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We study $\mathbb{Z}_3$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with $Z_3 \times Z_3$ symmetry.
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After completion of this manuscript, we became aware that H N H_{N} and ξ N \xi_{N} were reported in S. Rachel, D. Schuricht, B. Scharfenberger, R. Thomale, M. Greiter, J. Phys.: Conf. Ser. 200
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