We demonstrate that the presence of a localized state at the corner of an insulating domain is not always a predictor nor a direct consequence of a certain nontrivial higher-order topological invariant, even though they appear to coexist in the same Hamiltonian parameter space. Our analysis of $C_n$-symmetric crystalline insulators and their multilayer stacks reveals that topological corner states are not necessarily correlated with other well-established higher-order boundary observables, such as fractional corner charge or filling anomaly. In a $C_3$-symmetric breathing Kagome lattice, for example, we show that the bulk polarization, which successfully predicts the fractional corner anomaly, fails to be the relevant topological invariant for zero-energy corner states; instead, these corner states are explained by the decoration of topological edges. Also, while the corner states at the interface between $C_4$-symmetric topological crystalline insulators and their trivial counterpart have long been reported to be the result of the bulk polarization of the lowest band, we reveal that such embedded corner states are trivial defect states. By refining several bulk-corner correspondences in two-dimensional topological crystalline insulators, our work motivates further development of rigorous theoretical grounds for associating the existence of corner states with higher-order topology of host materials.

中文翻译：

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角局部状态的精确高阶体边界对应

我们证明了绝缘域角落处局部状态的存在并不总是预测因子，也不是某个非平凡的高阶拓扑不变量的直接结果，即使它们似乎共存于同一个哈密顿参数空间中。我们的分析$C_n$对称晶体绝缘体及其多层叠堆揭示了拓扑角状态不一定与其他完善的高阶边界可观测值相关，例如分数角电荷或填充异常。在一个$C_3$对称呼吸 Kagome 晶格，例如，我们表明，成功预测分数角异常的体极化不能成为零能量角状态的相关拓扑不变量；相反，这些角状态由拓扑边的装饰来解释。此外，虽然角状态在之间的界面$C_4$对称拓扑晶体绝缘体及其平凡的对应物长期以来一直被报道为最低带体极化的结果，我们揭示这种嵌入的角态是平凡的缺陷态。通过改进二维拓扑晶体绝缘体中的几个体角对应，我们的工作推动了严格的理论基础的进一步发展，以将角态的存在与主体材料的高阶拓扑相关联。