  Bulk properties of quantum phases should be independent of a specific choice
of boundary conditions as long as the boundary respects the symmetries. Based
on this physically reasonable requirement, we discuss the
Lieb-Schultz-Mattis-type ingappability in two-dimensional quantum magnets under
a boundary condition that makes evident a quantum anomaly underlying the
lattice system. In particular, we direct our attention to those on the
checkerboard lattice which are closely related to frustrated quantum magnets on
the square lattice and on the Shastry-Sutherland lattice. Our discussion is
focused on the adiabatic U(1) flux insertion through a closed path in a
boundary condition twisted by a spatial rotation and a reflection.
Two-dimensional systems in this boundary condition are effectively put on a
nonorientable space, namely the Klein bottle. We show that the translation
symmetry on the Klein-bottle space excludes the possibility of the unique and
gapped ground state. Taking advantage of the flux insertion argument, we also
discuss the ground-state degeneracy on magnetization plateaus of the Heisenberg
antiferromagnet on the checkerboard lattice.
