Transition between two-dimensional symmetry-protected topological phases on a Klein bottle

Manifolds with nontrivial topology play an essential role in the study of topological phases of matter. In this paper, we study the nontrivial symmetry response of the $2+1$D $Z_2$ symmetry-protected topological (SPT) phase when the system is put on a non-orientable manifold -- the Klein bottle. In particular, we find that when a symmetry defect is inserted along the orientation-reversing cycle of the Klein bottle, the ground state of the system gets an extra charge. This response remains well defined at transition points into the trivial SPT phase, resulting in an exact two-fold degeneracy in the ground state independent of the system size. We demonstrate the symmetry response using exactly solvable lattice models of the SPT phase, as well as numerical work across the transition. We explore the connection of this result to the modular transformation of the $3+1$D $Z_2$ gauge theory and the emergent nature of the parity symmetry in the $Z_2$ SPT phase.