To date, almost all of the discussions on topological insulators (TIs) have focused on two- and three-dimensional systems. One-dimensional (1D) TIs manifested in real materials, in which localized spin states may exist at the end or near the junctions, have largely been unexplored. Previous studies have considered the system of gapped graphene nanoribbons (GNRs) possessing spatial symmetries (e.g., inversion) with only termination patterns commensurate with inversion- or mirror-symmetric unit cells. In this work, we prove that a symmetry-protected topological classification exists for any type of termination. In these cases the Berry phase summed up over all occupied bands turns out to be π-quantized in the presence of the chiral symmetry. However, it does not always provide the correct corresponding as one would have expected. We show that only the origin-independent part of the Berry phase gives the correct bulk-boundary correspondence by its π-quantized values. The resulting invariant depends on the choice of the 1D unit cell (defined by the nanoribbon termination) and is shown to be connected to the symmetry eigenvalues of the wave functions at the center and boundary of the Brillouin zone. Using the cove-edged GNRs as examples, we demonstrate the existence of localized states at the end of some GNR segments and at the junction between two GNRs based on a topological analysis. The current results are expected to shed light on the design of electronic devices based on GNRs as well as the understanding of the topological features in 1D systems.

中文翻译：

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具有空间对称性的带缝石墨烯纳米带的拓扑性质

迄今为止，几乎所有有关拓扑绝缘体（TI）的讨论都集中在二维和三维系统上。在实际材料中表现出来的一维（1D）TI很大程度上尚未得到探索，在这些材料中，局部自旋态可能存在于结的末端或附近。先前的研究已经考虑了具有空间对称性（例如，倒位）且仅具有与倒位或镜像对称的晶胞相对应的终止模式的带间隙的石墨烯纳米带（GNR）系统。在这项工作中，我们证明了对于任何类型的终止都存在对称保护的拓扑分类。在这些情况下，在存在手性对称的情况下，在所有占据的谱带上加起来的贝里相被量化为π量化。但是，它并不总是提供正确的对应正如人们所期望的那样。我们表明，只有Berry相的原点无关部分才能通过其π量化值给出正确的体边界对应关系。产生的不变量取决于一维晶胞的选择（由纳米带末端定义），并显示为与布里渊区的中心和边界处的波函数的对称特征值相关。以最前沿的GNR为例，我们基于拓扑分析论证了在某些GNR段的末端以及两个GNR之间的交界处存在局部状态。当前的结果有望为基于GNR的电子设备的设计以及对一维系统拓扑特征的理解提供启发。