\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
We investigate the ground-state properties of a monolayer MoS$_2$ incorporating the Coulomb interaction together with a short-range inter-valley interaction between charge particles among two valleys within the Hartree-Fock approximation. We consider four variables as independent parameters, namely, homogenous charge (electron or hole) density, the averaged dielectric constant, the spin degree of freedom and finally the Hubbard repulsion coefficient which mostly originates from $4d$ orbits of MoS$_2$ atoms. We find the electronic charge compressibility within the mean-field approximation and show that a non-monotonic behavior of the compressibility as a function of carrier density which is rather different from those of the two-dimensional electron gas. We also explore a paramagnetic-to-ferromagnetic quantum phase transition for the wide range of the electron density in the parameter space.
\end{document}