Group-theoretical study of band nodes and the emanating nodal structures in crystalline materials

Abstract

Abstract Topological materials usually possess protected gapless states in either the boundary or bulk, exhibiting various properties such as spin-momentum locking, Klein tunneling, Fermi arcs and so on. Database searches using symmetry data at high-symmetry points have catalogued thousands of topological materials revealing a magnitude of band nodes (BNs) at high-symmetry points or lying within high-symmetry lines/planes. A complete mapping from symmetry data (namely, representation of little group) in any BN to the k ⋅ p $k\cdot p$ model characterizing low-energy Hamiltonian around the BN (and from the k ⋅ p $k\cdot p$ model to concrete BN, inversely), is expected to complete the characterization of all BNs and gapless states. Here we first review recent progress on classifying BNs by systematically and automatically constructing k ⋅ p $k\cdot p$ models based on recently completed tabulation of all irreducible (co-)representation matrices of little groups of the 1651 magnetic space groups. As one indispensable input in constructing a symmetry-allowed and generic k ⋅ p $k\cdot p$ model, the expansion order, has been carefully and systematically truncated for any BN to a reasonable nonzero integer, by comparing the emanating nodal structure (ENS, including nodal point, nodal line and nodal surface) near the BN obtained by the explicitly constructed k ⋅ p $k\cdot p$ model and that by pure symmetry analysis using compatibility relations (CRs). Owing to the progress, we are able to summarize all 25 different configurations of ENS near BN required by CRs, provide a complete mapping from k ⋅ p $k\cdot p$ model to its realization around BN, and the corresponding ENS by CRs in an accessible file, and also reveal the protection mechanism of additional nodal lines that escape conventional analysis by CRs and is only predictable by constructing k ⋅ p $k\cdot p$ model. The symmetry-based classification results on all BNs could facilitate large-scale materials prediction and hold promise for realizing topological semimetals suitable for device applications.