Fig. 1. A schematic diagram of the boundary modes of topological matter. The line with corresponds to conventional topological matters which host gapless modes whose dimensions are one-dimensional lower than the system dimension. The lines with correspond to higher-order topological matters which host gapless modes whose dimensions are n-dimensional lower than the system dimension. 拓扑物态的边界态示意图　 的行对应传统的拓扑物态, 其具有比系统维度低一维的无能隙边界态; 的行对应高阶拓扑物态, 其具有比维度低n维的无能隙边界态

Fig. 2. From first-order topological insulator to second-order topological insulator. (a) Energy spectra for a sample with open boundary condition in the x direction (the system size ) and periodic boundary condition in the y direction. Parameters are , , which corresponds to the original BHZ model. The energy spectra reflect the existence of gapless boundary modes. (b) the density profile of a boundary mode. The parameters are the same as in (a), but now open boundary conditions are taken both in the x and y directions. One can see that the density profile of the boundary mode distributes over the whole boundary. (c) the boundary conditions and parameters are the same as in (a), except now . One can see that the presence of the Λ term opens a gap for the boundary modes. (d) the density profiles of zero modes.The parameters are the same as in (c), but now open boundary conditions are taken both in the x and y directions. One can see that there are four zero-energy modes in the inset. Their wave functions are found to be localized around the corners. 从一阶拓扑绝缘体到二阶拓扑绝缘体　(a) 沿x方向取开放边界条件( ), 沿y方向取周期边界条件, 参数为 , , 对应BHZ模型, 能谱反映出无能隙边界态的存在; (b) 插图中红点对应的能量本征态的波函数分布, 参数同(a), 但沿x和y两个方向均取开放边界条件; 红色的深浅对应波函数分布概率的大小, 可以看出对一阶拓扑绝缘体, 波函数分布在整个边界上; (c) 边界条件和参数同(a), 除了此处 , 可看出Λ项的出现让边界态打开了能隙; (d) 零模的波函数分布, 参数同(c), 但沿x和y两个方向均取开放边界条件; 从插图中可发现存在四个零模, 这四个零模的波函数局域在四个角上

Fig. 3. Constructing two-dimensional second-order topological insulator by using one-dimensional topological insulator: (a) A schematic diagram of the SSH chain; (b) using the one-dimensional SSH chains to construct a two-dimensional second-order topological insulator, within each small square, there is a -flux. 从一维一阶拓扑绝缘体到二维二阶拓扑绝缘体　(a) 一维SSH链的示意图; (b) 利用一维SSH链构造二维二阶拓扑绝缘体, 每个单位元胞中有一个 磁通