Using first-principles calculations, we report that ZrO is a topological material with the coexistence of three pairs of type-II triply degenerate nodal points (TNPs) and three nodal rings (NRs), when spin–orbit coupling (SOC) is ignored. Noticeably, the TNPs reside around the Fermi energy with a large linear energy range along the tilt direction (>1 eV), and the NRs are formed by three strongly entangled bands. Under symmetry-preserving strain, each NR would evolve into four droplet-shaped NRs before fading away, producing distinct evolution compared with that in usual two-band NR. When SOC is included, TNPs would transform into type-II Dirac points while all of the NRs are gapped. Remarkably, the type-II Dirac points inherit the advantages of TNPs: residing around the Fermi energy and exhibiting a large linear energy range. Both features facilitate the observation of interesting phenomena induced by type-II dispersion. The symmetry protections and low-energy Hamiltonian for the nontrivial band crossings are discussed.

中文翻译：

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从II型三重简并节点和三带节点环到中心对称氧化锆中的II型狄拉克点

使用第一性原理计算，我们报告ZrO是一种拓扑材料，当忽略自旋-轨道耦合（SOC）时，三对II型三重简并节点（TNP）和三个节点环（NRs）并存。值得注意的是，TNP位于费米能量附近，沿着倾斜方向（> 1 eV）具有较大的线性能量范围，而NR由三个强纠缠带形成。在保持对称性的应变下，每个NR逐渐消失之前会演变成四个液滴状的NR，与普通的两波段NR相比，会产生不同的演化。当包含SOC时，TNP将转换为II型Dirac点，而所有NR都将留空。值得注意的是，II型狄拉克点继承了TNP的优点：驻留在费米能量附近并具有较大的线性能量范围。这两个功能都有助于观察由II型弥散引起的有趣现象。讨论了非平凡过境的对称保护和低能哈密顿量。