Electron correlations and $T$-breaking density wave order in a $\mathbb{Z}_2$ kagome metal

There have been extensive recent developments on kagome metals, such as T$_m$X$_n$ (T= Fe, Co and X= Sn, Ge) and $A$V$_3$Sb$_5$ ($A=$ Cs, K, Rb). An emerging issue is the nature of correlated phases when topologically \textit{non-trivial} bands cross the Fermi level. Here, we consider an extended Hubbard model on the kagome lattice in the presence of spin-orbit couplings, involving a Kramer's pair of bands that have opposite Chern numbers and are isolated in the band structure. We construct an effective model in a time-reversal (T) symmetric lattice description. We determine the correlated phases of this model and identify a density-wave order in the phase diagram. We show that this order is T-breaking, which originates from the Wannier orbitals lacking a common Wannier center -- a fingerprint of the underlying $Z_2$ topology. Implications of our results for the correlation physics of the kagome metals are discussed.