A diagnosis of explicit symmetry breaking in the tight-binding constructions for symmetry-protected topological systems

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If additional symmetries are present, σ O ​ ( r c ) \sigma_{O}(r_{c}) can be made arbitrarily small despite the non-onsite implementation of the protecting symmetry. One example is the inversion-symmetric Kane-Mele model in the absence of the sublattice potential. Here σ O ​ ( r c ) = 0 \sigma_{O}(r_{c})=0 regardless of the truncation distance

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