Density of states in graphene with vacancies: midgap power law and frozen multifractality

TL;DR

This paper investigates the density of states in graphene with vacancies, revealing a transition from a Gade-type singularity to a power-law behavior at very low energies, indicating complex multifractal phenomena.

Contribution

It provides numerical and analytical evidence for a crossover in the density of states in vacancy-disordered graphene, highlighting unconventional fixed points in bipartite random hopping models.

Findings

Compatibility with Gade-type singularity at intermediate energies

Transition to a power-law divergence at very low energies

Evidence for strong-coupling fixed points in bipartite random hopping models

Abstract

The density of states (DoS), $\varrho(E)$, of graphene is investigated numerically and within the self-consistent T-matrix approximation (SCTMA) in the presence of vacancies within the tight binding model. The focus is on compensated disorder, where the concentration of vacancies, $n_\text{A}$ and $n_\text{B}$, in both sub-lattices is the same. Formally, this model belongs to the chiral symmetry class BDI. The prediction of the non-linear sigma-model for this class is a Gade-type singularity $\varrho(E) \sim |E|^{-1}\exp(-|\log(E)|^{-1/x})$. Our numerical data is compatible with this result in a preasymptotic regime that gives way, however, at even lower energies to $\varrho(E)\sim E^{-1}|\log(E)|^{-\mathfrak{x}}$, $1\leq \mathfrak{x} < 2$. We take this finding as an evidence that similar to the case of dirty d-wave superconductors, also generic bipartite random hopping models may exhibit unconventional (strong-coupling) fixed points for certain kinds of randomly placed scatterers if these are strong enough. Our research suggests that graphene with (effective) vacancy disorder is a physical representative of such systems.