  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 the number of parameters of the
anisotropic tensor Heisenberg model to be significantly reduced, 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.
