Nodal Surface Research Articles

Carrier concentrations and optical conductivity of a band-inverted semimetal in two and three dimensions

Here we study the single-particle, electronic transport and optical properties of a gapped system described by a simple two-band Hamiltonian with inverted valence bands. We analyze its properties in the three-dimensional (3D) and the two-dimensional (2D) case. The insulating phase changes into a metallic phase when the band gap is set to zero. The metallic phase in the 3D case is characterized by a nodal surface. This nodal surface is equivalent to a nodal ring in two dimensions. Within a simple theoretical framework, we calculate the density of state, the total and effective charge carrier concentration, the Hall concentration and the Hall coefficient, for both 2D and 3D cases. The main result is that the three concentrations always differ from one another in the present model. These concentrations can then be used to resolve the nature of the electronic ground state. Similarly, the optical conductivity is calculated and discussed for the insulating phase. We show that there are no optical excitations in the metallic phase. Finally, we compare the calculated optical conductivity with the rule-of-thumb derivation using the joint density of states.

Ideal topological nodal-surface phonons in RbTeAu-family materials

By using first-principles calculations and symmetry analysis, we uncover that the rubidium-catena-telluridoaurate-family materials in space group (SG) 51 can exhibit ideal nodal-surface phonons separately in the three-dimensional Brillouin zone, characterized by nearly flat phonon bands in their phonon spectra. The nodal surfaces are protected both by the screw rotation symmetry and the time reversal symmetry. Moreover, our theoretical calculations demonstrate that the nodal surfaces are composed by twofold degenerate nodal points, which can form a straight singularity nodal line in the (001) surface, and that the (001) surface states display Dirac-type crossings and surface flat-band phononic states, confirming the topologically nontrivial nature of nodal-surface phonons. Our findings not only support that another class of nodal phononic states, i.e., one-node surface phonons, can exist in realistic materials in 230 SGs, but also put forward an effective way to search for nodal-surface phonons.

Coexistence of zero-, one-, and two-dimensional degeneracy in tetragonal SnO2 phonons

Based on the dimension of degeneracy, topological electronic systems can roughly be divided into three parts: nodal point, line and surface materials corresponding to zero-, one- and two-dimensional degeneracy, respectively. In parallel to electronic systems, the concept of topology was extended to phonons, promoting the birth of topological phonons. Till date, few nodal point, line and surface phonons candidates have been predicted in solid-state materials. In this study, based on symmetry analysis and first-principles calculation, for the first time, we prove that zero-, one- and two-dimensional degeneracy co-exist in the phonon dispersion of one single realistic solid-state material SnO$_2$ with \textit{P}4$_2$/\textit{mnm} structure. In contrast to the previously reported electronic systems, the topological phonons observed in SnO$_2$ are not restricted by the Pauli exclusion principle, and they experience negligible spin-orbit coupling effect. Hence, SnO$_2$ with multiple dimensions of degeneracy phonons is a good platform for studying the entanglement among nodal point, line and surface phonons. Moreover, obvious phonon surface states are visible, which is beneficial for experimental detection.

Symmetry-enforced ideal lanternlike phonons in the ternary nitride Li6WN4

Condensed matter systems contain both fermionic and bosonic quasiparticles. Owing to the constraint imposed by the Fermi level, ideal material candidate for the emergent particles with higher-dimensional degeneracy manifold (i.e., nodal lines and nodal surfaces) has not been found in electronic systems. This paper demonstrates that according to the first-principle calculations and symmetry analysis, realistic ternary nitride Li$_6$WN$_4$ features ideal (nearly flat) nodal-surface and nodal-line structures in its phonon spectra. These nodal degeneracies are shaped like lanterns, and their existence is guaranteed by nonsymmorphic symmetry. The corresponding topological phonon surface state covers exactly half the surface Brillouin zone (BZ) and can thereby be distinguished from those of conventional nodal-line and nodal-surface semimetals. The results of our study demonstrate the existence of ideal lantern-like phonons in realistic materials, which enriches the classification of topological quantum phases and provides a good basis for investigating the interaction between nodal-line and nodal-surface phonons in a single material.

