Octonionic Weyl Point Criterion

• The octonionic Weyl point criterion is a basis-free diagnostic for Weyl points in solids, replacing conventional surface-based methods with a local octonionic approach.
• It constructs a unit octonion field and leverages a G₂-invariant three-form to compute a pseudoscalar density, whose sign accurately identifies Weyl node chirality.
• A self-consistency test using the octonionic associator norm confirms the isolation of simple Weyl nodes and warns of potential band entanglement or multi-fold touchings.

Searching arXiv for the specified paper and closely related work to ground the article in the primary source and relevant context. The octonionic Weyl point criterion is a local, basis-free diagnostic for Weyl points in solids constructed from the octonionic structure on $\mathbb{R}^7$. Starting from a smooth two-band projector, it associates a unit octonion field and an octonionic connection; contracting three directional derivatives with the $\mathrm{G}_2$-invariant three-form produces a pseudoscalar density whose sign equals the Weyl chirality (Tantardini, 23 Sep 2025). In the linear regime, the construction is proved equivalent to both conventional diagnostics—the Chern charge on a small sphere and $\mathrm{sgn}\det v$—while avoiding enclosing surfaces, gauge smoothing, charting, and local frame transport (Tantardini, 23 Sep 2025).

1. Definition and conceptual setting

Conventional diagnostics of Weyl points are described as topologically sound but dependent on user choices, including sphere center and radius, gauge smoothing, charting, and local frame transport. These choices introduce algorithmic arbitrariness in first-principles workflows (Tantardini, 23 Sep 2025). The octonionic Weyl point criterion addresses this by replacing the usual enclosing-surface construction with a local computation at a candidate crossing point $k_\star$.

The formulation begins with a Bloch or Wannier-TB Hamiltonian $H(k)$ for which the $n$-th and $(n+1)$-th bands cross at $k_\star$. In a small neighborhood of $k_\star$, one assumes that a smooth rank-2 projector can be isolated,

$$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$

Within that two-dimensional subspace one defines the flattened sign-Hamiltonian

$\mathrm{G}_2$0

with $\mathrm{G}_2$1 and $\mathrm{G}_2$2 (Tantardini, 23 Sep 2025).

The criterion is called basis-free because its diagnostic quantities are invariant under $\mathrm{G}_2$3 gauge changes of the two-band subspace and under $\mathrm{G}_2$4 rotations of its completion (Tantardini, 23 Sep 2025). A plausible implication is that the method is intended to make chirality assignment less sensitive to implementation-specific gauge conventions than surface-flux pipelines.

2. Octonionic construction

The construction fixes an octonionic (Fano-plane) basis $\mathrm{G}_2$5 of $\mathrm{G}_2$6 and selects the distinguished associative, quaternionic subspace

$\mathrm{G}_2$7

The Pauli frame is embedded into $\mathrm{G}_2$8 by identifying $\mathrm{G}_2$9 for $\mathrm{sgn}\det v$0 (Tantardini, 23 Sep 2025).

One then chooses a unit-octonion field

$\mathrm{sgn}\det v$1

whose imaginary part lies, to leading order, in $\mathrm{sgn}\det v$2, and such that

$\mathrm{sgn}\det v$3

with $\mathrm{sgn}\det v$4 the octonionic conjugate. Any two choices of $\mathrm{sgn}\det v$5 differ by a pointwise $\mathrm{sgn}\det v$6 rotation in $\mathrm{sgn}\det v$7, and the subsequent constructions are $\mathrm{sgn}\det v$8-invariant (Tantardini, 23 Sep 2025).

The octonionic analogue of a Maurer-Cartan form is defined by the left-transports

$\mathrm{sgn}\det v$9

Because $k_\star$0 carries a canonical $k_\star$1-invariant 3-form $k_\star$2, with $k_\star$3 on the distinguished $k_\star$4, one obtains the octonionic pseudoscalar density

$k_\star$5

with $k_\star$6 (Tantardini, 23 Sep 2025).

The decisive statement is

$k_\star$7

so the sign of the local pseudoscalar density gives the Weyl chirality (Tantardini, 23 Sep 2025).

3. Associativity, closure, and the role of the associator

A central structural feature of the criterion is the use of octonionic associativity failure as a diagnostic of whether the local two-band geometry closes inside an associative three-plane. The octonionic associator is

$k_\star$8

A real 3-plane in $k_\star$9 is associative, equivalently quaternionic, if and only if the associator vanishes on triples drawn from it (Tantardini, 23 Sep 2025).

The criterion therefore supplements $H(k)$0 with the quantity

$H(k)$1

where the norm is the Euclidean norm in $H(k)$2 (Tantardini, 23 Sep 2025). At a simple Weyl node,

$H(k)$3

because the local two-band geometry sits inside an associative $H(k)$4 (Tantardini, 23 Sep 2025). By contrast, a large $H(k)$5 warns of entanglement or of a multi-fold touching (Tantardini, 23 Sep 2025).

