  We study the half-filled Hubbard model on the triangular lattice with
spin-dependent Kitaev-like hopping. Using the variational cluster approach, we
identify five phases: a metallic phase, a non-coplanar chiral magnetic order, a
$120^\circ$ magnetic order, a nonmagnetic insulator (NMI), and an interacting
Chern insulator (CI) with a nonzero Chern number. The transition from CI to NMI
is characterized by the change of the charge gap from an indirect band gap to a
direct Mott gap. Based on the slave-rotor mean-field theory, the NMI phase is
further suggested to be a gapless Mott insulator with a spinon Fermi surface or
a fractionalized CI with nontrivial spinon topology, depending on the strength
of Kitaev-like hopping. Our work highlights the rising field that interesting
phases emerge from the interplay of band topology and Mott physics.
