  It was commonly believed that a mirror Chern insulator (MCI) must require
spin-orbital coupling, since time-reversal symmetry for spinless systems
contradicts with the mirror Chern number. So MCI cannot be realized in spinless
systems which include the large field of topological artificial crystals. Here,
we disprove this common belief. The first point to clarify is that the
fundamental constraint is not from spin-orbital coupling but the symmetry
algebra of time reversal and mirror operations. Then, our theory is based on
the conceptual transformation that the symmetry algebras will be projectively
modified under gauge fields. Particularly, we show that the symmetry algebra of
mirror reflection and time-reversal required for MCI can be achieved
projectively in spinless systems with lattice $\mathbb{Z}_2$ gauge fields,
i.e., by allowing real hopping amplitudes to take $\pm$ signs. Moreover, we
propose the basic structure, the twisted $\pi$-flux blocks, to fulfill the
projective symmetry algebra, and develop a general approach to construct
spinless MCIs based on these building blocks. Two concrete spinless MCI models
are presented, which can be readily realized in artificial systems such as
acoustic crystals.
