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

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.