Symmetry-enforced band nodes in 230 space groups

Crystallographic symmetries enforcing band touchings (BTs) in the Brillouin zone (BZ) have been utilized to classify and predict the topological semimetals. Though the early proposed topological semimetals contain isolated nodal points in the BZ, the proposed nodal line semimetals later could host various structures of several nodal lines/loops: nodal chains, nodal nets or Hopf-links, etc. In this work, using compatibility relations, we first list all possible high symmetry lines (HSLs) that can be nodal lines itself, high symmetry planes (HSPLs) that can host nodal loops, high symmetry planes (HSPLs) that are nodal surfaces for all 230 SGs, with spin-orbit coupling and time-reversal symmetry considered or not. We then show how to diagnose a nodal loop from the band crossing in an HSL, or nodal line/surface from irreducible representation (irrep) of an high-symmetry point (HSP), while the rest cases correspond to nodal points. Among our results, those essential cases, for which the nodal points/lines/loops/surfaces must exist, are highlighted since they are promising for the realizations of (nearly) ideal nodal point/line/loop/surface semimetals, as well as systems with flexible tunability owning fixed structure of topological nodal points/lines/loops/surfaces. Based on our results, SGs allowing Hopf-link structure with one straight nodal line threading a nodal loop, or two nesting nodal loops lying in two respective high symmetry planes, are highlighted, with the predicted materials being B$_5$Pb$_2$IO$_9$ in SG 34 and SrAl$_2$Au$_3$ in SG 62, respectively. Our exhaustive results could serve as a useful guide for efficiently predicting and designing materials or artificial systems owning exotic geometric nodal structures of energy bands simply based on structure symmetries.

Linking structures of doubly charged nodal surfaces in centrosymmetric superconductors

In topological semimetals and nodal superconductors, band crossings between occupied and unoccupied bands form stable nodal points/lines/surfaces carrying quantized topological charges. In particular, in centrosymmetric systems, some nodal structures at the Fermi energy $E_F$ carry two distinct topological charges, and thus they are called doubly charged nodes. Here we show that doubly charged nodal surfaces of centrosymmetric superconductors in three-dimensions always develop peculiar linking structures with nodal points or lines formed between occupied bands below $E_F$. Such linking structures can naturally explain the inherent relationship between the charge of the node below $E_F$ and the two charges of the nodal surfaces at $E_F$. Based on the Altland-Zirnbauer (AZ)-type ten-fold classification of nodes with additional inversion $\mathcal{I}$ symmetry, which is called the AZ$+\mathcal{I}$ classification, we provide the complete list of linking structures of doubly charged nodes in centrosymmetric systems. The linking structures of doubly charged nodes clearly demonstrate that not only the local band structure around the node but also the global band structure play a critical role in characterizing gapless topological phases.

Spin-polarized type-II nodal loop and nodal surface states in hexagonal compounds XTiO2 ( X = Li, Na, K, Rb)

Topological semimetals and metals have been extensively studied in nonmagnetic materials, while those in magnetic materials are still quite rare. Here, based on first-principles calculations, we reveal that a family of hexagonal compounds $X{\mathrm{TiO}}_{2}$ ($X$ = Li, Na, K, Rb) with ferromagnetic ground states host excellent topological electronic structures. In their low-energy band structures, they show two Weyl nodal loops and two Weyl nodal surfaces, where each nodal loop and nodal surface is contributed by the electron states in the same spin channel, namely, spin-polarized nodal loops and nodal surfaces. In particular, the nodal loops in these materials show type-II band dispersions and are robust against spin-orbit coupling because of the protection of the mirror symmetry. These nodal loops are featured with spin-polarized drumhead surface states. We build an effective Hamiltonian, which can well describe the type-II nodal loops in the system. Besides the type-II nodal loops, these materials also possess two spin-polarized nodal surfaces, protected by specific nonsymmorphic symmetry. The findings in this paper can serve as good material platforms to study the fundamental physics of type-II nodal loop and nodal surface fermions in a time-reversal-breaking system.

Weyl nodal-line surface half-metal in CaFeO3

Manipulating the spin degrees of freedom of electrons affords an excellent platform for exploring novel quantum states in condensed-matter physics and material science. Based on first-principles calculations and analysis of crystal symmetries, we propose a fully spin-polarized composite semimetal state, which is combined with the one-dimensional nodal lines and two-dimensional nodal surfaces, in the half-metal material CaFeO$_3$. In the nodal line-surface states, the Baguenaudier-like nodal lines feature six rings linked together, which are protected by the three independent symmetry operations:$\mathcal{PT}$, $\mathcal{M}_{y}$, and $\mathcal{\widetilde{M}}_{z}$. Near the Fermi level, the spin-polarized nodal surface states are guaranteed by the joint operation $\mathcal{T}\mathcal{S}_{2i}$ in the $k_{i(i=x,y,z)}=\pi$ plane. Furthermore, high-quality CaFeO$_3$ harbors ultra-clean energy dispersion, which is rather robust against strong triaxial compressional strain and correlation effect. The realization of the Weyl nodal line-surface half-metal presents great potential for spintronics applications with high speed and low power consumption.

