Twisted (co)homology of non-orientable Weyl semimetals

TL;DR

This paper develops a coordinate-free topological framework for Weyl semimetals with non-orientable Brillouin zones, using twisted (co)homology to restore Poincaré duality and to classify bulk and boundary invariants via Mayer–Vietoris sequences. It demonstrates how orientation-reversing symmetries lead to mod-2 charge cancellation and introduces twisted Dirac strings and loops that underpin Fermi arcs and surface states, with explicit treatment of a Klein-bottle-like Brillouin zone $K^2\times S^1$. The work then extends the formalism to all four free-action non-orientable Brillouin zones, non-Hermitian systems, and inversion-symmetric Weyl semimetals, revealing a rich invariant structure (including a $\mathbb{Z}^3\oplus\mathbb{Z}_2^4$ classification) and offering a unified, physical interpretation of surface states and symmetry-protected features. These insights have implications for experimental probes and for extending topological classifications to broader symmetry settings and dissipative systems.

Abstract

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 $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ 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 provide a complete survey of all possible non-orientable Brillouin zones and their associated invariants, and extend our formalism into the realm of non-Hermitian topological physics and inversion-symmetric Weyl semimetals. Our work exemplifies the vast potential of fundamental mathematical descriptions to not only aid the corresponding physical intuition, but also predict novel and hitherto overlooked phenomena of great relevance throughout the physics research forefront.

Figures (10)

• Figure 1: A schematic comparison between (a) the conventional physical interpretation of Weyl point topology and (b) the corresponding (co)homology picture. In (a), a positively (negatively) charged Weyl point is depicted as a source (sink) of the Berry field inside the Brillouin zone. After projection to a surface, these nodes form the endpoints of a physical Fermi arc. In the cohomology picture (b), the charges correspond to second cohomology classes on spheres surrounding the Weyl points. The oriented Dirac string connecting the two nodes represents a first homology counterpart to this: by Poincaré duality, the signed intersections of this string with the spheres agree with the cohomological charges. The Dirac string projects naturally onto a first homology class on the surface, which manifests as an oriented Fermi arc. Poincaré duality then ensures that the surface projections of the Weyl nodes have charges in terms of first cohomology classes on circles surrounding them.
• Figure 2: Illustration of the effect of glide symmetry on a two-dimensional Brillouin zone. (a) The symmetry acts on objects in the Brillouin zone by simultaneous reflection and translation, changing their handedness. On the full torus $\mathbb{T}^2$, the chiralities of the resulting objects are still well defined. (b) After reduction to the shaded fundamental domain, the Brillouin zone becomes non-orientable. The notion of handedness is lost in the process: the left palm traversing the orientation-reversing line at $k_y=0$ emerges as a right palm at $k_y=-\pi$. (c) Equivalent behaviour appears on the surface Brillouin zone of a glide symmetric Weyl semimetal: from the perspective of the shaded fundamental domain, a Fermi arc which crosses $k_y=0$ appears to connect similarly charged Weyl nodes.
• Figure 3: Top view of the three-dimensional Brillouin torus (or two-dimensional surface torus) for a given Weyl semimetal subject to the glide symmetry in Eq. \ref{['eq:3DGlide']}. A simple configuration of Weyl nodes is shown, with oriented Dirac strings (Fermi arcs) connecting them. Different parametrisations of the fundamental domain are shaded in teal, all of which take on $K^2\times S^1$ ($K^2$) topology after identifying their boundaries along the indicated arrows: (a) the domain outlined in Ref. fonseca2024; (b) the same domain shifted in the $k_y$ direction; (c) a domain spanning the $k_y$ direction. Consequently, the notions of both relative and absolute chirality of the two Weyl nodes in the fundamental domain are affected by the choice of parametrisation, indicating that these notions do not have phenomenological implications.
• Figure 4: Schematic illustration of the heuristic argument determining which of the (co)homology groups to twist. The top left shows how Weyl point charges [determined by $H^2(M\setminus W)$] and Dirac strings [representing invariants in $H_1\left(M,W\right)$] are altered by a change in orientation (represented by the dashed mirror axis). Being a chiral feature, the Weyl point charge is reversed; meanwhile, the internal orientation of the Dirac string is maintained, despite the string itself being mirrored (i.e. its image is also oriented away from the "hook" at the end). As a result, use of ordinary (co)homology groups leads to constructions (shown in the shaded area) where the orientation Dirac string may disagree with the chiralities of the two Weyl nodes it connects -- this is a result of Poincaré duality breaking. The duality is restored by twisting either the cohomology or the homology, effectively undoing the chirality reversal or reversing the orientation of the Dirac string, respectively. The correct choice between the two then depends on the way in which the orientation-reversing symmetry is observed to act on Weyl nodes in the system: symmetric pairs of opposite (same) chirality indicate twisted (co)homology.
• Figure 5: Top view of the $\mathbb{T}^3$ Brillouin torus. Each of the three basic Dirac loops $\ell_i$ are shown contained within an appropriately chosen $K^2\times S^1$ fundamental domain (shaded teal), and doubled by glide symmetry to the other half of the torus. The combined structure (shown in red) can be regarded as a twisted Dirac loop $\tilde{\ell}_i$ [representing an invariant in $H_1(K^2\times S^1;\widetilde{\mathbb{Z}})$] in each instance. (a) For $\ell_x$, the parity change induced by $k_x \mapsto -k_x$ is cancelled out by the orientation reversal induced by the twist in the homology. The resulting twisted Dirac string $\tilde{\ell}_x$ thus has a consistent orientation, and the associated invariant $\nu_x\in\mathbb{Z}$ induces an even Chern number $C_x = 2\nu_x\in 2\mathbb{Z}$ on the full torus. (b) For the case of $\ell_y$, the corresponding twisted Dirac loop $\tilde{\ell}_y$ is trivial: the mirroring along $k_x=0$ and subsequent orientation reversal result in a set of two oppositely orientated loops. These loops can be brought together along the dashed arrows in a symmetry-preserving fashion, eventually cancelling when they meet on the glide plane. As a result, there is no invariant $\nu_y$. (c) The Dirac loop $\ell_z$ running in the positive $k_z$ direction ("out of the page") in the fundamental domain is doubled to a second loop running in the negative $k_z$ direction ("into the page"). These loops cannot be merged in a fashion consistent with the symmetry, but they can instead be permuted along the dashed arrows. As a consequence, the resulting twisted Dirac string $\tilde{\ell}_z$ is equivalent to its own inverse $-\tilde{\ell}_z$; that is, $\tilde{\ell}_z$ generates a $\mathbb{Z}_2$ invariant $\nu_z$.