Valley and spin polarized broken symmetry states of interacting electrons in gated MoS$_2$ quantum dots

Understanding strongly interacting electrons enables the design of materials, nanostructures and devices. Developing this understanding relies on the ability to tune and control electron-electron interactions by, e.g., confining electrons to atomically thin layers of 2D crystals with reduced screening. The interplay of strong interactions on a hexagonal lattice with two nonequivalent valleys, topological moments, and the Ising-like spin-orbit interaction gives rise to a variety of phases of matter corresponding to valley and spin polarized broken symmetry states. In this work we describe a highly tunable strongly interacting system of electrons laterally confined to monolayer transition metal dichalcogenide MoS$_2$ by metalic gates. We predict the existence of valley and spin polarized broken symmetry states tunable by the parabolic confining potential using exact diagonalization techniques for up to $N=6$ electrons. We find that the ground state is formed by one of two phases, either both spin and valley polarized or valley unpolarised but spin intervalley antiferromagnetic, which compete as a function of electronic shell spacing. This finding can be traced back to the combined effect of Ising-like spin-orbit coupling and weak intervalley exchange interaction. These results provide an explanation for interaction-driven symmetry-breaking effects in valley systems and highlight the important role of electron-electron interactions for designing valleytronic devices.