Symmetry-enforced Band Nodes in 230 Space Groups

TL;DR

This paper systematically classifies all possible symmetry-enforced band nodes in 230 space groups, providing a comprehensive guide for predicting topological semimetals with various nodal structures based on crystallographic symmetries.

Contribution

It introduces a complete list of high-symmetry lines, planes, and points that can host nodal features, and offers diagnostic methods to identify different types of band crossings, advancing the design of topological materials.

Findings

Identified all high-symmetry lines and planes capable of hosting nodal lines or surfaces.

Provided criteria to diagnose nodal loops and points from band crossings and irreducible representations.

Highlighted specific space groups and materials likely to host exotic nodal structures.

Abstract

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.