Straight Photonic Nodal Lines with Quadrupole Berry Curvature Distribution and Superimaging "Fermi Arcs".

Topological metals or semimetals have attracted great research attention and interest in condensed matter physics and chemistry due to their exotic properties. Different from the conventional topological insulators, topological metals or semimetals are characterized by distinct topological surface states, such as a Fermi arc or a drumhead surface state, which are often used in experiments to verify the corresponding topological properties. However, the current study in this field is strongly limited in the experimental characterization because of the extreme lack of perfect material candidates with a clean band structure and clear surface states. In this work, based on theoretical calculations, we propose a new topological semimetal TiS2, which has an orthorhombic structure and exhibits excellent stability. Calculated electronic band structures reveal that there is a single Weyl nodal ring in the ky = 0 plane. A detailed symmetry analysis is provided and the corresponding surface state is calculated, which exhibits both a large energy variation of 1.5 eV and wide space distribution without and with the spin orbit coupling effect. Besides, the surface states are well separated from the bulk state. These ideal features together make TiS2 a promising nodal line semimetal for experimental investigation. In combination with the other two isostructural compounds TiSe2 and TiTe2 with similar properties, their further experimental synthesis and characterization can be highly expected and the corresponding study for the topological nodal line state can thus be greatly facilitated.

Recently, topological semimetals featured with symmetry-protected nodal line degeneracies in three-dimensional momentum space have attracted great attention. In this work, we have constructed a three-dimensional acoustic metacrystal that hosts topological nodal lines pinning to the edges of a Brillouin zone. Markedly different from the nodal rings or chains observed previously, here the nodal lines are geometrically straight and fully stabilized by the crystal symmetry, which are highly unique and favored for detection. In addition to the symmetry-enforced nodal lines identified by transmission measurements, exotic waterslidelike surface states have been unveiled through scanning surface fields. Excellent agreements are found between our experiments and simulations. Our study may provoke new possibilities for controlling sound, such as realizing unusual sound radiation and scattering.

We consider a two-orbital tight-binding model defined on a layered three-dimensional hexagonal lattice to investigate the properties of topological nodal lines and their associated drumhead surface states. We examine these surface states in centrosymmetric systems, where the bulk nodal lines are of Dirac type (i.e., four-fold degenerate), as well as in non-centrosymmetric systems with strong Rashba and/or Dresselhaus spin-orbit coupling, where the bulk nodal lines are of Weyl type (i.e., two-fold degenerate). We find that in non-centrosymmetric systems the nodal lines and their corresponding drumhead surface states are fully spin polarized due to spin-orbit coupling. We show that unique signatures of the topologically nontrivial drumhead surface states can be measured by means of quasiparticle scattering interference, which we compute for both Dirac and Weyl nodal line semimetals. At the end, we analyze the possible crystal structures with a symmetry that supports flat surface states which are effectively ringlike.

Multiple fermionic states with clear nontrivial surface signature in CsCl-type compound ErAs

Based on first-principles calculations and symmetry analysis, we propose that trigonal CaI2 with the space group P3̄m1 possesses straight and twisted open nodal-line phonon states with linear dispersion. The symmetry analysis indicates that joint symmetry PT and rotational symmetry C3z protect the straight nodal lines along Γ-A and K-H while PT and mirror symmetry M010 (M110) maintain the twisted nodal lines that traverse Γ-M (Γ-K) and A-L (A-H). The calculated π Berry phase suggests that all the nodal lines are nontrivial and the corresponding drumhead-like surface states are clearly visible in the observation window, which is less than 6 THz, suggesting a significant chance for them to be measured using meV-resolution inelastic X-ray scattering. The distribution of the nodal lines in the Brillouin zone is also confirmed by the phononic tight-binding model. Furthermore, the isostructural compounds MgBr2 and MgI2 show similar phonon spectra and topological nontrivial surface states. This work provides promising candidates for investigating straight and twisted open nodal-line phonon states in a single material, which will facilitate future experimental observation.

Three-dimensional topological nodal lines, the touching curves of two bands in momentum space, which give rise to drumhead surface states, provide an opportunity to explore a variety of exotic phenomena. However, solid evidence for a flat drumhead surface state remains elusive. In this paper, we report a realization of three-dimensional nodal line dispersions and drumhead surface states in phononic crystal. Profiting from its macroscopic nature, the phononic crystal permits a flexible and accurate fabrication for materials with ring-like nodal lines and drumhead surface states. Phononic nodal rings of the lowest two bands and, more importantly, topological drumhead surface states are unambiguously demonstrated. Our system provides an ideal platform to explore the intriguing properties of acoustic waves endowed with extraordinary dispersions.

