We report on the status of ongoing Hybrid-Monte-Carlo simulations of the tight-binding model of mono-layer graphene. We present results concerning the semimetal-insulator phase transition, whereby two-body interactions are modeled by a partially screened Coulomb potential which takes into account screening by electrons in the lower $sigma$-orbitals. We obtain evidence that finite-size effects may still be present in the current estimate of the critical coupling strength $alpha_C$, which was previously extracted from simulations on lattice-sizes up to $N_x=N_y=18$. We also present preliminary results concerning the Neck-disrupting Lifshitz transition which occurs at finite Fermion-density in the limit of vanishing two-body interactions. A sign-problem is circumvented by using a spin-dependent chemical potential in our simulations.
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