  We study the quantum phase diagram of spinful fermions on kagome lattice with
half-filled lowest flat bands. To understand the competition between magnetism,
flat band frustration, and repulsive interactions, we adopt an extended $t$-$J$
model, where the hopping energy $t$, antiferromagnetic Heisenberg interaction
$J$, and short-range neighboring Hubbard interaction $V$ are considered. In the
weak $J$ regime, we identify a fully spin-polarized phase, which can further
support the spontaneous Chern insulating phase driven by the short-range
repulsive interaction. This phase still emerges with in-plane ferromagnetism,
whereas the non-interacting Chern insulator disappears constrained by symmetry.
As $J$ gradually increases, the ferromagnetism is suppressed and the system
first becomes partially-polarized with large magnetization and then enters a
non-polarized phase with the ground state exhibiting vanishing magnetization.
We identify this non-polarized phase as an insulator with a nematic charge
density wave. In the end, we discuss the potential experimental observations of
our theoretical findings.
