Twisted (co)homology of non-orientable Weyl semimetals

The quasi-particle excitations in Weyl semimetals, known as Weyl fermions, are usually forced to emerge in charge-conjugate pairs by the Nielsen–Ninomiya theorem. When the Brillouin zone is non-orientable, this constraint is replaced by a \mathbb{Z}_2 ℤ 2 charge cancellation, as a result of the chirality becoming ill defined on such manifolds; this results in configurations with seemingly non-zero total chirality. Here, we set out to explain this behaviour from a purely topological perspective, and provide a classification of non-orientable Weyl semimetal topology in terms of exact sequences of twisted (co)homology groups. This leads to several discoveries of direct physical importance: in particular, we recover the \mathbb{Z}_2 ℤ 2 charge cancellation in a coordinate-independent way, allowing meaningful limits to be set on its physical interpretation. A detailed discussion is provided on a specific Klein bottle-like topology induced by a momentum-space glide symmetry, including a full review of the insulating and semimetallic invariants of the system and a classification of the surface states on the non-orientable boundary. Beyond this, we survey all possible non-orientable Brillouin zones and their semimetallic invariants, and extend our formalism into the realm of non-Hermitian topological physics and inversion-symmetric Weyl semimetals. Our work exemplifies the potential of fundamental mathematical descriptions to aid the physical intuition, predict novel and hitherto overlooked phenomena, and find concrete applications in areas such as acoustics and photonics.