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The marked interplay between the crystalline, electronic, and magneticstructure of atomically thin magnets has been regarded as the key feature fordesigning next-generation magneto-optoelectronic devices. In this respect, adetailed understanding of the microscopic interactions underlying the magneticresponses of these crystals is of primary importance. Here, we combine modelHamiltonians with multi-reference configuration interaction wavefunctions toaccurately determine the strength of the spin couplings in the prototypicalsingle-layer magnet CrI$_3$. Our calculations identify the (ferromagnetic)Heisenberg exchange interaction $J = -1.44$ meV as the dominant term, being theinter-site magnetic anisotropies substantially {weaker}. We also find thatsingle-layer CrI$_3$ features an out-of-plane easy axis ensuing from asingle-ion anisotropy $A = -0.10$ meV, and predict $g$-tensor in-planecomponents $g_{xx} = g_{yy} = 1.90$ and out-of-plane component $g_{zz} = 1.92$.In addition, we assess the performance of a dozen widely used densityfunctionals against our accurate correlated wavefunctions {calculations} andavailable experimental data, thereby establishing reference results for futurefirst-principles investigations. Overall, our findings offer a firm theoreticalground to experimental observations.

Magnetic exchange interactions in monolayer CrI$_3$ from many-body wavefunction calculations