Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems

Abstract

Weyl points in three spatial dimensions are characterized by a $\mathbb{Z}$-valued charge -- the Chern number -- which makes them stable against a wide range of perturbations. A set of Weyl points can mutually annihilate only if their net charge vanishes, a property we refer to as robustness. While nodal loops are usually not robust in this sense, it has recently been shown using homotopy arguments that in the centrosymmetric extension of the $\text{AI}$ symmetry class they nevertheless develop a ${\mathbb{Z}}_2$ charge analogous to the Chern number. Nodal loops carrying a non-trivial value of this ${\mathbb{Z}}_2$ charge are robust, i.e. they can be gapped out only by a pairwise annihilation and not on their own. As this is an additional charge independent of the Berry $\pi$-phase flowing along the band degeneracy, such nodal loops are, in fact, doubly charged. In this manuscript, we generalize the homotopy discussion to the centrosymmetric extensions of all Atland-Zirnbauer classes. We develop a taylored mathematical framework dubbed the AZ+$\mathcal{I}$ classification and show that in three spatial dimensions such robust and multiply charged nodes appear in four of such centrosymmetric extensions, namely AZ+$\mathcal{I}$ classes $\text{CI}$ and $\text{AI}$ lead to doubly charged nodal lines, while $\text{D}$ and $\text{BDI}$ support doubly charged nodal surfaces. We remark that no further crystalline symmetries apart from the spatial inversion are necessary for their stability. We provide a description of the corresponding topological charges, and develop simple tight-binding models of various semimetallic and superconducting phases that exhibit these nodes. We also indicate how the concept of robust and multiply charged nodes generalizes to other spatial dimensions.

23 pages, 11 figures, 6 tables

References in corpus (27)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Chiral tunneling and the Klein paradox in graphene
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological quantum chemistry
• Weyl semimetal phase in non-centrosymmetric transition metal monophosphides
• Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase
• Dirac materials
• Classification of stable three-dimensional Dirac semimetals with nontrivial topology
• Topological Node-Line Semimetal in Three Dimensional Graphene Networks
• Topological nodal line semimetals
• Topology of crystalline insulators and superconductors
• Wannier representation of Z_2 topological insulators
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
• Evidence for time-reversal symmetry breaking in the non-centrosymmetric superconductor LaNiC${}_2$
• Bogoliubov Fermi surfaces in superconductors with broken time-reversal symmetry
• Exotic Superconducting Properties in the Electron-Hole Compensated Heavy Fermion `Semimetal' URu2Si2
• Topological surface states in nodal superconductors
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Composite Weyl nodes stabilized by screw symmetry with and without time reversal
• Topological Blount's theorem of odd-parity superconductors
• Inflated nodes and surface states in superconducting half-Heusler compounds
• Weyl semimetal from spontaneous inversion symmetry breaking in pyrochlore oxides
• Chern-Simons Theory and Wilson Loops in the Brillouin Zone
• Topological Crystalline Metal in Orthorhombic Perovskite Iridates

Cited by in corpus (149)

