  Magnetism in recently discovered van der Waals materials has opened new
avenues in the study of fundamental spin interactions in truly two-dimensions.
A paramount question is what effect higher-order interactions beyond bilinear
Heisenberg exchange have on the magnetic properties of few-atom thick
compounds. Here we demonstrate that biquadratic exchange interactions, which is
the simplest and most natural form of non-Heisenberg coupling, assume a key
role in the magnetic properties of layered magnets. Using a combination of
nonperturbative analytical techniques, non-collinear first-principles methods
and classical Monte Carlo calculations that incorporate higher-order exchange,
we show that several quantities including magnetic anisotropies, spin-wave gaps
and topological spin-excitations are intrinsically renormalized leading to
further thermal stability of the layers. We develop a spin Hamiltonian that
also contains antisymmetric exchanges (e.g. Dzyaloshinskii-Moriya interactions)
to successfully rationalize numerous observations currently under debate, such
as the non-Ising character of several compounds despite a strong magnetic
anisotropy, peculiarities of the magnon spectrum of 2D magnets, and the
discrepancy between measured and calculated Curie temperatures. Our results lay
the foundation of a universal higher-order exchange theory for novel 2D
magnetic design strategies.
