  We investigate spinon band topology and engineering from the interplay
between long-ranged magnetic order and fractionalized spinons, as well as
Zeeman coupling under external magnetic fields, in honeycomb lattice magnets.
The synergism of N\'eel order and magnetic fields could reconstruct the spinon
bands and drive a topological phase transition from the coexisting phase of
long-ranged order and chiral spin liquid with semion topological order to the
conventional magnetic order. Our prediction can be immediately tested through
thermal Hall transport measurements among the honeycomb lattice magnets that
are tuned to be proximate to the quantum critical point. Our theory should also
shed light on the critical behavior of honeycomb Kitaev materials with emergent
Majorana fermion bands. We suggest a possible relevance to the spin-1/2
honeycomb spin liquid candidate material In$_3$Cu$_2$VO$_9$.