• Nodal surface semimetals: Theory and material realization
• Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe${}_2$ (X=Mo,W)
• Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Non-Abelian band topology in noninteracting metals
• Topological exceptional surfaces in non-Hermitian systems with parity-time and parity-particle-hole symmetries
• All Topological Bands of All Nonmagnetic Stoichiometric Materials
• Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\text{ZrTe}$
• Topological Euler class as a dynamical observable in optical lattices
• Topological Classification of Non-Hermitian Hamiltonians
• Nonsymmorphic-symmetry-protected hourglass Dirac loop, nodal line, and Dirac point in bulk and monolayer $X_3$SiTe${}_6$ ($X=$ Ta, Nb)
• Geometric approach to fragile topology beyond symmetry indicators
• Knotted Non-Hermitian Metals
• Bogoliubov Fermi surfaces: General theory, magnetic order, and topology
• Exceptional rings protected by emergent symmetry for mechanical systems
• Jones polynomial and knot transitions in topological semimetals
• Boundary criticality of $PT$-invariant topology and second-order nodal-line semimetals
• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
• Topological correspondence between magnetic space group representations
• Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates
• Hyperbolic Matter in Electrical Circuits with Tunable Complex Phases
• Symmetry representation approach to topological invariants in $C_{2z}T$-symmetric systems
• Stiefel-Whitney classes and topological phases in band theory
• Non-abelian topology of nodal-line rings in $\mathcal{P}\mathcal{T}$-symmetric systems
• Unified understanding of symmetry indicators for all internal symmetry classes
• Topological superconducting phase in high-Tc superconductor MgB2 with Dirac-nodal-line fermions
• Higher-Order Topological Superconductivity of Spin-Polarized Fermions
• Tailoring $T_c$ by symmetry principles: The concept of Superconducting Fitness
• Topological carbon materials: a new perspective
• Robust parity-mixed superconductivity in disordered monolayer transition metal dichalcogenides
• Circumventing the no-go theorem: A single Weyl point surrounded by nodal walls
• Exceptional Non-Abelian Topology in Multiband Non-Hermitian Systems
• Switching spinless and spinful topological phases with projective $PT$ symmetry
• Systematic investigation of emergent particles in type-III magnetic space groups
• Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet
• ${\mathbb{Z}}_2$-enriched symmetry indicators for topological superconductors in the 1651 magnetic space groups
• Weyl nodal surfaces
• Discovering topological surface states of Dirac points
• Unconventional superconducting gap structure protected by space group symmetry
• Topological acoustic triple point
• Conversion rules for Weyl points and nodal lines in topological media
• Exceptional band touching for strongly correlated systems in equilibrium
• Classification of topological crystalline superconducting nodes on high-symmetry lines: Point nodes, line nodes, and Bogoliubov Fermi surfaces
• Topologically non-trivial photonic surface degeneracy in a photonic metamaterial
• Realization and Topological Properties of Third-Order Exceptional Lines Embedded in Exceptional Surfaces
• Magnetic Weyl semimetal in K${}_2$Mn${}_3$(AsO${}_4$)${}_3$ with the minimum number of Weyl points
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• From triple-point materials to multiband nodal links
• Non-Abelian hyperbolic band theory from supercells
• Two topologically distinct Dirac-line semimetal phases and topological phase transitions in rhombohedrally stacked honeycomb lattices
• d-wave superconductivity and Bogoliubov-Fermi surfaces in Rarita-Schwinger-Weyl semimetals
• Theory of optical responses in clean multi-band superconductors
• Superconductivity without inversion and time-reversal symmetries
• Experimental consequences of Bogoliubov Fermi surfaces
• Symmetry-enforced band crossings in tetragonal materials: Dirac and Weyl degeneracies on points, lines, and planes
• Pseudo Dirac Nodal Sphere: Unusual Electronic Structure and Material Realization
• Stiefel-Whitney topological charges in a three-dimensional acoustic nodal-line crystal
• Bogoliubov Fermi Surfaces in Spin-1/2 Systems: Model Hamiltonians and Experimental Consequences
• Bogoliubov Fermi surfaces stabilized by spin-orbit coupling
• Symmetry-Enforced Nodal Chain Phonons
• Symmetry-protected line nodes and Majorana flat bands in nodal crystalline superconductors
• Hyperbolic non-Abelian semimetal
• Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric
• Topology and entanglement in quench dynamics
• Classification of crystalline insulators without symmetry indicators: atomic and fragile topological phases in twofold rotation symmetric systems
• Topological continuum charges of acoustic phonons in 2D
• Euler Obstructed Cooper Pairing: Nodal Superconductivity and Hinge Majorana Zero Modes
• Symmetry-based approach to nodal structures: Unification of compatibility relations and gapless point classifications
• Nodal Andreev Spectra in Multi-Majorana Three-Terminal Josephson Junctions
• MagneticKP: A package for quickly constructing $\mathbf{k}\cdot \mathbf{p}$ models of magnetic and non-magnetic crystals
• Euler Obstructed Cooper Pairing in Twisted Bilayer Graphene: Nematic Nodal Superconductivity and Bounded Superfluid Weight
• Topological d+s wave superconductors in a multi-orbital quadratic band touching system
• Supercurrent-induced topological phase transitions
• Symmetry, nodal structure, and Bogoliubov Fermi surfaces for nonlocal pairing
• Nodal higher-order topological superconductivity from a ${\mathcal{C}}_4$-symmetric Dirac semimetal
• Quantum Valley Hall effect without Berry curvature
• Universal higher-order bulk-boundary correspondence of triple nodal points
• Landau quantization of nearly degenerate bands, and full symmetry classification of avoided Landau-level crossings
• Weyl nodal point-line Fermion in ferromagnetic Eu${}_5$Bi${}_3$
