Magnetism and Charge Order in the Honeycomb Lattice.

Despite being relevant to better understand the properties of honeycomblike systems, as graphene-based compounds, the electron-phonon interaction is commonly disregarded in theoretical approaches. That is, the effects of phonon fields on interacting Dirac electrons is an open issue, in particular when investigating long-range ordering. Thus, here we perform unbiased quantum Monte Carlo simulations to examine the Hubbard-Holstein model (HHM) in the half-filled honeycomb lattice. By performing careful finite-size scaling analysis, we identify semimetal-to-insulator quantum critical points, and determine the behavior of the antiferromagnetic and charge-density wave phase transitions. We have, therefore, established the ground state phase diagram of the HHM for intermediate interaction strength, determining its behavior for different phonon frequencies. Our findings provide quantitative and qualitative descriptions of the model at intermediate coupling strengths, and may shed light on the emergence of many-body properties in honeycomblike systems.