The interplay between symmetry breaking and topological electronic structure is crucial to design anomalous transport properties in materials. Materials with strong or quantum electromagnetic responses have an extensive impact on the development of data storage, information processing, energy conversion, etc. In magnetic materials, the anomalous transport of anomalous Hall effect, anomalous Nernst effect, and magneto-optical effect et al. can be understood from the Berry curvature of the electronic band structures. Two typical band structures of Weyl points and nodal line band structures host strong local Berry curvature. Since the Berry curvature is time-reversal symmetry odd, such strong Berry curvatures can lead to strongly enhanced anomalous transport signals. With this guiding principle, we studied the anomalous Hall effect in magnetic Weyl semimetal Co3Sn2S2 [1-3] and magnetic nodal line semimetal in Heusler compound Co2Mn(Ga/Al) [4].With a mirror symmetry, the inverted band structure forms a nodal loop in the absence of spin-orbital coupling. This nodal line can be broken by spin-orbital coupling and a bandgap opens, which generates non-zero Berry curvature in the bandgap and forms a hot loop, see Figure 1a-b. Such strong Berry curvature in the magnetic system can lead to a strongly enhance or even quantized anomalous Hall effect. We applied this idea to real materials of magnetic Heusler compounds Co2Mn(Ga/Al). Protected by mirror symmetries the band inversion between the bands with opposite mirror eigenvalue forms three gapless nodal lines in the kx=0, ky=0, and kz=0 mirror planes, respectively. With spin-orbital coupling, the symmetry of the system is reduced. Taking magnetic along z, the mirror symmetries in kx=0 and ky=0 planes are broken, which leads to band anti-crossings with strong local Berry curvature locating in the opened bandgap around original nodal lines, see Figure 1d. Integral of the Berry curvature in the whole k-space gives a large intrinsic anomalous Hall conductivity reaching ~1500 to ~2000 S/cm [4].Weyl points is another typical band structure and present as the Berry curvature monopole, and therefore naturally results in a strong anomalous Hall effect. In ideal models with only one pair of Weyl points locating at the Fermi level, the intrinsic anomalous Hall conductivity can be presented as the combination distance of Weyl points and the quantized anomalous Hall conductance. Inspired by these excellent relations, we studied the anomalous Hall effect in Co3Sn2S2, and a new record of three-dimensional anomalous Hall angel (~20%) was observed, which offers the 1st three-dimensional material with both strong anomalous Hall conductivity and anomalous Hall angle [1]. It indeed shows as a Weyl semimetal from electronic band structure analysis. One crucial symmetry in Co3Sn2S2 is the three mirror planes parallel to the c direction, which results in three pairs of nodal lines connected by a c3z rotation symmetry. Because the magnetization is aligned along the z-direction, the mirror symmetries are broken by spin-orbital coupling. Meanwhile, one pair of Weyl points with opposite chirality remains along each of the former nodal lines, leading to the so large anomalous Hall effect.Though the strong anomalous Hall effect provides a promising signature for the existence of Weyl and nodal line band structure. Our transport work about Co3Sn2S2 and Co2MnGa/Al inspired the direct band structure detection by ARPES and STM [5-7], and they are in turn became the 1st experimentally verified magnetic Weyl semimetal and nodal line semimetal, respectively.Applying temperature gradient instead of the electrical field, the Weyl points and nodal lines induced Berry curvature can also lead to strongly enhanced anomalous Nernst effect. From our theoretical calculations and experimental measurements, the anomalous Nernst conductivity can reach around 3 and 6 A/(m-K) in Co3Sn2S2 [8] and Co2MnGa [9], respectively, with Co2MnGa keeping the record. Owing to the large anisotropy, Co3Sn2S2 is, so far, the only material with a large anomalous Nernst effect with zero magnetic fields. In addition, very recently, a giant magneto-optical response was observed in Co3Sn2S2 with the applied field from polarized light [10].Very recently, a strong interest in antiferromagnets is rising. In an antiferromagnet without such kind of joint TO symmetry to reverse Berry curvature, it allows the existence of anomalous Hall effect, anomalous Nernst effect, magneto-optical responses, and special spin current, etc. The nonzero anomalous Hall effect in antiferromagnets was proposed as early as 2001 in distorted non-linear magnetic structures [11]. However, its experimental realization was not successful until 2015 [12-16]. This understanding can be further expanded into collinear antiferromagnets. Different from non-linear antiferromagnets, the collinear antiferromagnetic structure can be usually understood from two sublattices connected by translation of inversion operation. Therefore, there are mainly two ways to break the joint symmetry, to replace the magnetic atoms connected by the joint TO symmetry, or change the the nonmagnetic sites. With this understanding, we predicted the anomalous Hall and Nernst effect in anti-Heusler Weyl semimetal Ti2MnAl [17-18]. **

