The influence of a weak magnetic field in the Renormalization-Group functions of (2 + 1)-dimensional Dirac systems

The experimental observation of the renormalization of the Fermi velocity v _F as a function ofdoping has been a landmark for confirming the importance of electronic interactions ingraphene. Although the experiments were performed in the presence of a perpendicularmagnetic field B, the measurements are well described by arenormalization-group (RG) theory that did not include it. Here we clarify this issue, forboth massive and massless Dirac systems, and show that for the weak magnetic fields atwhich the experiments are performed, there is no change in the renormalization-groupfunctions. Our calculations are carried out in the framework of the Pseudo-quantumelectrodynamics (PQED) formalism, which accounts for dynamical interactions. We includeonly the linear dependence in B, and solve the problem using two differentparametrizations, the Feynman and the Schwinger one. We confirm the results obtainedearlier within the RG procedure and show that, within linear order in the magnetic field,the only contribution to the renormalization of the Fermi velocity for the massive casearises due to electronic interactions. In addition, for gapped systems, we observe arunning of the mass parameter.