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

Spinless Mirror Chern Insulator from Projective Symmetry Algebra