Bound States in Graphene

TL;DR

This paper analyzes the quantum behavior of charge impurities in graphene, revealing how strong interactions lead to bound states and spectral instabilities, with implications for understanding graphene's electronic properties.

Contribution

It provides a detailed quantum analysis of impurity states in graphene, including the effects of supercritical charges and renormalization group flow, which were not previously fully understood.

Findings

Spectral quantization occurs when the impurity charge exceeds a critical value.

Bound states with imaginary energies appear in the subcritical regime.

Renormalization drives the Coulomb interaction to its critical strength.

Abstract

We present a quantum analysis of the massless excitations in graphene with a charge impurity. When the effective charge exceeds a certain critical value, the spectrum is quantized and is unbounded from below. The corresponding eigenstates are square-integrable at infinity and have a rapidly oscillatory behaviour in the short distance, which can be interpreted as a fall to the centre. Using a cutoff regularization, we show that the effective Coulomb interaction strength is driven to its critical value under the renormalization group flow. In the subcritical region, we find bound states with imaginary values of the energy for certain range of the system parameters. The physical significance of these bound states with imaginary eigenvalues is discussed.