Weyl Points on Non-Orientable Manifolds

Weyl fermions are chiral, massless particles that play an important role in quantum field theory and the Standard Model of particle physics. Although elusive as fundamental particles, they have been observed as quasiparticle excitations in various domains, from solid-state materials to photonic and acoustic crystals. In these contexts, they are referred to as Weyl points, with their chirality determined by whether they act as sources or sinks of Berry curvature. According to the Nielsen-Ninomiya theorem, Weyl points in three-dimensional lattice systems always occur in pairs with opposite chirality, implying that there are exactly as many sources as sinks. This fundamental constraint ultimately arises from the properties of the space in which Weyl fermions exist, specifically a toroidal Brillouin zone.<br/><br/>Here, we demonstrate that by modifying the topology of the underlying space, it is possible to circumvent this fundamental theorem. Specifically, we achieve this by transforming the underlying momentum space fundamental domain into a non-orientable one. Furthermore, we show that this modification leads to the emergence of novel Z2 topological charges carried by Weyl points, and the conservation of these charges results in a different no-go theorem. We experimentally demonstrate all aspects of the theory using a photonic platform with synthetic degrees of freedom.