Losing the sense of orientation of Weyl semimetal topology

by Dr Marcus Stålhammar (Bäcklund) (Utrecht University and Uppsala University)

Thursday 22 Jan 2026, 10:00 → 12:00 Europe/Stockholm

Albano 3: 6228 - Mega (22 seats) (Albano Building 3)

Weyl semimetals are topological materials hosting accidental, yet stable, point-like crossings between the valence and conduction bands. These are bound to satisfy the Nielsen-Ninomiya theorem, stating that each quasi-particle charge (chirality) assigned to the Weyl nodes must be accompanied by one of opposite charge, ensuring a global charge cancellation in the Brillouin zone, or, equivalently, a net-zero chirality. A recent work has, however, suggested that allowing the Brillouin zone to be non-orientable provides a way to circumvent this, as the charge cancellation in Z is replaced by one in Z_2 [1]. In this talk, I will argue that the Z_2 charge cancellation derived in Ref. [1] requires a different interpretation, stemming from the fact that the notion of chirality becomes ill-defined on non-orientable manifolds. Being dependent on either a choice of orientation, or an underlying induced notion of orientation, the physical interpretation of chirality, and hence the quasi-particle charge associated to Weyl points, has to be understood from a coordinate-free framework. Based on a recent work together with Thijs Douwes [2], a master student at Utrecht University, I will explain how to use (co)homology theory and other tools commonly used within algebraic topology, to recover a correct physical interpretation of Weyl semimetal topology on non-orientable manifolds.

I aim to present this in a self-contained way, and will hence provide a background on how the algebraic topology-language connects to actual physical systems. However, I won’t provide a complete mathematically rigorous explanation of (co)homology, but rather try to provide a non-technical description of them, and focus on the physical interpretation of the somewhat mathematically abstract calculations rather than their full details, to make sure that people lacking a background in algebraic topology are still able to appreciate the content.

[1] A.G. Fonseca, S. Vaidya, T. Christensen, M.C. Rechtsman, T.L. Hughes, and M. Soljačic, Weyl points on non-orientable manifolds, PRL 132 266601 (2024).
[2] T. Douwes, and M. Stålhammar, Twisted (co)homology of non-orientable Weyl semimetals, arXiv:2511.22303.