Higher-order renormalization of graphene many-body theory

TL;DR

This paper demonstrates that the many-body theory of graphene with Coulomb interactions is renormalizable in the ladder approximation, allowing for consistent calculations of physical quantities and identifying a critical interaction strength for chiral symmetry breaking.

Contribution

It provides a detailed proof of the renormalizability of graphene's many-body theory beyond the large-N approximation using ladder diagrams, and computes the critical coupling for gap opening.

Findings

Graphene many-body theory is renormalizable in the ladder approximation.

A scale-invariant anomalous dimension is computed to all orders.

The critical coupling for chiral symmetry breaking matches previous gap equation results.

Abstract

We study the many-body theory of graphene Dirac quasiparticles interacting via the long-range Coulomb potential, taking as a starting point the ladder approximation to different vertex functions. We test in this way the low-energy behavior of the electron system beyond the simple logarithmic dependence of electronic correlators on the high-energy cutoff, which is characteristic of the large-N approximation. We show that the graphene many-body theory is perfectly renormalizable in the ladder approximation, as all higher powers in the cutoff dependence can be absorbed into the redefinition of a finite number of parameters (namely, the Fermi velocity and the weight of the fields) that remain free of infrared divergences even at the charge neutrality point. We illustrate this fact in the case of the vertex for the current density, where a complete cancellation between the cutoff dependences of vertex and electron self-energy corrections becomes crucial for the preservation of the gauge invariance of the theory. The other potentially divergent vertex corresponds to the staggered (sublattice odd) charge density, which is made cutoff independent by a redefinition in the scale of the density operator. This allows to compute a well-defined, scale invariant anomalous dimension to all orders in the ladder series, which becomes singular at a value of the interaction strength marking the onset of chiral symmetry breaking (and gap opening) in the Dirac field theory. The critical coupling we obtain in this way matches with great accuracy the value found with a quite different method, based on the resolution of the gap equation, thus reassuring the predictability of our renormalization approach.