Mirror real Chern insulator in two and three dimensions

A real Chern insulator (RCI) featuring a real Chern number and a second-order boundary mode appears in a two-dimensional (2D) system with the space-time inversion symmetry $\left(\mathrm{PT}\right)$. Here, we propose a kind of RCI: the mirror real Chern insulator (MRCI), which emerges from the system having additional horizontal mirror symmetry $M_z$. The MRCI generally is characterized by two independent real Chern numbers, respectively defined in the two mirror subsystems of the system. Hence, the MRCI may host the second-order boundary modes different from the conventional RCI. We show that for spinless systems, the definition of the MRCI is straightforward, as $\mathrm{PT}$ keeps each mirror subsystem invariant. For the spinful systems with both $\mathrm{PT}$ and $M_z$, the real Chern number for the total system remain well defined, as $M_z\mathrm{PT}=C_{2z}T$, and ${\left(C_{2z}T\right)}^2=1$. However, since $C_{2z}T$ exchanges the two mirror subsystems, the definition of the MRCI in spinful systems requires the help of projective symmetry algebra. We also discuss the MRCIs in 3D systems, where the MRCI is defined on certain mirror-invariant 2D planes. Compared with its 2D counterpart, the 3D MRCI can exhibit more abundant physics when the systems have additional nonsymmorphic operators. Several concrete MRCI models, including 2D and 3D, spinless and spinful models are constructed to further demonstrate our ideas.