Exceptional points (EPs) are prominent non-Hermitian band degeneracies that give rise to a variety of intriguing and unconventional phenomena. Similar to Weyl and Dirac points, EPs carry topological charges and comply with the celebrated fermion doubling theorems in lattices. Beyond these characteristics, EPs exhibit more exotic topological properties, particularly non-Abelian braiding topologies not seen in conventional degeneracies. However, these core concepts of EPs have been established under the assumption of toroidal Brillouin zones. Here, we investigate EPs in two-dimensional non-Hermitian lattices where the fundamental domain of the Brillouin zone is a Klein bottle, rather than a torus assumed in previous studies. We find that EPs do not necessarily appear in pairs with opposite topological charges in the Brillouin Klein bottle, thus violating the fermion doubling theorem. The violation occurs because, without crossing the boundary, the sum of the topological charges of EPs is in fact an even number rather than zero. Moreover, we uncover unique braiding topologies of EPs that cannot be captured by existing theories. Specifically, the composite braidings around all EPs equals the braiding along the boundary of the Brillouin Klein bottle. This braiding topology further confirms the failure of the fermion doubling theorem and allows us to explore the non-Abelian braidings of EPs beyond the scope of topological charges. Our work highlights the fundamental role of Brillouin-zone topology in non-Hermitian systems.

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• Cold atoms & matter waves
• Open quantum systems & decoherence
• Phononic crystals
• Photonic crystals
• Topological phases of matter
• Non-Hermitian systems
• Mathematical physics methods
• Topology