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Electronic correlations in monolayer VS$_2$

The layered transition metal dichalcogenide vanadium disulfide (VS$_2$), which nominally has one electron in the $3d$ shell, is potent for strong correlation physics and is possibly another realization of an effective one-band model beyond the cuprates. Here monolayer VS$_2$ in both the trigonal prismatic and octahedral phases is investigated using density functional theory plus Hubbard $U$ (DFT+$U$) calculations. Trigonal prismatic VS$_2$ has an isolated low-energy band that emerges from a confluence of crystal field splitting and direct V--V hopping. Within spin density functional theory, ferromagnetism splits the isolated band of the trigonal prismatic structure, leading to a low-band-gap $S=1/2$ ferromagnetic Stoner insulator; the octahedral phase is higher in energy. Including the on-site interaction $U$ increases the band gap, leads to Mott insulating behavior, and for sufficiently high values stabilizes the ferromagnetic octahedral phase. The validity of DFT and DFT+$U$ for these two-dimensional materials with potential for strong electronic correlations is discussed. A clear benchmark is given by examining the experimentally observed charge density wave (CDW) in octahedral VS$_2$, for which DFT grossly overestimates the bond length differences compared to known experiments; the presence of CDWs is also probed for the trigonal prismatic phase. Finally, we investigate why only the octahedral phase has been observed in experiments and discuss the possibility of realizing the trigonal prismatic phase. Our work suggests trigonal prismatic VS$_2$ is a promising candidate for strongly correlated electron physics that, if realized, could be experimentally probed in an unprecedented fashion due to its monolayer nature.