac models, revealing both qualitative agreements and lattice-specific phenomena like bound states.

Contribution

It provides a non-perturbative analysis of Coulomb impurities in graphene using both lattice and continuum approaches, highlighting their agreement and differences.

Findings

Dirac model accurately describes low-energy behavior

Lattice solutions show bound states and van Hove singularity renormalization

Continuum and lattice results are qualitatively consistent

Abstract

We address the problem of an unscreened Coulomb charge in graphene, and calculate the local density of states and displaced charge as a function of energy and distance from the impurity. This is done non-perturbatively in two different ways: (1) solving the problem exactly by studying numerically the tight-binding model on the lattice; (2) using the continuum description in terms of the 2D Dirac equation. We show that the Dirac equation, when properly regularized, provides a qualitative and quantitative low energy description of the problem. The lattice solution shows extra features that cannot be described by the Dirac equation, namely bound state formation and strong renormalization of the van Hove singularities.