Complete classification of band nodal structures

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.

arXiv commentThe Supplementary Information I-V can be downloaded in: https://drive.google.com/file/d/1b7q5vNn4NLwdy_NjTSA_wNpPgOYebSZ6/view?usp=sharing

References in corpus (62)

• Electric Field Effect in Atomically Thin Carbon Films
• Berry Phase Effects on Electronic Properties
• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Weyl and Dirac Semimetals in Three Dimensional Solids
• Chiral tunneling and the Klein paradox in graphene
• Classification of topological quantum matter with symmetries
• Dirac semimetal and topological phase transitions in A3Bi (A=Na, K, Rb)
• Dirac semimetal in three dimensions
• Topological nodal semimetals
• Three Dimensional Dirac Semimetal and Quantum Transports in Cd3As2
• Topological quantum chemistry
• Colloquium: Topological Band Theory
• Theoretical Discovery/Prediction: Weyl Semimetal states in the TaAs material (TaAs, NbAs, NbP, TaP) class
• Weyl semimetal phase in non-centrosymmetric transition metal monophosphides
• Chiral Anomaly and Classical Negative Magnetoresistance of Weyl Metals
• Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals
• Quantum interference and Klein tunneling in graphene heterojunctions
• The (High Quality) Topological Materials In The World
• Dirac materials
• Topological nodal line semimetals with and without spin-orbital coupling
• Catalogue of Topological Electronic Materials
• Classification of stable three-dimensional Dirac semimetals with nontrivial topology
• Dirac Line Nodes in Inversion Symmetric Crystals
• Topological Nodal Line Semimetal and Dirac Semimetal State in Antiperovskite Cu${}_3$PdN
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Dirac Cone Protected by Non-Symmorphic Symmetry and 3D Dirac Line Node in ZrSiS
• Towards ideal topological materials: Comprehensive database searches using symmetry indicators
• Multi-Weyl Topological Semimetals Stabilized by Point Group Symmetry
• The space group classification of topological band insulators
• Topological classification of crystalline insulators through band structure combinatorics
• Discovery of topological Weyl fermion lines and drumhead surface states in a room temperature magnet
• Topological quantum properties of chiral crystals
• Multiple types of topological fermions in transition metal silicides
• High-throughput calculations of magnetic topological materials
• Double Dirac Semimetals in Three Dimensions
• Nodal surface semimetals: Theory and material realization
• Double-Weyl phonons in transition-metal monosilicides
• Nodal-link semimetals
• Structure and Topology of Band Structures in the 1651 Magnetic Space Groups
• Topological Hopf and chain link semimetal states and their application to Co2MnGa (Theory and Materials Prediction)
• Topological semimetals with double-helix nodal link
• Existence of bulk chiral fermions and crystal symmetry
• Nodal-knot semimetals
• Wallpaper Fermions and the Nonsymmorphic Dirac Insulator
• Topological Semimetals carrying Arbitrary Hopf Numbers: Hopf-Link, Solomon's-Knot, Trefoil-Knot and Other Semimetals
• Filling-Enforced Gaplessness in Band Structures of the 230 Space Groups
• CaTe: a new topological node-line and Dirac semimetal
• Twofold quadruple Weyl nodes in chiral cubic crystals
• Weyl-link semimetals
• Dualities and non-Abelian mechanics
• Multifold nodal points in magnetic materials
• Hopf-link multi-Weyl-loop topological semimetals
• Systematic investigation of emergent particles in type-III magnetic space groups
• Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet
• Encyclopedia of emergent particles in type-IV magnetic space groups
• Exhaustive constructions of effective models in 1651 magnetic space groups
• Three-Dimensional Majorana Fermions in Chiral Superconductors
• Topological Materials Discovery using Electron Filling Constraints
• Symmetry-based approach to nodal structures: Unification of compatibility relations and gapless point classifications
• Symmetry-protected Nodal Points and Nodal Lines in Magnetic Materials
• Topology in abundance
• XFe4Ge2 (X = Y, Lu) and Mn3Pt: Filling-enforced magnetic topological metals

Cited by in corpus (21)

• Enumeration of spin-space groups: Towards a complete description of symmetries of magnetic orders
• Symmetry-Enforced Nodal Chain Phonons
• Symmetry invariants and classes of quasiparticles in magnetically ordered systems having weak spin-orbit coupling
• Encyclopedia of emergent particles in 528 magnetic layer groups and 394 magnetic rod groups
• Hierarchy of quasi-symmetries and degeneracies in chiral crystal materials CoSi
• Group Theory on Quasisymmetry and Protected Near Degeneracy
• Plasmons in a two-dimensional nonsymmorphic nodal-line semimetal
• Higher-order topological and nodal superconducting transition-metal sulfides MS (M = Nb and Ta)
• Fundamental laws of chiral band crossings: local constraints, global constraints, and topological phase diagrams
• Magnetic hourglass fermions: from exhaustive symmetry conditions to high-throughput materials predictions
• All hourglass bosonic excitations in the 1651 magnetic space groups and 528 magnetic layer groups
• Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect
• Electronic structures near unmovable nodal points and lines in two-dimensional materials
• Existence of Mexican-hat dispersion and symmetry group of a layer
• Movable but unavoidable nodal lines through high-symmetry points in two-dimensional materials
• Representation Theory for Massless Quasiparticles in Bogoliubov-de Gennes Systems
• Low temperature phase of AuSn${}_4$ induced by the van der Waals interactions
• High-chirality and multiquaternion Weyl nodes in hexagonal ReO${}_3$
• Realization of Hopf-link structure in phonon spectra: Symmetry guidance and High-throughput investigation
• ToMSGKpoint: A user-friendly package for computing symmetry transformation properties of electronic eigenstates of nonmagnetic and magnetic crystalline materials
• Quantum geometry and low-frequency optical conductivity of nodal planes