Degeneracy and topology for the magnetic space group of CuBi$_2$O$_4$

In the presence of antiferromagnetic order CuBi$_2$O$_4$ belongs to one of $\#56$ magnetic space groups. Inversion-time-reversal symmetry leads to 2-fold degeneracy of all the energy bands in the entire Brillouin zones. It surprises us that non-symmorphic symmetries in the subgroup $\#56$ space group protect 4-fold degeneracy in a symmetric line. We further investigate the topology and surface states of CuBi$_2$O$_4$ stemming from its magnetic space group.