Variational nodal method for three-dimensional multigroup neutron diffusion equation based on arbitrary triangular prism

With the development of nuclear technology, new concepts of reactor core have been continuously proposed, and new types of geometric assembly and reactor core configuration would be adopted to achieve better characteristics. Traditional nodal method based on rectangle or hexagonal nodal is no longer applicable for complex geometry modeling. Given that triangles have the advantage of constructing arbitrarily complex geometry, in this study, variational nodal method of three-dimensional multigroup neutron diffusion theory based on arbitrary triangular prism was developed. ANSYS code is used to divide calculation region into arbitrary triangle mesh, which is then transformed into regular triangle mesh by coordinate transformation, and a set of orthogonal polynomial basis function is built. Variables such as nodal neutron scalar flux and neutron current on nodal surface are expanded based on those polynomial basis functions. The response relationship between neutron partial neutron current and neutron flux can be obtained by variational principle, and traditional power method is employed to solve those response matrix system. Based on above, TriVNM code was developed, and 4 types of typical benchmarks including rectangle, hexagonal assembly reactor and reactor with curved boundary were employed to verify TriVNM. Both power distributions and keff calculated by TriVNM code agree well with reference results, and the maximum relative error of all the benchmarks’ power doesn’t exceed 1% and error of keff doesn’t exceed 50 pcm. TriVNM code achieves the same or higher level of calculation accuracy compared with other diffusion codes. Mesh sensitivity analysis shows that for large mesh size TriVNM code still has good accuracy.

Coexistence of fully spin-polarized Weyl nodal loop, nodal surface, and Dirac point in a family of quasi-one-dimensional half-metals

Recent investigation has connected topology with magnetization, which has been attracting great research interest. Here, based on symmetry analysis and first-principle calculations, we reveal rich band crossings in ferromagnetic quasi-one-dimensional compounds ${XYZ}_{3}$ ($X=\mathrm{Cs},\mathrm{Rb}$, $Y=\mathrm{Cr},\mathrm{Cu}$, $Z=\mathrm{Cl},\mathrm{I}$). We show that these materials host hourglass Weyl loops, nodal surfaces, and Dirac points in the absence of spin-orbital coupling. Notably, these topological phases only occur in a single spin channel, leading to fully spin-polarized phases. Based on symmetry analysis, we present that the hourglass-like Weyl loop is enforced by nonsymmorphic symmetries, which give rise to drumhead surface states. Particularly, the Weyl loop can persist when a magnetic direction is normal to the plane where it lies on. Additionally, we also find the nodal surfaces protected by the combination of screw rotation and time reversal symmetry. Besides, there exists a Dirac point enabled by nonsymmorphic symmetry and time reversal symmetry. Notably, this Dirac point shows a linear dispersion in one direction and quadratic dispersion in the normal plane, showing two Fermi arcs in its surface energy spectrum. By tuning the direction of magnetization, the Dirac point can be reduced to a pair of Weyl points. Our work suggests a concrete platform for exploring the fascinating physics associated with nonsymmorphic band crossings in quasi-one-dimensional ferromagnetic systems.

Coexistence of different dimensional topological states in stable ternary compound PrSbPt

The focus of topological materials has been recently expanded to metals and semimetals and, according to the dimensionaliy of their topological states, they can be divided into three types, i.e., zero-dimension nodal point, one-dimension nodal line and two-dimension nodal surface. So far, most of the studies have found singe topological element of different types and the coexistence of double states in a sole material is very scarce, not to mention the three types together. In this study, we present a realistic material PrSbPt with hexagonal crystal structure, which remarkably exhibits different topolgoial dimensitilities simutaneouly and, especially, the very rare Type-II nodal ring and critical type Dirac points have been also included. This rich topological diversity in a single material can be very interesting and promising for the future study of different fermionic states and their possible correlation effect.

COMPARISONS AMONG THREE COUPLING FORMS OF CYLINDRICAL NODAL EXPANSION METHODS

Nodal expansion method (NEM) is a typical nodal method in solving neutron diffusion equation in coarse mesh spatial discretization. NEM has been extended to cylindrical geometry in previous studies. Cylindrical NEM can be realized by different forms, where the variables in the nodal coupling equations are classified as net current, flux or partial current at the nodal surface, respectively. The three types of coupling forms of NEM have been implemented in PANGU code for the high temperature gas-cooled reactor (HTGR) physics analysis. This paper derives the three types of coupling forms of NEM, and analyzes their performance in solving the HTGR model containing void region.

