Symmetry-enforced topological parity: revisiting 2ℤ classifications beyond cellwise symmetry actions
Abstract
Momentum-dependent symmetry actions can turn the even-integer constraint of a 2ℤ classification of topological insulators and superconductors into an odd-integer constraint without changing the symmetry algebra. For internal and order-two crystalline symmetries in up to four spatial dimensions, we derive parity relations between the Chern or winding number of a gapped Hamiltonian and topological invariants of its momentum-dependent antiunitary symmetry matrices. Whenever the symmetry action enforces odd parity, any symmetry-preserving gapped phase must be topologically nontrivial. Using cellwise actions, which are represented by momentum-independent matrices in a fixed unit-cell basis, as a reference, we determine how locality constrains the realization of odd topological numbers. Explicit model constructions and no-go theorems distinguish three cases: odd values can be realized by finite-range symmetry actions, can be realized by exponentially decaying quasilocal actions but not by finite-range actions, or are impossible in finite-dimensional Bloch systems.
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BibTeX
@misc{shiozaki2026symmetryenforced,
title = {{Symmetry-enforced topological parity: revisiting $2\mathbbZ$ classifications beyond cellwise symmetry actions}},
author = {Ken Shiozaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37106},
eprint = {2609.37106},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37106},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetry-enforced topological parity: revisiting 2ℤ classifications beyond cellwise symmetry actions
AU - Ken Shiozaki
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37106
UR - https://arxiv.org/abs/2609.37106
AB - Momentum-dependent symmetry actions can turn the even-integer constraint of a 2ℤ classification of topological insulators and superconductors into an odd-integer constraint without changing the symmetry algebra. For internal and order-two crystalline symmetries in up to four spatial dimensions, we derive parity relations between the Chern or winding number of a gapped Hamiltonian and topological invariants of its momentum-dependent antiunitary symmetry matrices. Whenever the symmetry action enforces odd parity, any symmetry-preserving gapped phase must be topologically nontrivial. Using cellwise actions, which are represented by momentum-independent matrices in a fixed unit-cell basis, as a reference, we determine how locality constrains the realization of odd topological numbers. Explicit model constructions and no-go theorems distinguish three cases: odd values can be realized by finite-range symmetry actions, can be realized by exponentially decaying quasilocal actions but not by finite-range actions, or are impossible in finite-dimensional Bloch systems.
ER -