Nonorientable exceptional points in the Klein Brillouin zone

Nonorientable manifolds, such as the M"obius strip and the Klein bottle, defy conventional geometric intuition through their periodic boundary condition with a twist. As a result, topological defects on nonorientable manifolds give rise to novel physical phenomena. We study the adiabatic transport of exceptional points (EPs) along nonorientable closed loops and uncover distinct topological responses arising from the lack of global orientation. Notably, we demonstrate that the winding number associated with encircling an EP depends sensitively on the loop orientation, leading to inequivalent results for clockwise and counterclockwise encirclement; this feature is unique to nonorientable manifolds. Orientation-dependent geometric quantities cannot be consistently defined due to the absence of a global orientation. However, when a boundary is introduced, such quantities become well defined within the local interior, even though the global manifold remains nonorientable. We further demonstrate the adiabatic evolution of EPs and the emergence of orientation-sensitive observables in a Klein Brillouin zone, described by an effective non-Hermitian Hamiltonian that preserves momentum-space glide symmetry. Finally, we numerically implement these ideas in a microdisk cavity with embedded scatterers using synthetic momenta.