Flux Phases and Competing Orders in a Honeycomb Lattice Model

We investigate correlated electronic phases in a two-dimensional honeycomb lattice with nearest-neighbor ($V_1$) and next-nearest-neighbor ($V_2$) interactions. Using a self-consistent mean-field framework combined with numerical optimization methods, we map out the phase diagram in the $(V_1,V_2)$ plane. Our results reveal a competition between charge-ordered states and flux phases characterized by complex bond order parameters and previously not investigated in the literature. The analysis provides insight into how interaction-driven instabilities emerge in honeycomb systems and offers a broader perspective on correlated phases in materials such as graphene and its moiré superlattices.