A Dirac nodal surface semi-metallic carbon-based structure as a universal anode material for metal-ion batteries with high performance.

The rapid development of electronic devices requires high power storage batteries. However, reported 3D carbon-based materials are semiconductors or metals and are used in Li- or Na-ion batteries with low capacities. Thus, it is of interest to discover whether there is a universal semi-metallic material for use in high performance Li-, Na-, and K-ion batteries. Inspired by the recent synthesis of 3D carbon-based materials, in the research reported here, a 3D regular porous structure (bct-C56) is designed using graphene sheets. The porous carbon-based material has mechanical, dynamic, thermal, and mechanical stabilities. Interestingly, bct-C56 exhibits semi-metallic features with two Dirac nodal surfaces with mirror symmetry, as well as high Fermi velocities, indicating high electron-transport abilities. More excitingly, its theoretical capacities are 743.8, 478.2, and 425.0 mA h g-1, with diffusion barriers of 0.05-0.12, 0.07-0.12, and 0.03-0.05 eV, average OCVs of 0.31, 0.45, and 0.59 V, and volume expansion levels of 1.2%, 0.02%, and 3.1%, in Li-, Na-, and K-ion batteries, respectively. All these excellent characteristics suggest that semi-metallic bct-C56 is a universal anode material for use in metal-ion batteries with a fast charge-discharge rate. In this research, not only was a new material with a Dirac nodal surface feature designed, but it also offers an approach for the creation of high performance and universal metal-ion battery anodes with 3D porous carbon materials.

Prognostic value of radiologic extranodal extension in patients with hypopharyngeal cancer treated with primary chemoradiation

Background and purposeWe aimed to evaluate the prognostic value of radiologic extranodal extension (rENE) in patients with hypopharyngeal cancer (HPC) treated with primary chemoradiation. Materials and methodsCancer registry data were reviewed from 2005 to 2014. Inclusion criteria included HPC, clinical N1–3 disease (AJCC staging system, 7th edition), and receiving radiotherapy. Patients with M1 diseaseor with synchronous/metachronous cancer were excluded. Staging images were reviewed by two radiologists. rENE was defined as infiltration of adjacent fat/muscles, irregular nodal surface, or irregular capsular enhancement. Clinical stage, rENE status, and clinical outcome were analyzed. ResultsOverall, 355 patients were included. Patients with rENE had lower 3-year overall survival (OS) and recurrence-free survival (RFS) rates. Univariate analysis showed that clinical T4 or N3 stage, overall stage IV, and rENE correlated with OS and RFS. In multivariate analysis, clinical T4 or N3 stage correlated with poor OS, while clinical T4 or N3 stage and rENE were independent predictors of poor RFS. N1/2 without rENE was designated as Group 1, N1/2 with rENE as Group 2, and N3 with/without rENE as Group 3. The 3-year RFS rates in Groups 1, 2, and 3 were 55.8%, 41.0%, and 29.3%, respectively. The 3-year RFS rate in Group 1 was significantly higher than that in the other two groups. ConclusionsrENE is an adverse prognostic factor for survival in patients with HPC treated with primary chemoradiation. It correlated with inferior RFS regardless of N stage. rENE may be used as a criterion for clinical ENE in future staging systems.

The role of high-order electron correlation effects in a model system for non-valence correlation-bound anions.

The diffusion Monte Carlo (DMC), auxiliary field quantum Monte Carlo (AFQMC), and equation-of-motion coupled cluster (EOM-CC) methods are used to calculate the electron binding energy (EBE) of the non-valence anion state of a model (H2O)4 cluster. Two geometries are considered, one at which the anion is unbound and the other at which it is bound in the Hartree-Fock (HF) approximation. It is demonstrated that DMC calculations can recover from the use of a HF trial wave function that has collapsed onto a discretized continuum solution, although larger EBEs are obtained when using a trial wave function for the anion that provides a more realistic description of the charge distribution and, hence, of the nodal surface. For the geometry at which the cluster has a non-valence correlation-bound anion, both the inclusion of triples in the EOM-CC method and the inclusion of supplemental diffuse d functions in the basis set are important. DMC calculations with suitable trial wave functions give EBE values in good agreement with our best estimate EOM-CC result. AFQMC using a trial wave function for the anion with a realistic electron density gives a value of the EBE nearly identical to the EOM-CC result when using the same basis set. For the geometry at which the anion is bound in the HF approximation, the inclusion of triple excitations in the EOM-CC calculations is much less important. The best estimate EOM-CC EBE value is in good agreement with the results of DMC calculations with appropriate trial wave functions.

