Symmorphic Intersecting Nodal Rings in Semiconducting Layers

Observing Relative Homotopic Degeneracy Conversions with Circuit Metamaterials

Making nodal lines (NLs) deterministic is quite challenging because directly probing them requires bulk momentum resolution. Here, based on the general scattering theory, we show that the Bloch modes of the circuit metamaterials can be selectively excited with a proper source. Consequently, the transport measurement for characterizing the circuit band structure is momentum resolved. Facilitated by this bulk resolution, we systematically demonstrate the degeneracy conversions ruled by the relative homotopy, including the conversions between Weyl points (WPs) and NLs, and between NLs. It is experimentally shown that two WPs with opposite chirality in a two-band model surprisingly convert into an NL rather than annihilating. And the multiband anomaly (due to the delicate property) in the NL-to-NL conversions is also observed, which in fact is captured by the non-Abelian relative homotopy. Additionally, the physical effects owing to the conversions, like the Fermi arc connecting NLs and the parallel transport of eigenstates, are discussed as well. Other types of degeneracy conversions, such as those induced by spin-orbit coupling or symmetry breaking, are directly amenable to the proposed circuit platform.

Nodal-loop half metallicity in a two-dimensional Fe4N2 pentagon crystal with room-temperature ferromagnetism

Two-dimensional (2D) materials with fully spin-polarized nodal-loop band crossing are a class of topological magnetic materials, holding promise for high-speed low-dissipation spintronic devices. Recently, several 2D nodal-loop materials have been reported in theory and experiment, such as Cu₂Si, Be₂C, CuSe, and Cr₂S₃ monolayers, adopting triangular, tetragonal, hexagonal, or complex lattices. However, a 2D nodal-loop half metal with room-temperature magnetism is still less reported. Here, we report that the 2D Fe₄N₂ pentagon crystal is a nodal-loop half metal with room-temperature magnetism over 428 K and a global minimum structure via first-principles calculations and global structure search. The Dirac nodal lines in Fe₄N₂ form a flat nodal loop at the Fermi level and a spin-polarized type-II nodal-loop above the Fermi level, which are protected by mirror symmetry. Our results establish Fe₄N₂ as a platform to obtain nodal-loop half metallicity in the 2D pentagon lattice and provide opportunities to build high-speed low-dissipation spintronics in the nanoscale.

Intrinsic in-plane nodal chain and generalized quaternion charge protected nodal link in photonics

Nodal lines are degeneracies formed by crossing bands in three-dimensional momentum space. Interestingly, these degenerate lines can chain together via touching points and manifest as nodal chains. These nodal chains are usually embedded in two orthogonal planes and protected by the corresponding mirror symmetries. Here, we propose and demonstrate an in-plane nodal chain in photonics, where all chained nodal lines coexist in a single mirror plane instead of two orthogonal ones. The chain point is stabilized by the intrinsic symmetry that is specific to electromagnetic waves at the Г point of zero frequency. By adding another mirror plane, we find a nodal ring that is constructed by two higher bands and links with the in-plane nodal chain. The nodal link in momentum space exhibits non-Abelian characteristics on a C₂T - invariant plane, where admissible transitions of the nodal link structure are determined by generalized quaternion charges. Through near-field scanning measurements of bi-anisotropic metamaterials, we experimentally mapped out the in-plane nodal chain and nodal link in such systems.

Observation of Non-Abelian Nodal Links in Photonics

In crystals, two bands may cross each other and form degeneracies along a closed loop in the three-dimensional momentum space, which is called nodal line. Nodal line degeneracy can be designed to exhibit various configurations such as nodal rings, chains, links, and knots. Very recently, non-Abelian band topology was proposed in nodal link systems, where the nodal lines formed by consecutive pairs of bands exhibit interesting braiding structures and the underlying topological charges are described by quaternions. Here, we experimentally demonstrate non-Abelian nodal links in a biaxial hyperbolic metamaterial. The linked nodal lines threading through each other are formed by the crossings between three adjacent bands. Based on the non-Abelian charges, we further analyze various admissible nodal link configurations for the three-band system. On the interface between the metamaterial and air, surface bound states in the continuum are observed, which serves as the symmetry-enforced derivative of drumhead surface states from the linked nodal lines. Our work serves as a direct observation of the global topological structures of nodal links, and provides a platform for studying non-Abelian topological charge in the momentum space.

Jones Polynomial and Knot Transitions in Hermitian and non-Hermitian Topological Semimetals

Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this Letter, we use the Jones polynomial as a general topological invariant to capture the global knot topology of the oriented nodal lines. We show that every possible change in Jones polynomial is attributed to the local evolutions around every point where two nodal lines touch. As an application of our theory, we show that nodal chain semimetals with four touching points can evolve to a Hopf link. We extend our theory to 3D non-Hermitian multiband exceptional line semimetals. Our work provides a recipe to understand the transition of the knot topology for protected nodal lines.

Critical topological nodal points and nodal lines/rings in Kagome graphene

Critical topological phases, possessing flat bands, provide a platform to study unique topological properties and transport phenomena under a many-body effect. Here, we propose that critical nodal points and nodal lines or rings can be found in Kagome lattices. After the C3 rotation symmetry of a single-layer Kagome lattice is eliminated, a quadratic nodal point splits into two critical nodal points. When the layered Kagome lattices are stacked into a three-dimensional (3D) structure, critical nodal lines or rings can be generated by tuning the interlayer coupling. Furthermore, we use Kagome graphene as an example to identify that these critical phases could be obtained in real materials. We also discuss flat-band-induced ferromagnetism. It is found that the flat band splits into two spin-polarized bands by hole-doping, and as a result the Dirac-type critical phases evolve into Weyl-type phases.

