Phase diagram of the interacting Haldane model with spin-dependent sublattice potentials

Using the exact-diagonalization (ED) and mean-field (MF) approaches, we investigate the ground-state phase diagram of the interacting Haldane model on the honeycomb lattice, incorporating spin-dependent sublattice potentials Δ_σ,α. Here α=A,B and σ=↑,↓ denote the sublattice and spin components, respectively. Setting Δ_σ,A=+Δ (-Δ) and Δ_σ,B=-Δ (+Δ) for σ=↑ (↓) results in the system favoring a spin ordered state. Conversely, introducing the nearest-neighbor Coulomb interaction can induce charge ordering in the system. Due to the competition between these factors, we observe that in both ED and MF approaches, an exotic state with Chern number $C=1$ survives amidst two locally ordered phases and a topologically ordered phase with $C=2$. In the ED method, various properties, such as the fidelity metric, the excitation gaps and the structure factors, are employed to identify critical points. In the MF method, using a sufficiently large lattice size, we define the local order parameters and band gaps to characterize the phase transitions. The interacting Haldane model and the spin-dependent lattice potential may be experimentally realized in an ultracold atom gas, providing a potential means to detect this intriguing state.