  Despite serious effort, the nature of the magnetic interactions and the role
of electron-correlation effects in magnetic two-dimensional (2D) van der Waals
materials remain elusive. Using CrI$_3$ as a model system, we show that the
calculated electronic structure including nonlocal electron correlations yields
spin excitations consistent with inelastic neutron scattering measurements.
Remarkably, this approach identifies an unreported correlation-enhanced
interlayer super-superexchange, which rotates the magnon Dirac lines off, and
introduces a gap along, the high-symmetry $\Gamma$-$K$-$M$ path. This discovery
provides a different perspective on the gap opening mechanism observed in
CrI$_3$, which was previously associated with spin-orbit coupling through the
Dzyaloshinskii-Moriya interaction or Kitaev interaction. Our observation
elucidates the critical role of electron correlations on the spin ordering and
spin dynamics in magnetic van der Waals materials and demonstrates the
necessity of explicit treatment of electron correlations in the broad family of
2D magnetic materials.
