A basis-free, octonionic criterion for Weyl points in solids

Abstract

Standard ways of locating and characterizing Weyl points--Berry-flux integrals on small spheres and local $k\cdot p$ fits--are topologically sound but depend on user choices (sphere center/radius, gauge smoothing, surface charting, and Pauli-frame transport) that complicate high-throughput workflows. We introduce a local, basis-free criterion that avoids these knobs by using the octonionic geometry of $\operatorname{Im}\mathbb{O}\cong\mathbb{R}^7$. From a smooth two-band projector we build a unit octonion field $u(\mathbf{k})$ and its octonionic connection $\mathcal{A}_i=\operatorname{Im}(\bar u\,\partial_{k_i}u)$. Contracting the three directional derivatives with the $\mathrm{G}_2$-invariant three-form produces a pseudoscalar density whose sign gives the node's chirality. A companion quantity--the octonionic associator--vanishes at leading order precisely when the local three directions close inside an associative (quaternionic) three-plane; its smallness thus provides an intrinsic, pointwise self-consistency check that the two-band reduction is valid. The construction is invariant under $\mathrm{SU}(2)$ gauge changes of the two-band subspace and under $\mathrm{G}_2$ rotations of its completion, and it eliminates enclosing surfaces and gauge seams. In the linear regime the density reduces to the oriented volume set by the velocity matrix, thereby agreeing with the Chern number on a small sphere and with the chirality obtained from a $k\cdot p$ linearization. We outline a practical stencil-based algorithm compatible with Wannier tight-binding Hamiltonians and demonstrate robustness to benign numerical choices, while providing an intrinsic warning signal near band entanglement or multi-fold touchings.