We use a variational technique to study the many-body phase diagram of electrons and holes in $n$-doped and $p$-doped monolayer transition metal dichalcogenides (TMDs). We find a total of four different phases. (i) A fully spin polarized and valley polarized ferromagnetic state. (ii) A state with no global spin polarization but with spin polarization in each valley separately, i.e., spin-valley locking. (iii) A state with spin polarization in one of the valleys and little to no spin polarization in the other valley. (iv) A paramagnetic state with no valley polarization. These phases are separated by first-order phase transitions and are determined by the particle density and the dielectric constant of the substrate. We find that in the presence of a perpendicular magnetic field the four different phases persist. In the case of $n$-doped ${\mathrm{MoS}}_2$, a fifth phase, which is completely valley polarized but not spin polarized, appears for magnetic fields larger than 7 T and for magnetic fields larger than 23 T completely replaces the second phase.

• Received 13 July 2018
• Revised 3 September 2018

DOI:https://doi.org/10.1103/PhysRevB.98.115432