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In the Van der Waal material $\rm NiPS_3$, Ni atoms have spin S=1 and realizea honeycomb lattice. Six sulfur atoms surround each Ni and split their dmanifold into three filled and two unfilled bands. Aimed to determine the spinHamiltonian of $\rm NiPS_3$, we study its exchange mechanisms using a two-bandhalf-filled Hubbard model. Hopping between d orbitals is mediated by p orbitalsof sulfur and gives rise to bilinear and biquadratic spin couplings in thelimit of strong electronic correlations. The microscopic model exposed aferromagnetic biquadratic spin interaction $\rm K_1$ allowing the completion ofa minimal $\rm J_1-J_3-K_1$ spin Hamiltonian for $\rm NiPS_3$. In bulk, aferromagnetic first nearest neighbor $\rm J_1$ and a more significantantiferromagnetic third nearest neighbor spin coupling $\rm J_3$ agreed withthe literature, while in monolayer $\rm J_1$ is positive and very small incomparison. Using a variational scheme we found that a zig-zagantiferromagnetic order is the ground state of bulk samples. The zig-zagpattern is adjacent to commensurate and incommensurate spin spirals, whichcould hint at the puzzling results reported in $\rm NiPS_3$ monolayers.

Spin model for the Honeycomb $\rm NiPS_3$