Mirror Chern insulators in two-dimensional altermagnetic Tc$_2$Cl$_2$O and Tc$_2$Br$_2$O
The interplay between altermagnetism and crystalline band topology provides an intriguing avenue for realizing unconventional topological phases with distinctive spin-dependent properties. Here, based on first-principles calculations and theoretical analysis, we identify monolayer $\mathrm{Tc}_2X_2\mathrm{O}$ ($X$ = Cl, Br) as a family of two-dimensional altermagnetic mirror Chern insulators. In the absence of spin--orbit coupling (SOC), both monolayers exhibit robust altermagnetism with mirror-spin coupling and host two symmetry-protected Weyl points in each spin channel near the Fermi level. The Weyl points in opposite spin channels carry distinct mirror-symmetry eigenvalues, $m_z=\pm i$. Upon inclusion of SOC, the Weyl points are gapped, and the two mirror sectors acquire opposite Chern numbers, ${\cal {C}}_{+}=1$ and ${\cal {C}}_{-}=-1$, resulting in a nonzero mirror Chern number ${\cal {C}}_m=1$. A low-energy $k\cdot p$ model captures the symmetry protection of the Weyl points and elucidates their SOC-induced mass gaps and topological character. Furthermore, the resulting mirror Chern insulating phases host helical edge states within the bulk band gap and exhibit a quantized spin Hall conductivity. Our work establishes a direct connection between altermagnetism and mirror Chern topology and provides a promising platform for exploring unconventional topological and spin-dependent phenomena in two-dimensional altermagnetic materials.