Exceptional Points on Nonorientable Manifolds

Nielsen and Ninomiya's fermion doubling theorem arises from the topological classification of chiral fermions in lattice systems, mandating that their total chirality across the Brillouin zone must vanish. It has recently been shown to break down in two distinct ways. First, in non-Hermitian systems, the fermions are exceptional points that are topologically classified by non-Abelian braids. This complicates the total chirality requirement, leading to the possibility of monopoles. Second, in systems with non-symmorphic symmetries, the underlying Brillouin zone reduces to a non-orientable manifold, weakening the notion of chirality itself.

Here we combine these two groundbreaking approaches and investigate the interplay of exceptional points and non-orientable manifolds. We fully classify both gapped and gapless phases in the boundary braid framework and construct exemplary non-Hermitian tight-binding models with achiral monopoles on both Klein bottle and real projective Brillouin zones. This research extends existing topological classifications and opens new avenues for exploring the rich landscape of non-Hermitian physics on nonorientable manifolds, with potential implications for future quantum materials and photonic applications.