For conventional topological phases, the boundary gapless modes are determined by bulk topological invariants. Based on developing an analytic method to solve higher-order boundary modes, we present $\mathcal{PT}$-invariant 2D topological insulators and 3D topological semimetals that go beyond this bulk-boundary correspondence framework. With unchanged bulk topological invariants, their first-order boundaries undergo transitions separating different phases with second-order boundary zero modes. For the 2D topological insulator, the helical edge modes appear at the transition point for two second-order topological insulator phases with diagonal and off-diagonal corner zero modes, respectively. Accordingly, for the 3D topological semimetal, the criticality corresponds to surface helical Fermi arcs of a Dirac semimetal phase. Interestingly, we find that the 3D system generically belongs to a novel second-order nodal-line semimetal phase, possessing gapped surfaces but a pair of diagonal or off-diagonal hinge Fermi arcs.

中文翻译：

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PT不变拓扑和二阶节点线半金属的边界临界性

对于常规拓扑阶段，边界无间隙模式由整体拓扑不变量确定。在开发解析方法以解决高阶边界模式的基础上，我们提出$\mathcal{PT}$不变的2D拓扑绝缘体和3D拓扑半金属超出了该体边界对应框架。拥有不变的本体拓扑不变性，它们的一阶边界经历了以二阶边界零模式分离不同相的过渡。对于2D拓扑绝缘子，螺旋边缘模式出现在两个二阶拓扑绝缘子相的过渡点，分别具有对角线和非对角线拐角零模式。因此，对于3D拓扑半金属，临界度对应于Dirac半金属相的表面螺旋费米弧。有趣的是，我们发现3D系统通常属于一种新颖的二阶节点线半金属相，具有间隙表面，但有一对对角或非对角铰链费米弧。