The sublattice symmetry on a bipartite lattice is commonly regarded as the chiral symmetry in the AIII class of the tenfold Altland–Zirnbauer classification. Here, we reveal the spatial nature of sublattice symmetry and show that this assertion holds only if the periodicity of primitive unit cells agrees with that of the sublattice labeling. In cases where the periodicity does not agree, sublattice symmetry is represented as a glide reflection in energy–momentum space, which inverts energy and simultaneously translates some k by π, leading to substantially different physics. Particularly, it introduces novel constraints on zero modes in semimetals and completely alters the classification table of topological insulators compared to class AIII. Notably, the dimensions corresponding to trivial and nontrivial classifications are switched, and the nontrivial classification becomes \({{\mathbb{Z}}}_{2}\) instead of \({\mathbb{Z}}\). We have applied these results to several models, including the Hofstadter model both with and without dimerization.

二分晶格上的亚晶格对称性通常被认为是十重 Altland-Zirnbauer 分类的 AIII 类中的手性对称性。在这里，我们揭示了亚晶格对称性的空间本质，并表明只有当原始晶胞的周期性与亚晶格标记的周期性一致时，这一断言才成立。在周期性不一致的情况下，亚晶格对称性被表示为能量动量空间中的滑移反射，这会反转能量并同时将一些k平移π ，从而导致截然不同的物理现象。特别是，它引入了对半金属中零模态的新颖约束，并且与 AIII 类相比完全改变了拓扑绝缘体的分类表。值得注意的是，平凡和非平凡分类对应的维度被交换，非平凡分类变成了\({{\mathbb{Z}}}_{2}\)而不是\({\mathbb{Z}}\) 。我们已将这些结果应用于多个模型，包括具有和不具有二聚化的 Hofstadter 模型。