Tuning ferromagnetic BaFe$_2$(PO$_4$)$_2$ through a high Chern number topological phase

There is strong interest in discovering or designing wide gap Chern insulators. Here we follow a Chern insulator to trivial Mott insulator transition versus interaction strength U in a honeycomb lattice Fe-based transition metal oxide, discovering that a spin-orbit coupling energy scale $\xi$=40 meV can produce and maintain a topologically entangled Chern insulating state against large band structure changes arising from an interaction strength U up to 60 times as large. Within the Chern phase the minimum gap switches from the zone corner K to the zone center $\Gamma$ while maintaining the topological structure. At a critical strength $U_c$, the continuous evolution of the electronic structure encounters a gap closing then reopening, upon which the system reverts to a trivial Mott insulating phase. This Chern insulator phase of honeycomb lattice Fe$^{2+}$ BaFe$_2$(PO$_4$)$_2$ corresponds to a large Chern number C=-3 that will provide enhanced anomalous Hall conductivity due to the associated three edge states threading through the bulk gap of 80 meV.