This establishes a two-part local test. A nonzero density identifies a Weyl point, while vanishing of the octonionic associator at leading order certifies closure inside an associative three-plane (Tantardini, 23 Sep 2025). A common misconception would be to interpret $H(k)$6 as merely an auxiliary numerical error bar. In the formulation of the criterion, it is instead an intrinsic warning signal connected to the geometry of the local band problem itself.

4. Computational workflow in Wannier tight binding

The proposed implementation is described as a practical algorithm compatible with Wannier tight-binding Hamiltonians and includes self-consistency checks based on stencil refinement and the associator norm (Tantardini, 23 Sep 2025). The workflow is organized as a local stencil computation around a candidate $H(k)$7.

Step	Operation	Output
0	Start from a Wannierized Hamiltonian $H(k)$8; use the coarse-mesh gap scan in Sec. II of the paper to locate $H(k)$9 as a candidate	Candidate crossing point
1	Evaluate smooth two-band projectors $n$0 and form $n$1	Stencil projectors and flattened Hamiltonians
2	Build unit quaternions $n$2 and embed them as $n$3	Unit octonions
3	Compute central finite-difference octonionic connections $n$4	Local connection data
4	Compute $n$5 and $n$6	Chirality density and associator norm
5	Require $n$7 and $n$8; assign $n$9	Decision
6	Check stencil refinement, window stability, and local $(n+1)$0 invariance	Self-consistency

All derivatives are taken by central finite differences of step $(n+1)$1 (Tantardini, 23 Sep 2025). At the central point,

$(n+1)$2

and then

$(n+1)$3

The stated decision rule is to require $(n+1)$4 and $(n+1)$5 for chosen tolerances, then assign $(n+1)$6 (Tantardini, 23 Sep 2025).

The self-checks are explicit. Under $(n+1)$7, one verifies that $(n+1)$8 is stable up to $(n+1)$9 and that $k_\star$0. One also rebuilds $k_\star$1 with slightly shifted disentanglement windows and checks that $k_\star$2 is unchanged, and applies a random local $k_\star$3 rotation at $k_\star$4 to verify that $k_\star$5 and $k_\star$6 are invariant (Tantardini, 23 Sep 2025). This suggests that the method is designed not only to classify nodes but also to expose when the assumptions of the local two-band description are unreliable.

5. Relation to conventional Weyl diagnostics

Near $k_\star$7, the effective Hamiltonian is linearized as

$k_\star$8

In the embedding $k_\star$9, the criterion yields, to leading order,

$k_\star$0

and hence

$k_\star$1

(Tantardini, 23 Sep 2025).

The same linear analysis reproduces the flux diagnostic. Integrating $k_\star$2 over a small sphere gives the Chern number

$k_\star$3

so the criterion coincides with both the flux-on-sphere test and the usual $k_\star$4 in the linear regime (Tantardini, 23 Sep 2025).

This equivalence is significant because it does not replace the established topology with a different invariant; rather, it repackages the same chirality information into a local octonionic scalar density. A plausible implication is that the novelty lies primarily in the elimination of surface construction and gauge-seam handling rather than in a different classification of Weyl nodes.

6. Symmetry properties, conventions, and scope

The criterion is invariant under local changes of basis in the two-band subspace. Any local change of two-band basis $k_\star$5 acts by the adjoint on the Pauli-quaternion frame, written as

$k_\star$6

and this induces an $k_\star$7 rotation of the triple $k_\star$8 in the associative subspace $k_\star$9 (Tantardini, 23 Sep 2025). Because $$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$0 is $$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$1-invariant and $$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$2,

$$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$3

The consequence stated in the paper is that no gauge-patching, Wilson-loop, or two-cap construction on an enclosing $$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$4 sphere is needed (Tantardini, 23 Sep 2025).

The construction is also invariant under $$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$5 rotations of the completion of the two-band subspace (Tantardini, 23 Sep 2025). However, it does not eliminate every convention. The only remaining convention is the global orientation of $$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$6 (Tantardini, 23 Sep 2025). Under an orientation-reversing linear change $$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$7 with $$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$8,

$$P_2(k)=\sum_{a=1}^2 \ket{u_a(k)}\bra{u_a(k)}, \qquad P_2(k)^2=P_2(k),\quad \mathrm{Tr}\,P_2=2.$$9

so $\mathrm{G}_2$00 flips sign exactly as the Chern flux and $\mathrm{G}_2$01 do when one reverses the outward normal on $\mathrm{G}_2$02 (Tantardini, 23 Sep 2025).

The intended computational scope is high-throughput searches, where the method is said to streamline chirality assignment and to provide an intrinsic warning signal in the presence of band entanglement or proximity to multi-fold touchings (Tantardini, 23 Sep 2025). A misconception would be to regard the procedure as universally gauge-free in an absolute sense; the stated formulation is intrinsically gauge-free only up to the unavoidable choice of orientation. Another misconception would be to read a nonzero $\mathrm{G}_2$03 without checking $\mathrm{G}_2$04; the paper explicitly uses the associator norm as a self-consistency test for whether the local problem truly behaves as an isolated simple Weyl node (Tantardini, 23 Sep 2025).