We present a study of "nodal semimetal" phases, in which non-degenerate conduction and valence bands touch at points (the "Weyl semimetal") or lines (the "line node semimetal") in three-dimensional momentum space. We discuss a general approach to such states by perturbation of the critical point between a normal insulator (NI) and a topological insulator (TI), breaking either time reversal (TR) or inversion symmetry. We give an explicit model realization of both types of states in a NI--TI superlattice structure with broken TR symmetry. Both the Weyl and the line-node semimetals are characterized by topologically-protected surface states, although in the line-node case some additional symmetries must be imposed to retain this topological protection. The edge states have the form of "Fermi arcs" in the case of the Weyl semimetal: these are chiral gapless edge states, which exist in a finite region in momentum space, determined by the momentum-space separation of the bulk Weyl nodes. The chiral character of the edge states leads to a finite Hall conductivity. In contrast, the edge states of the line-node semimetal are "flat bands": these states are approximately dispersionless in a subset of the two-dimensional edge Brillouin zone, given by the projection of the line node onto the plane of the edge. We discuss unusual transport properties of the nodal semimetals, and in particular point out quantum critical-like scaling of the DC and optical conductivity of the Weyl semimetal, and similarities to the conductivity of graphene in the line node case.

Here we systematically investigate the structure, phase stability, half-metallicity, and topological electronic structure for a topological spintronic material $\mathrm{Na}{\mathrm{V}}_{2}{\mathrm{O}}_{4}$. The material has a tetragonal structure with excellent dynamical and thermal stabilities. It shows a half-metallic ground state, where only the spin-up bands are present near the Fermi level. These bands are demonstrated to form a nodal line with the double degeneracy on the ${k}_{z}=0$ plane. The nodal line is robust against spin-orbit coupling, under the protection of the mirror symmetry. The nodal line band structure is very clean, thus the drumhead surface states can be clearly identified. Remarkably, the nodal line and drumhead surface states have the $100%$ spin polarization, which are highly desirable for spintronics applications. In addition, by shifting the magnetic field in-plane, we find that the nodal line can transform into a single pair of Weyl nodes. The nodal-line and Weyl-node fermions in the bulk, as well as the drumhead fermions on the surface are all fully spin-polarized, which may generate interesting physical properties and promising applications.

Theoretical study of compounds XSb (X = La, Pr, Nd): Realization of inner nodal chains, nodal line frame, and Dirac points

Topological nodal-line semimetals are exotic conductors that host symmetry-protected conducting nodal-lines in their bulk electronic spectrum and nontrivial drumhead states on the surface. Based on first-principles calculations and an effective model analysis, we identify the presence of topological nodal-line semimetal states in the TT'X family of compounds (T, T' = transition metal, X= Si, or Ge) in the absence of spin-orbit coupling (SOC). Taking ZrPtGe as an exemplar system, we show that this material harbors a single nodal line on the $k_y=0$ plane, which is protected by the $M_y$ mirror plane symmetry. Surface electronic structure calculations further reveal the existence of a drumhead surface state nested inside the nodal line projection on the (010) surface with a saddle-like energy dispersion. When the SOC is included, the nodal line gaps out and the system transitions to a strong topological insulator state with $Z_2=(1;000)$. The topological surface state evolves from the drumhead surface state via the sharing of its saddle-like energy dispersion within the bulk energy gap. These features differ remarkably from those of the currently known topological surface states in topological insulators such as Bi$_2$Se$_3$ with Dirac-cone-like energy dispersions.

The study of topological semimetals has been extended to more general topological nodal systems such as metamaterials and artificial periodic structures. Among various nodal structures, triply degenerate nodal line (TDNL) is rare and, hence, has received little attention. In this work, we have proposed a simple tight-binding (TB) model, which hosts a topological non-trivial TDNL. This TDNL not only has the drumhead surface states (DSSs) as usual nodal line systems but also has surface states that form a contracted-drumhead shape. The shape and area of this contracted drumhead can be tuned by the hopping parameters of the model. This provides an effective way to modulate surface states and their density of states, which can be important in future applications of topological nodal systems.

For topological materials with a coexistence of Weyl nodes and nodal rings, their unique surface-state configuration and connection still need to be studied and discussed. In this Letter, we predict a ferromagnetic (FM) material, Cs_{2}MoCl_{6}, with a coexistence of Weyl and nodal ring fermions in its spinful FM electronic band structure, which is unusual since FM materials are very rare in nature and nodal ring band crossings will usually open a gap when spin-orbit coupling is taken into consideration. We find that the surface states of Cs_{2}MoCl_{6} show different properties along different directions, i.e., the surface states are in a drumhead shape showing the nodal ring property on the (001) surface and in a helicoid shape showing the Weyl property on the (010) surface. Interestingly, both the drumhead surface states and the helicoid surface states will cross the projected points of the Weyl and nodal ring along different directions. In particular, helicoid surface states on the (010) surface will meet the nodal ring tangentially, with their shapes changing abruptly as a function of the energy. We implement both first-principles calculation and an analytical model to understand the unique surface-state connection for systems with a coexistence of Weyl nodes and nodal rings (or nodal lines). This result is universal and irrespective of the presence or absence of time-reversal symmetry (T).

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.

High-order one-dimensional (1D) fermion in ferromagnetic RbFeF3