  We propose that non-collinear magnetic order in quantum magnets can harbor a
novel higher-order topological magnon phase with non-Hermitian topology and
hinge magnon modes. We consider a three-dimensional system of interacting local
moments on stacked-layers of honeycomb lattice. It initially favors a collinear
magnetic order along an in-plane direction, which turns into a non-collinear
order upon applying an external magnetic field perpendicular to the easy axis.
We exploit the non-Hermitian nature of the magnon Hamiltonian to show that this
field-induced transition corresponds to the transformation from a topological
magnon insulator to a higher-order topological magnon state with a
one-dimensional hinge mode. As a concrete example, we discuss the
recently-discovered monoclinic phase of the thin chromium trihalides, which we
propose as the first promising material candidate of the higher-order
topological magnon phase.
