  In this work, we study topological properties of magnons via creating spin
excitations in both ferromagnets and antiferromagnets in presence of an
external magnetic field on a two-dimensional square lattice. It is known that
Dzyaloshinskii-Moriya interaction (DMI) plays an important role in coupling
between different particle (spin excitation) sectors, here we consider an
anisotropic DMI and ascertain the role of the anisotropy parameter in inducing
topological phase transitions. While the scenario, for dealing with
ferromagnets, albeit with isotropic DMI is established in literature, we have
developed the formalism for studying magnon band topology for the
antiferromagnetic case. The calculations for the ferromagnetic case are
included to facilitate a comparison between the two magnetically ordered
systems. Owing to the presence of a two-sublattice structure of an
antiferromagnet, a larger number of magnon bands participate in deciding upon
the topological properties. However, in both the cases, an extended trivial
region is observed even with the DMI to be non-zero, which is surprising since
the DMI is the origin of the finite Berry curvature in presence of external
magnetic field. Furthermore, in an antiferromagnet, a smaller anisotropy is
capable of inducing a gap-closing transition from a topological to a trivial
phase compared to that for a ferromagnet. The nature of the phases in both the
cases and the phase transitions therein are characterized by the band
structure, presence (or absence) of the chiral edge modes observed in a
semi-infinite nano-ribbon geometry, computation of the thermal Hall effect,
etc. Moreover, the strength of the magnetic field is found to play a decisive
role in controlling the critical point that demarcates various topological
phases.
