  We study the orbital angular momentum of magnons for collinear ferromagnet
(FM) and antiferromagnetic (AF) systems with nontrivial networks of exchange
interactions. The orbital angular momentum of magnons for AF and FM zig-zag and
honeycomb lattices becomes nonzero when the lattice contains two inequivalent
sites and is largest at the avoided-crossing points or extremum of the
frequency bands. Hence, the arrangement of exchange interactions may play a
more important role at producing the orbital angular momentum of magnons than
the spin-orbit coupling energy and the resulting non-collinear arrangement of
spins.
