  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 (PT ). Here, we propose a kind of RCI: mirror real Chern
insulator (MRCI) which emerges from the system having additional horizontal
mirror symmetry Mz. 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 PT keeps each mirror subsystem invariant. For the
spinful systems with both PT and Mz, the real Chern number for the total system
remain well defined, as MzPT = C2zT , and (C2zT )2= 1. However, since C2zT
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.
