Recently, it has been discovered theoretically that there exist novel types of 3-dimensional topological insulators (TIs) whose gapless states manifest themselves at 1-dimensional hinges. They are called second-order topological insulators (SOTIs). Most mathematical models of SOTIs compose of conventional strong topological insulators (STIs) and additional mass terms In this paper, we investigate whether the models made based on weak topological insulators (WTIs) can have hinge states as the same as those made based on the STIs. We found that the models based on WTIs have only trivial index determined by the Wilson loop formalism unlike the models based on STIs. However, they can be regarded as stacking of 2-dimensional SOTIs along the z direction. Thus, we propose that there are topologically three different phases in our considering system: ordinary insulator, strong SOTI, and weak SOTI phases. This classification suggests the existence of other topological invariants besides the index. Finally, we propose new indices which can distinguish weak SOTIs from ordinary insulators.

最近，从理论上已经发现，存在着新型的3维拓扑绝缘体（TI），它们的无间隙状态表现在1维铰链上。它们被称为二阶拓扑绝缘体（SOTI）。SOTI的大多数数学模型由常规的强拓扑绝缘体（STI）和附加质量项组成。在本文中，我们研究了基于弱拓扑绝缘体（WTI）制作的模型是否可以具有与基于STI的模型相同的铰链状态。我们发现，与基于STI的模型不同，基于WTI的模型仅具有由Wilson循环形式主义确定的琐碎索引。但是，它们可以看作是沿z方向的二维SOTI的堆叠方向。因此，我们建议在考虑的系统中存在拓扑上三个不同的阶段：普通绝缘子，强SOTI和弱SOTI阶段。该分类表明除索引外还存在其他拓扑不变式。最后，我们提出了新的指数，可以区分较弱的SOTI和普通绝缘子。