  We demonstrate the emergence of anti-chiral edge states in a Heisenberg
ferromagnet with Dzyaloshinskii-Moriya interaction(DMI) on a honeycomb lattice
with in-equivalent sub-lattices. The DMI, which acts between atoms of the same
species, differs in magnitude for the two sub-lattices, resulting in a shifting
of the energy of the magnon bands in opposite directions at the two Dirac
points. The chiral symmetry is broken and for sufficiently strong asymmetry,
the band shifting leads to anti-chiral edge states (in addition to the normal
chiral edge states) in a rectangular strip where the magnon current propagates
in the same direction along the two edges. This is compensated by a
counter-propagating bulk current that is enabled by the broken chiral symmetry.
We analyze the resulting magnon current profile across the width of the system
in details and suggest realistic experimental probes to detect them. Finally,
we propose a material that can potentially exhibit such anti-chiral edge
states.
