标题： 三重态节点线和陈带在XCuCl$_{3}$（X= K，Tl）中 显示英文标题

标题： Triplet nodal lines and Chern bands in XCuCl$_{3}$ (X= K, Tl)

摘要： 我们研究了XCuCl$_{3}$(X= K, Tl) 三重态激发的对称性强制线节点，表明它们由非平移对称性保护，并且不受微观细节的影响，例如相互作用和各向异性强度，只要基态和对称性群保持不变。 扩展了通常用于XCuCl$_{3}$的各向同性自旋模型，我们的分析包括所有对称性允许的各向异性，并详细说明了它们在三重态能带拓扑中的作用。 我们表明三重态线节点具有非平凡的贝里相位，并计算了它们的$Z_2$拓扑指数。 为了研究破坏保护三重态节点的非平移对称性的影响，我们将一个倾斜于高对称$(010)$轴的磁场施加到系统上。 我们发现虽然g张量各向异性表现为一个平凡的质量项使三重态禁带化，但交换各向异性提供了非平凡的动量依赖质量项。 类似于哈德纳最初的模型，这些质量项的竞争决定了XCuCl$_{3}$中能带拓扑的性质。 为了能够对能带拓扑进行解析研究，我们推导了一个有效的狄拉克哈密顿量，并通过从线性键波形式计算节点线和禁带相中的能带结构和拓扑指数来验证它。 显示英文摘要

摘要： We investigate the symmetry-enforced line nodes of the triplet excitations of XCuCl$_{3}$ (X= K, Tl), showing that they are protected by the nonsymmorphic symmetries and are unaffected by the microscopic details, such as interaction and anisotropy strength, as long as the ground state and the symmetry group remain unaltered. Extending the conventionally used isotropic spin model for XCuCl$_{3}$, our analysis includes all the symmetry-allowed anisotropies and gives a detailed account of the role they play in the band topology of triplets. We show that the triplet line nodes carry nontrivial Berry phases and compute their $Z_2$ topological indices. To investigate the effect of breaking the nonsymmorphic symmetry protecting the triplet nodes, we apply a magnetic field tilted away from the high symmetry $(010)$ axis. We find that while the g-tensor anisotropy behaves as a trivial mass gapping out the triplets, exchange anisotropies supply a nontrivial momentum-dependent mass term. Analogous to Haldane's original model, the competition of these mass terms determines the nature of the band topology in XCuCl$_{3}$. To enable an analytic study of the band topology we derive an effective Dirac Hamiltonian and validate it by computing the band structure and topological indices in the nodal line and gapped phases from the linear bond-wave formalism.