-
近期, 高阶拓扑绝缘体和高阶拓扑超导体的概念激发了广泛关注和研究兴趣. 由于新的体-边对应关系, 在同一维度高阶拓扑绝缘体和高阶拓扑超导体的边界态的维度要低于一阶(或称传统)拓扑绝缘体和拓扑超导体的边界态. 本文阐述了高阶拓扑物态和一阶拓扑物态的联系. 具体展示了在同一维度上如何利用对称性的破缺从一阶拓扑物态转变为高阶拓扑物态, 以及如何利用低维的一阶拓扑物态构造高维的高阶拓扑物态; 回顾了高阶拓扑绝缘体和高阶拓扑超导体的研究进展. 通过对近期的研究进展的回顾, 可以看出这一新兴领域虽然研究进展迅速, 但对电子型的高阶拓扑绝缘体和高阶拓扑超导体的性质的理论研究和实验研究均处在非常初级的阶段, 要对这一新兴领域有更深更全面的理解认识还有待更多的研究投入.
Very recently, higher-order topological insulators and superconductors have attracted wide attention and aroused the great interest of researchers. Owing to their unconventional bulk-boundary correspondence, higher-order topological insulators and superconductors possess novel boundary modes whose dimensions are always lower than the first-order (or say conventional) topological insulators and superconductors, provided that their bulk dimensions are the same. The essence of higher-order topological phase is the formation of Dirac-mass domain walls on the gapped one-dimensional lower boundary. Roughly speaking, the origins of the formation can be classified as " intrinsic” and " extrinsic” type. For the former case, the formation of domain walls is forced by symmetry, suggesting that the resulting higher-order topological phases can be taken as topological crystalline phases. For this case, the domain walls are quite robust if the corresponding symmetry is preserved. For the latter case, the domain walls are formed simply because the one-dimensional lower boundary modes are gapped in a nontrivial way, however, the nontrivial way is not forced by symmetry. For this case, the domain walls are also stable against perturbations as long as the separations between them are large enough. The domain walls can have various patterns, which indicates that the higher-order topological phases are very rich. In this paper, we first reveal the connection between the higher-order topological phase and the first-order topological phase. Concretely, we show how to realize a higher-order topological phase by breaking some symmetries of a first-order topological phase, as well as stacking lower-dimensional first-order topological systems in an appropriate way. After these, we review the recent progress of theoretical and experimental study of higher-order topological insulators and superconductors. For the higher-order topological insulators, we find that the electronic materials are still laking though a lot of experimental realizations have been achieved. For higher-order topological superconductors, we find that their experimental realization and investigation are still in the very primary stage though quite a lot of relevant theoretical studies have been carried out. In order to comprehensively understand this newly-emerging field there are still many things to be done. -
Keywords:
- higher-order topological insulator /
- higher-order topological superconductor /
- bulk-boundary correspondence
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图 1 拓扑物态的边界态示意图
$n = 1$ 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;$n \geqslant 2$ 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. 1. A schematic diagram of the boundary modes of topological matter. The line with
$n = 1$ corresponds to conventional topological matters which host gapless modes whose dimensions are one-dimensional lower than the system dimension. The lines with$n \geqslant 2$ correspond to higher-order topological matters which host gapless modes whose dimensions are n-dimensional lower than the system dimension.
图 2 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(
${L_x} = 100$ ), 沿y方向取周期边界条件, 参数为$M = B = A = 1$ ,$\varLambda = 0$ , 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$\varLambda = 0.5$ , 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. 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
${L_x} = 100$ ) and periodic boundary condition in the y direction. Parameters are$M = B = A = 1$ ,$\varLambda = 0$ , 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$\varLambda = 0.5$ . 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.
图 3 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
${\text{π}}$ 磁通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
${\text{π}}$ -flux. -
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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