Coulomb interactions and renormalization of semi-Dirac fermions near a topological Lifshitz transition

We aim to understand how the spectrum of semi-Dirac fermions is renormalized due to long-range Coulomb electron-electron interactions at a topological Lifshitz transition where two Dirac cones merge. At the transition, the electronic spectrum is characterized by massive quadratic dispersion in one direction, whereas it remains linear in the other. We have found that, to lowest order, the unconventional ${\mathrm{log}}²$ (double logarithmic) correction to the quasiparticle mass in bare perturbation theory leads to resummation into strong mass renormalization in the exact full solution of the perturbative renormalization-group equations. This behavior effectively wipes out the curvature of the dispersion and leads to Dirac cone restoration at low energy: The system flows towards Dirac dispersion which is anisotropic but linear in momentum with interaction-depended logarithmic modulation. The Berry phase associated with the restored critical Dirac spectrum is zero---a property guaranteed by time-reversal symmetry and unchanged by renormalization. Our results are in contrast with the behavior that has been found within the large-$N$ approach.