Localized fermions on the triangular lattice with Ising-like interactions

The model of localized fermions on the triangular lattice is analyzed in means of the Monte Carlo simulations in the grand canonical ensemble. The Hamiltonian of the system has a form of the extended Hubbard model (at the atomic limit) with nearest-neighbor Ising-like magnetic $J$ interactions and onsite Coulomb $U$ interactions. The model is investigated for both signs of $J$, arbitrary $U$ interaction and arbitrary chemical potential $\mu$ (or, equivalently, arbitrary particle concentration $n$). Based on the specific heat capacity and sublattice magnetization analyses, the phase diagrams of the model are determined. For ferromagnetic case ($J<0$), the transition from the ordered phase (which is a standard ferromagnet and can be stable up to $k_{B}T/|J| \approx 0.61$) is found to be second-order (for sufficiently large temperatures $k_{B}T/|J| \gtrsim 0.2$) or first-order (for $-1<U/|J|<-0.65$ at the half-filling, i.e., $n=1$). In the case of $J>0$, the ordered phase occurs in a range of $-1/2<U/|J|<0$ (for $n=1$), while for larger $U$ the state with short-range order is also found (also for $n \neq 1$). The ordered phase is characterized by an antiferromagnetic arrangement of magnetic moments in two sublattices forming the hexagonal lattice. The transition from this ordered phase, which is found also for $\mu \neq 0$ ($n \neq 1$) and $U/|J|>-1/2$ is always second-order for any model parameters. The ordered phase for $J>0$ can be stable up to $k_{B}T/|J| \approx 0.06$.