Field theoretic renormalization study of reduced quantum electrodynamics and applications to the ultra-relativistic limit of Dirac liquids

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 graphene-like 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 BPHZ prescription allows for a simple and clear understanding of the structure of the model. In particular, in relation with the ultra-relativistic limit of graphene, we straightforwardly recover the results for both the interaction correction to the optical conductivity: ${\mathcal{C}}^{\ast }=(92-9{\pi }^2)\mathrm{/}(18\pi )$ and the anomalous dimension of the fermion field: $\gamma_{\psi}(\bar{\alpha},\xi) = 2 \bar{\alpha},(1-3\xi)/3 -16,\left( \zeta_2 N_F + 4/27 \right), \bar{\alpha}^2 + O(\bar{\alpha}^3)$,where$\bar{\alpha} = e^2/(4\pi)^2$and$\xi$ is the gauge-fixing parameter.