The nontrivial band topologies protected by certain symmetries have attracted significant interest in condensed matter physics. The discoveries of nontrivial topological phases in real materials provide a series of archetype materials to further explore the topological physics. Ternary borides M₂XB₂ (M = W, Mo; X = Co, Ni) have been widely investigated as the wear-resistant and high-hardness materials. Based on first-principles calculations, we find the nontrivial topological properties in these materials. Taking W₂NiB₂ as an example, this material shows the nodal line semimetal state in the absence of spin-orbit coupling. Two types of nodal lines appear near the Fermi level simultaneously. One is protected by the combined space-inversion and time-reversal symmetry, and the other is by the mirror symmetry. Part of these two-type nodal lines form nodal chains. When spin-orbit coupling is included, these nodal lines are fully gapped and the system becomes a strong topological insulator with nontrivial Z ₂ index (1;000). Our calculations demonstrate that a nontrivial spin-momentum locked surface Dirac cone appears on the (1¯10) surface. We also find that other isostructural ternary borides Mo₂NiB₂, Mo₂CoB₂, and W₂CoB₂ possess similar topological band structures. Therefore, our work not only enriches the understanding of band topology for ternary borides, but also lays the foundation for the further study of topological phases manipulation and potential spintronic applications in realistic materials.

中文翻译：

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三元硼化物 M2XB2 中的非平凡拓扑相（M=W、Mo；X=Co、Ni）

受某些对称性保护的非平凡能带拓扑引起了凝聚态物理学的极大兴趣。真实材料中非平凡拓扑相的发现为进一步探索拓扑物理提供了一系列原型材料。三元硼化物M ₂ XB ₂ (M = W, Mo; X = Co, Ni) 作为耐磨和高硬度材料已被广泛研究。基于第一性原理计算，我们发现了这些材料中非平凡的拓扑性质。以W ₂ NiB ₂为例，该材料在没有自旋轨道耦合的情况下表现出节线半金属态。费米能级附近同时出现两种类型的节线。一种是受到空间反演和时间反演对称性组合的保护，另一种是受到镜像对称性的保护。这两种类型的节点线的一部分形成节点链。当包括自旋轨道耦合时，这些节点线完全有间隙，并且系统成为具有非平凡的强拓扑绝缘体Z ₂索引 (1;000)。我们的计算表明，一个非平凡的自旋动量锁定表面狄拉克锥出现在 （1￣10） 表面。我们还发现其他同构三元硼化物Mo ₂ NiB ₂、Mo ₂ CoB ₂和W ₂ CoB ₂具有相似的拓扑能带结构。因此，我们的工作不仅丰富了对三元硼化物能带拓扑的理解，而且为进一步研究拓扑相操纵和在现实材料中潜在的自旋电子应用奠定了基础。