Equivalence between the Axion Invariant and the S4 Symmetry Indicator
Abstract
The equivalence between the axion invariant and the S4 symmetry indicator is established for three-dimensional S4-symmetric axion insulators with vanishing three-dimensional Chern numbers. Starting from the Chern-Simons expression for the magnetoelectric polarizability, 2P3=θ/π is rewritten in terms of the S4 sewing matrix. After stable reduction to determinant-one two-band blocks, the invariant is expressed as the degree of a map from the Brillouin zone to SU(2). The degree modulo two is then evaluated from the S4 eigenvalues at the four S4-invariant momenta and is shown to coincide with the symmetry indicator z2. A minimal tight-binding model with S4 symmetry verifies the correspondence between 2P3 and z2 throughout its phase diagram. The result closes a gap between the topological-field-theory description of the axion response and the topological-band-theory classification by symmetry indicators. It also extends the known response-indicator equivalence from antiunitary settings such as CnT symmetry to the unitary, orientation-reversing roto-inversion symmetry S4.
Versions
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BibTeX
@misc{zhang2026equivalence,
title = {{Equivalence between the Axion Invariant and the $S_4$ Symmetry Indicator}},
author = {Mengyao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.05719},
eprint = {2607.05719},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.05719},
}
RIS
TY - UNPB
TI - Equivalence between the Axion Invariant and the S_4 Symmetry Indicator
AU - Mengyao Zhang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2607.05719
UR - https://arxiv.org/abs/2607.05719
AB - The equivalence between the axion invariant and the S_4 symmetry indicator is established for three-dimensional S_4-symmetric axion insulators with vanishing three-dimensional Chern numbers. Starting from the Chern-Simons expression for the magnetoelectric polarizability, 2P_3=θ/π is rewritten in terms of the S_4 sewing matrix. After stable reduction to determinant-one two-band blocks, the invariant is expressed as the degree of a map from the Brillouin zone to SU(2). The degree modulo two is then evaluated from the S_4 eigenvalues at the four S_4-invariant momenta and is shown to coincide with the symmetry indicator z_2. A minimal tight-binding model with S_4 symmetry verifies the correspondence between 2P_3 and z_2 throughout its phase diagram. The result closes a gap between the topological-field-theory description of the axion response and the topological-band-theory classification by symmetry indicators. It also extends the known response-indicator equivalence from antiunitary settings such as C_nT symmetry to the unitary, orientation-reversing roto-inversion symmetry S_4.
ER -