Field theoretic renormalization study of reduced quantum electrodynamics and applications to the ultrarelativistic limit of Dirac liquids

Abstract

The field theoretic renormalization study of reduced quantum electrodynamics (QED) is performed up to two loops. In the condensed matter context, reduced QED constitutes a very natural effective relativistic field theory describing (planar) Dirac liquids, e.g., graphene and graphenelike materials, the surface states of some topological insulators, and possibly half-filled fractional quantum Hall systems. From the field theory point of view, the model involves an effective (reduced) gauge field propagating with a fractional power of the d'Alembertian in marked contrast with usual QEDs. The use of the Bogoliubov-Parasiuk-Hepp-Zimmermann prescription allows for a simple and clear understanding of the structure of the model. In particular, in relation with the ultrarelativistic limit of graphene, we straightforwardly recover the results for both the interaction correction to the optical conductivity ${\mathcal{C}}^{*}=(92\ensuremath{-}9{\ensuremath{\pi}}^{2})/(18\ensuremath{\pi})$ and the anomalous dimension of the fermion field ${\ensuremath{\gamma}}_{\ensuremath{\psi}}(\overline{\ensuremath{\alpha}},\ensuremath{\xi})=2\overline{\ensuremath{\alpha}}(1\ensuremath{-}3\ensuremath{\xi})/3\ensuremath{-}16({\ensuremath{\zeta}}_{2}{N}_{F}+4/27){\overline{\ensuremath{\alpha}}}^{2}+\mathrm{O}({\overline{\ensuremath{\alpha}}}^{3})$, where $\overline{\ensuremath{\alpha}}={e}^{2}/(4\ensuremath{\pi}{)}^{2}$ and $\ensuremath{\xi}$ is the gauge-fixing parameter.