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Equivalence between the Axion Invariant and the $S_4$ Symmetry Indicator
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
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