  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.
