Hubbard interaction at finite $T$ on a hexagonal lattice

The temporal finite volume induces significant effects in Monte Carlo simulations of systems in low dimensions, such as graphene, a 2-D hexagonal system known for its unique electronic properties and numerous potential applications. In this work, we explore the behavior of fermions on a hexagonal sheet with a Hubbard-type interaction characterized by coupling $U$. This system exhibits zero or near zero-energy excitations that are highly sensitive to finite temperature effects. We compute corrections to the self-energy and the effective mass of low-energy excitations, arriving at a quantization condition that includes the temporal finite volume. These analyses are then conducted for both zero and finite temperatures. Our findings reveal that the first-order $\mathcal{O}(U)$ contributions are absent, leading to non-trivial corrections starting at $\mathcal{O}(U^2)$. We validate our calculations against exact and numerical results obtained from Hybrid Monte Carlo simulations on small lattices.