A discrete Stiefel-Whitney invariant with twofold rotation symmetry
Abstract
We review Stiefel--Whitney invariants and give a discrete formulation of an additional ℤ2 invariant for two-dimensional spinful insulators with twofold rotation and time-reversal symmetries (layer group p1121'). A singular gauge transformation specifies a real bundle over the quotient of the Brillouin torus, whose second Stiefel--Whitney number defines the invariant. We derive its dependence on the choice of auxiliary gauge function and prove invariance under changes of Bloch frame and additivity under direct sums. The discrete formula uses overlaps of independently chosen Bloch frames and symmetry-compatible Pin+ lifts on the boundary of a half Brillouin zone. Atomic and topological-insulator models illustrate the formula and distinguish the additional invariant from the Kane--Mele index. Together with three quantized ℤ2 partial polarizations and the filling number, it labels the known stable classification ℤ⊕ℤ2⊕ 4.
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BibTeX
@misc{shiozaki2026discrete,
title = {{A discrete Stiefel-Whitney invariant with twofold rotation symmetry}},
author = {Ken Shiozaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09723},
eprint = {2609.09723},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09723},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A discrete Stiefel-Whitney invariant with twofold rotation symmetry
AU - Ken Shiozaki
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09723
UR - https://arxiv.org/abs/2609.09723
AB - We review Stiefel--Whitney invariants and give a discrete formulation of an additional ℤ_2 invariant for two-dimensional spinful insulators with twofold rotation and time-reversal symmetries (layer group p1121'). A singular gauge transformation specifies a real bundle over the quotient of the Brillouin torus, whose second Stiefel--Whitney number defines the invariant. We derive its dependence on the choice of auxiliary gauge function and prove invariance under changes of Bloch frame and additivity under direct sums. The discrete formula uses overlaps of independently chosen Bloch frames and symmetry-compatible Pin_+ lifts on the boundary of a half Brillouin zone. Atomic and topological-insulator models illustrate the formula and distinguish the additional invariant from the Kane--Mele index. Together with three quantized ℤ_2 partial polarizations and the filling number, it labels the known stable classification ℤ⊕ℤ_2^⊕ 4.
ER -