• Coulomb-induced instabilities of nodal surfaces
• Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index
• Nodal topological superconductivity in nodal-line semimetals
• Magneto-transport and Shubnikov-de Haas oscillations in the layered ternary telluride Ta3SiTe6 topological semimetal
• Third-order charge transport in a magnetic topological semimetal
• Multiband Quantum Criticality of Polar Metals
• Photonic Dirac Nodal Line Semimetals Realized by Hyper-crystal
• Mirror real Chern insulator in two and three dimensions
• Pair-Density-Wave Order and Paired Fractional Quantum Hall Fluids
• Bogoliubov-Fermi surface with inversion symmetry and electron-electron interactions: relativistic analogies and lattice theory
• Triple nodal points characterized by their nodal-line structure in all magnetic space groups
• Alice string in Non-Hermitian Systems
• Bulk-edge correspondence of Stiefel-Whitney and Euler insulators through the entanglement spectrum and cutting procedure
• Multipolar superconductivity in Luttinger semimetals
• Band geometry from position-momentum duality at topological band crossings
• Intra-unit-cell singlet pairing mediated by altermagnetic fluctuations
• Topological insulators and higher-order topological insulators from gauge-invariant 1D lines
• Projected spin texture as a bulk indicator of fragile topology
• Hermitian Topologies originating from non-Hermitian braidings
• Generic phase diagram for Weyl superconductivity in mirror-symmetric superconductors
• Unconventional Majorana fermions on the surface of topological superconductors protected by rotational symmetry
• Symmetry-protected Spinful Magnetic Weyl Nodal Loops and Multi-Weyl Nodes in $5d^n$ Cubic Double Perovskites $(n=1,2)$
• Filling-enforced Dirac loops and their evolutions under various perturbations
• Nodal topology in $d$-wave superconducting monolayer FeSe
• Supersymmetry on the lattice: Geometry, Topology, and Flat Bands
• Surface density of states and tunneling spectroscopy of a spin-3/2 superconductor with Bogoliubov Fermi Surfaces
• Nematic Bogoliubov Fermi surfaces from magnetic toroidal order in FeSe${}_{1-x}$S${}_x$
• Topological charge pumping in spin-dependent superlattices with glide symmetry
• Braiding topology of symmetry-protected degeneracy points in non-Hermitian systems
• Topological nodal superconducting phases and topological phase transition in the hyperhoneycomb lattice
• Observing the nodal-line conversion determined by the relative homotopy
• Diagnosis of topological nodal lines with nontrivial monopole charge in the presence of rotation symmetries
• Time reversal symmetry breaking and $d$-wave superconductivity of triple-point fermions
• Topological Modes Protected by Chiral and Two-Fold Rotational Symmetry in a Spring-Mass Model with a Lieb Lattice Structure
• Topological gapless points in superconductors: From the viewpoint of symmetry
• Linking structures of doubly charged nodal surfaces in centrosymmetric superconductors
• Crystal-symmetry-protected gapless vortex-line phases in superconducting Dirac semimetals
• Supercurrent-Induced Weyl Superconductivity
• Multiple types of unconventional quasiparticles in the chiral crystal CsBe${}_2$F${}_5$
• Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect
• Coexistence of Chiral Majorana Edge States and Bogoliubov Fermi Surfaces in Two-Dimensional Nonsymmorphic Dirac Semimetal/Superconductor Heterostructures
• Chiral Topological Phononic Quasiparticles in Enantiomeric Crystals SrSi${}_2$ and BaSi${}_2$
• Triplet-Superconductivity in Triple-Band Crossings
• Spontaneous surface current in multi-component cubic superconductors with time-reversal symmetry breaking
• Layer Construction of Three-Dimensional Z2 Monopole Charge Nodal Line Semimetals and prediction of the abundant candidate materials
• Ideal Spin-Orbit-Free Dirac Semimetal and Diverse Topological Transitions in Pr${}_8$CoGa${}_3$ Family
• Bulk-entanglement spectrum correspondence in $PT$- and $PC$-symmetric topological insulators and superconductors
• Unconventional Topological Insulators from Extended Topological Band Degeneracies
• Tensor Theory for Higher Dimensional Chern Insulators with Large Chern Numbers
• Morse theory study on the evolution of nodal lines in $\mathcal{P}\mathcal{T}$-symmetric nodal-line semimetals
• Geometric semimetals and their simulation in synthetic matter
• Unremovable linked nodal structures protected by crystalline symmetries in stacked bilayer graphene with Kekulé texture
• Second-order topological insulator and fragile topology in topological circuitry simulation
• Many-body selection rule for quasiparticle pair creations in centrosymmetric superconductors
• Emergent topology in thin films of nodal line semimetals
• Realization of Hopf-link structure in phonon spectra: Symmetry guidance and High-throughput investigation
• Two-dimensional hourglass Weyl nodal loop in monolayer Pb(ClO${}_2$)${}_2$ and Sr(ClO${}_2$)${}_2$
• Nuclear spin relaxation rate of nonunitary Dirac and Weyl superconductors
• Disorder and non-Hermiticity in Kitaev spin liquids with a Majorana Fermi surface
• Topological strongly correlated phases in orthorhombic diamond lattice compounds
• Circular-polarization-selective perfect reflection from chiral superconductors
• Euler band topology in superfluids and superconductors
• Hinge modes of three-dimensional Euler insulators
• Quantum geometry and low-frequency optical conductivity of nodal planes
• Essentially degenerate hidden nodal lines in two-dimensional magnetic layer groups
• Possible topological phases in quantum anomalous Hall insulator/unconventional superconductor hybrid systems
• Thermoelectric effect in a superconductor with Bogoliubov Fermi surfaces
• Proposal for realizing and probing topological crystalline insulators in optical lattices
• Conductivity scaling and absence of localization in disordered nodal line semimetals
• Systematic study for two-dimensional $Z_2$ topological phase transitions at high-symmetry points in all layer groups