Strong intervalley correlation induced a magnetic order transition in monolayer $\text{MoS}_{2}$

In this work, we study a model for monolayer molybdenum disulfide with including the intravalley and intervalley electron-electron interaction. We solve the model at a self-consistent mean-field level and get three solutions $L_{0}$, $L_{+}$ and $L_{-}$. As for $L_{0}$, the spin polarizations are opposite at $\textbf{K}$ and $\textbf{K}^{\prime}$ valley and the total magnetization is zero. $L_{\pm}$ describe two degenerate spin-polarized states, and the directions of polarization are opposite for the states of $L_{+}$ and $L_{-}$. Based on these results, the ground state can be deduced to be spin polarized in domains in which their particular states can be randomly described by $L_{+}$ or $L_{-}$. Therefore, a zero net magnetization is induced for zero external magnetic field $\mathbf{B}$, but a global ferromagnetic ground state for a nonzero $\mathbf{B}$. We estimate the size of domains as several nanometers. As the increase of the chemical potential, the ground state changes between $L_{0}$ and $L_{\pm}$, indicating first order phase transitions at the borders, which is coincident with the observation of photoluminescence experiments in the absence of the external magnetic field [J. G. Roch $\it{et}$ $\it{al.}$, Phys. Rev. Lett. ${\bf 124}$, 187602 (2020)].