Fourth order Heisenberg models with minimal number of parameters for two-dimensional magnetic crystals

In this work we investigated adequacy of the Heisenberg model application to novel two-dimensional magnetic materials, on an example of monolayer CrI3. We introduced the concept of the mean tensor invariant under symmetry operations of the magnetic structure, which allows to add fourth-order terms to the anisotropic Heisenberg model and simultaneously to significantly reduce the number of independent parameters, while maintaining the compliance with the results of ab-initio calculations. We derived the expressions for fourth-order corrections to Heisenberg Hamiltonian and to Dzyaloshinskii-Moriya interaction in the form of quartic symmetry invariants with minimal number of parameters. We tested the physical adequacy of such approach in the case of monolayer CrI3, utilizing an alternative to four-states energy mapping - the all-parameters least square fit.