Nodal Line Spin-Gapless Semimetals and High-Quality Candidate Materials

Spin-gapless semimetals (SGSMs), which generate 100% spin polarization, are viewed as promising semi-half-metals in spintronics with high speed and low consumption. We propose and characterize a new Z_{2} class of topological nodal line (TNL) in SGSMs. The proposed TNLSGSMs are protected by space-time inversion symmetry or glide mirror symmetry with two-dimensional (2D) fully spin-polarized nearly flat surface states. Based on first-principles calculations and effective model analysis, a series of high-quality materials with R3[over ¯]c and R3c space groups are predicted to realize such TNLSGSMs (chainlike). The 2D fully spin-polarized nearly flat surface states may provide a route to achieving equal spin pairing topological superconductivity as well as topological catalysts.

Nodal-line semimetal superlattices

Spatial modulations, such as superlattices, to realize topological materials have recently been studied in theoretical and experimental works. In this paper, we investigate properties of the superlattices of the nodal-line semimetal and the normal insulator. We consider two types of superlattices, with the stacking direction being perpendicular or parallel to the plane where the nodal line lies. In particular, we show that when the stacking direction is parallel to the plane, the nodal lines remain but they change their shapes because of the folding of the Brillouin zone. We also study the superlattices with magnetization. One can expect that the quantum anomalous Hall (QAH) phase emerges in some cases, depending on the direction of the magnetization. If the magnetization is along the C ₂-invariant axis, the superlattice becomes the Weyl semimetal phase if the C ₂-invariant axis intersects the nodal lines, and otherwise it becomes the QAH phase.

Band Topology and Linking Structure of Nodal Line Semimetals with Z_{2} Monopole Charges

We study the band topology and the associated linking structure of topological semimetals with nodal lines carrying Z_{2} monopole charges, which can be realized in three-dimensional systems invariant under the combination of inversion P and time reversal T when spin-orbit coupling is negligible. In contrast to the well-known PT-symmetric nodal lines protected only by the π Berry phase, in which a single nodal line can exist, the nodal lines with Z_{2} monopole charges should always exist in pairs. We show that a pair of nodal lines with Z_{2} monopole charges is created by a double band inversion process and that the resulting nodal lines are always linked by another nodal line formed between the two topmost occupied bands. It is shown that both the linking structure and the Z_{2} monopole charge are the manifestation of the nontrivial band topology characterized by the second Stiefel-Whitney class, which can be read off from the Wilson loop spectrum. We show that the second Stiefel-Whitney class can serve as a well-defined topological invariant of a PT-invariant two-dimensional insulator in the absence of Berry phase. Based on this, we propose that pair creation and annihilation of nodal lines with Z_{2} monopole charges can mediate a topological phase transition between a normal insulator and a three-dimensional weak Stiefel-Whitney insulator. Moreover, using first-principles calculations, we predict ABC-stacked graphdiyne as a nodal line semimetal (NLSM) with Z_{2} monopole charges having the linking structure. Finally, we develop a formula for computing the second Stiefel-Whitney class based on parity eigenvalues at inversion-invariant momenta, which is used to prove the quantized bulk magnetoelectric response of NLSMs with Z_{2} monopole charges under a T-breaking perturbation.

Conversion Rules for Weyl Points and Nodal Lines in Topological Media

According to a widely held paradigm, a pair of Weyl points with opposite chirality mutually annihilate when brought together. In contrast, we show that such a process is strictly forbidden for Weyl points related by a mirror symmetry, provided that an effective two-band description exists in terms of orbitals with opposite mirror eigenvalue. Instead, such a pair of Weyl points convert into a nodal loop inside a symmetric plane upon the collision. Similar constraints are identified for systems with multiple mirrors, facilitating previously unreported nodal-line and nodal-chain semimetals that exhibit both Fermi-arc and drumhead surface states. We further find that Weyl points in systems symmetric under a π rotation composed with time reversal are characterized by an additional integer charge that we call helicity. A pair of Weyl points with opposite chirality can annihilate only if their helicities also cancel out. We base our predictions on topological crystalline invariants derived from relative homotopy theory, and we test our predictions on simple tight-binding models. The outlined homotopy description can be directly generalized to systems with multiple bands and other choices of symmetry.

Nodal-chain network, intersecting nodal rings and triple points coexisting in nonsymmorphic Ba3Si4

Coexistence of topological elements in topological metals/semimetals (TMs) has gradually attracted attention. However, non-topological factors always interfere with the Fermi surface and cover interesting topological properties. Here, we find that Ba3Si4 is a "clean" TM which contains coexisting nodal-chain networks, intersecting nodal rings (INRs) and triple points, in the absence of spin-orbit coupling (SOC). Moreover, the nodal rings in the topological phase exhibit diverse types: from type-I and type-II to type-III rings according to band dispersions. All of the topological elements are generated by crossings of three energy bands, and thus they are correlated rather than mutually independent. When some structural symmetries are eliminated by an external strain, the topological phase evolves into another phase including a Hopf link, a one-dimensional nodal chain and new INRs.

Spindle nodal chain in three-dimensional α′ boron

A three-dimensional 3D-α′ boron is proposed and a novel spindle nodal chain is found in this material.