  We investigated the topological property of magnon bands in the collinear
magnetic orders of zigzag and stripy phases for the antiferromagnetic honeycomb
lattice and identified Berry curvature and symmetry constraints on the magnon
band structure. Different symmetries of both zigzag and stripy phases lead to
different topological properties, in particular, the magnon bands of the stripy
phase being disentangled with a finite Dzyaloshinskii-Moriya (DM) term with
non-zero spin Chern number. This is corroborated by calculating the spin Nernst
effect. Our study establishes the existence of the non-trivial magnon band
topology for all observed collinear antiferromagnetic honeycomb lattice in the
presence of the DM term.
