Crystallographic symmetries enforcing band touchings (BTs) in the Brillouin zone (BZ) have been utilized to classify and predict the topological semimetals. Though the early proposed topological semimetals contain isolated nodal points in the BZ, the proposed nodal line semimetals later could host various structures of several nodal lines/loops: nodal chains, nodal nets or Hopf-links, etc. In this work, using compatibility relations, we first list all possible high symmetry lines (HSLs) that can be nodal lines itself, high symmetry planes (HSPLs) that can host nodal loops, high symmetry planes (HSPLs) that are nodal surfaces for all 230 SGs, with spin-orbit coupling and time-reversal symmetry considered or not. We then show how to diagnose a nodal loop from the band crossing in an HSL, or nodal line/surface from irreducible representation (irrep) of an high-symmetry point (HSP), while the rest cases correspond to nodal points. Among our results, those essential cases, for which the nodal points/lines/loops/surfaces must exist, are highlighted since they are promising for the realizations of (nearly) ideal nodal point/line/loop/surface semimetals, as well as systems with flexible tunability owning fixed structure of topological nodal points/lines/loops/surfaces. Based on our results, SGs allowing Hopf-link structure with one straight nodal line threading a nodal loop, or two nesting nodal loops lying in two respective high symmetry planes, are highlighted, with the predicted materials being B$_5$Pb$_2$IO$_9$ in SG 34 and SrAl$_2$Au$_3$ in SG 62, respectively. Our exhaustive results could serve as a useful guide for efficiently predicting and designing materials or artificial systems owning exotic geometric nodal structures of energy bands simply based on structure symmetries.
Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery
我们确定了所有的自旋-轨道耦合和非自旋-轨道耦合的斜方晶体中的电子强迫带交叉，并讨论了它们的拓扑性质。我们发现，正交晶体可以主机大量不同的带简并，包括可移动的外尔和狄拉克点与沙漏色散，四倍双外尔点，外尔和狄拉克节线，几乎可移动的节线，节链，拓扑节面。有趣的是，在空间群18、36、44、45和46中的自旋轨道耦合材料可以具有在整个布里渊区中只有两个外尔点的带对。这导致费米弧的连接性更简单，并且比具有四个或更多外尔点的材料具有更明显的拓扑响应。此外，我们表明，空间群56，61和62的对称性，强制非平凡的弱$\mathbb{Z}_2$拓扑结构的材料具有强的自旋轨道耦合，导致螺旋表面状态。有了这些分类结果在手，我们进行广泛的数据库搜索正交材料结晶在相关的空间群。我们发现Sr $_2$ Bi $_3$和IrPDMATHABMN Si具有跨越费米能的带，具有对称性强制的非平凡的$\mathbb{Z}_2$不变量，CuIrB在费米能附近具有节链，Pd $_7$ Se $_4$和Ag $_2$ Se表现出四重双外尔点，即使在没有自旋轨道耦合的情况下，后一个也是如此，而AuTlSb的四重简并是由相交的节线组成的。对于这些例子中的每一个，我们计算从头算能带结构，讨论它们的拓扑结构，并在某些情况下还计算表面状态。
Superconductivity and electronic structure evolution in the enforced semimetal Fe-doped ZrTe$_2$
Escola de Engenharia de Lorena - DEMAR, Universidade de São Paulo（洛雷纳工程学院-DEMAR，圣保罗大学）；Laboratory for Quantum Matter under Extreme Conditions, Instituto de Física, Universidade de São Paulo（极端条件下量子物质实验室，物理研究所，圣保罗大学）
Magnetically tunable symmetry-enforced nodal lines producing huge anomalous Hall conductivity in altermagnetic $α$-MnTe
International Research Centre Magtop, Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所磁顶国际研究中心）；Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所）；New Technologies-Research Center, University of West Bohemia in Pilsen（比尔森西波希米亚大学新技术研究中心）；Photon Science Division, Paul Scherrer Institut（保罗谢尔研究所光子科学部）；Institute of Semiconductor and Solid State Physics, Johannes Kepler University of Linz（林茨约翰内斯开普勒大学半导体与固体物理研究所）；Solaris National Synchrotron Radiation Centre（索拉里斯国家同步辐射中心）
Unusual Dual Flat Bands and two-dimensional Dirac-node Arc State in Kagome Metal Ni3In2S2
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所凝聚态物理国家实验室）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；National Laboratory of Solid State Microstructures and School of Physics, Nanjing University（南京大学固体微结构物理国家重点实验室和物理学院）；Department of Physics and Guangdong Basic Research Center of Excellence for Quantum Science, Southern University of Science and Technology（南方科技大学物理系及广东省量子科学基础研究卓越中心）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences（中国科学院上海微系统与信息技术研究所）；National Synchrotron Radiation Laboratory and School of Nuclear Science and Technology, University of Science and Technology of China（中国科学技术大学核科学技术学院及国家同步辐射实验室）
