Complete classification of band nodal structures

TL;DR

This paper provides a comprehensive classification of all symmetry-related band nodal structures in magnetic space and layer groups, including nearly nodal structures, using compatibility relations and $k ext{ extperiodcentered}p$ models, aiding material design.

Contribution

It systematically enumerates all strict and nearly nodal structures in 1651 MSGs and 528 MLGs, introducing a cutoff $k ext{ extperiodcentered}p$ order and exploring nearly NSrs via lower-order models.

Findings

All NSrs are exhaustively enumerated based on irreducible representations.

Nearly NSrs are prevalent and can occur in low-symmetry MSGs.

Higher-order $k ext{ extperiodcentered}p$ models can predict more strict NSrs than compatibility relations.

Abstract

Nodal structures (NSrs) where energy bands meet to be degenerate in the Brillouin zone (BZ) in the form of point, line or surface, received immense research interest in the past decade. However, the nearly NSrs with negligible gaps, can also own many exotic quantum responses and nontrivial topological properties and deserve a systematic investigation. Here, we provide a complete list of all symmetry-diagonalizable strictly and nearly NSrs in the 1651 magnetic space groups (MSGs) and 528 magnetic layer groups (MLGs) in both spinful and spinless settings. We first apply compatibility relations (CRs) which encode how bands split from a band node (BN) (located at a symmetric point of BZ), to obtain all NSrs (emanating from the BN). The NSrs by CRs are definitely strict and are exhaustively enumerated based on all irreducible representations of the little groups for all BNs. We then construct $k\cdot p$ models around BNs and prove that there is a cutoff $k\cdot p$ order (6 at most) for any BN which is sufficient to predict the corresponding strictly NSrs and including higher-order $k\cdot p$ terms cannot gap the NSrs. We provide all $k\cdot p$ models up to the cutoff orders around all the BNs, and comprehensively explore all the nearly NSrs by lowering the $k\cdot p$ order. Our results reveal that the same as BNs, the NSrs, especially the nearly NSrs, are also ubiquitous. For example, while strictly nodal surface can only exist in some nonsymmorphic MSGs, nearly nodal surface can occur even in many low symmetric MSGs. Moreover, we also find that for some BNs, the method of $k\cdot p$ modeling could give more strictly NSrs than CRs. With our complete list, one can conveniently obtain all possible NSrs for any two/three-dimensional and nonmagnetic/magnetic material, and given a target NSr, one can also design/search for materials realizations based on the MSGs/MLGs.