Room Temperature Electron-Hole Liquid in quasi-2D monolayer MoS$_2$

Strong correlations between electrons and holes can drive the system to the electron-hole liquid (EHL) state typically at high carrier densities and low temperatures. We study the formation of such a correlated electron-hole liquid state in monolayer MoS$_2$, a quasi-2D system where interactions between charge carriers are governed by the Keldysh potential. We calculate the free energy at finite temperatures using the \textit{linked cluster expansion} method incorporating ring diagrams. The variation of thermodynamic conjugate variables indicate a first-order phase transition with a critical temperature larger than room temperature. We construct the phase diagram in the temperature-density phase space by Maxwell construction of thermodynamically unstable region demarcating phase space regions corresponding to exciton/electron-hole gas, electron-hole liquid and the region of coexistence.