Hexagonal Zr3X (X = Al, Ga, In) Metals: High Dynamic Stability, Nodal Loop, and Perfect Nodal Surface States.

In recent years, topological semimetals/metals, including nodal point, nodal line, and nodal surface semimetals/metals, have been studied extensively because of their potential applications in spintronics and quantum computers. In this study, we predict a family of materials, Zr3X (X = Al, Ga, In), hosting the nodal loop and nodal surface states in the absence of spin–orbit coupling. Remarkably, the energy variation of the nodal loop and nodal surface states in Zr3X are very small, and these topological signatures lie very close to the Fermi level. When the effect of spin–orbit coupling is considered, the nodal loop and nodal surface states exhibit small energy gaps (<25 and 35 meV, respectively) that are suitable observables that reflect the spin-orbit coupling response of these topological signatures and can be detected in experiments. Moreover, these compounds are dynamically stable, and they consequently form potential material platforms to study nodal loop and nodal surface semimetals.

Stochastic nodal surfaces in quantum Monte Carlo calculations.

Treating the fermionic ground state-problem as a constrained stochastic optimization problem, a formalism for fermionic quantum Monte Carlo is developed that makes no reference to a trial wave function. Exchange symmetry is enforced by nonlocal terms appearing in the Green's function corresponding to an additional walker propagation channel. Complemented by a treatment of diffusion that encourages the formation of a stochastic nodal surface, we find that an approximate long-range extension of walker cancellations can be employed without introducing significant bias, reducing the number of walkers required for a stable calculation. A proof-of-concept implementation is shown to give a stable fermionic ground state for simple harmonic and atomic systems.

Unconventional topological insulators from extended topological band degeneracies

A general and beautiful picture for the realization of topological insulators is that the mass term of the Dirac model has a nodal surface wrapping one Dirac point. We show that this geometric picture based on Dirac points can be generalized to extended band degeneracies with nontrivial topological charges. As the nontrivial topological charges force the extended band degeneracies to be created or annihilated in pairs, when the nodal surface of mass wraps one such extended band degeneracy, the resulting gapped phase must be topologically nontrivial since it cannot adiabatically be deformed into a topologically trivial atomic insulator without closing the energy gap. We use nodal lines carrying a nontrivial $Z_{2}$ monopole charge in three dimensions to illustrate the physics. Notably, because the wrapping surfaces for an extended band degeneracy are diverse, we find that this generalization can bring topological insulators with unconventional pattern of boundary states.

Hypothetical P63/mmc-Type CsCrCl3 Ferromagnet: Half-Metallic Property and Nodal Surface State

This work is focusing on the electronic, magnetic, half-metallic behaviors, and interesting surface state of hypothetical CsCrCl3 ferromagnet with P63/mmc space group that are investigated under the spin-polarized Generalized Gradient Approach (GGA) and spin-polarized GGA plus Hubbard U correction within the framework of the Density Functional Theory (DFT). The calculated total magnetic moment of P63/mmc CsCrCl3 is ~7.397 µB, and the mainly contribution to the total magnetism is coming from the Cr atoms. Based on obtained spin-polarized band structures, we confirmed that this hypothetical CsCrCl3 is a half-metal with 100% spin-polarization. Without spin-orbit coupling, i.e., the spin and the orbit degrees of freedom are dependents, we found that there should be a interesting nodal surface state at kz = plane due to P63/mmc type lattice structure enjoys a two-fold screw axis S2z. Our study can be seen as a strong evidence to exhibit that the protected nodal surface state can be achieved in magnetic materials.

Pure Zirconium: Type II Nodal Line and Nodal Surface States.

Type II nodal line states have novel properties, such as direction-reliant chiral anomalies and high anisotropic negative magneto-resistance. These type II nodal line states have been widely investigated. Compared to nodal line materials, there are far fewer proposed nodal surface materials, and furthermore, a very recent challenge is to find a realistic material that co-exhibits both nodal line and nodal surface states. In this manuscript, we present the study of the electronic and topological states of pure zirconium within the density functional theory. We found that pure Zr is an interesting material that rarely exhibits both the type II nodal line state (in kz = 0 plane) and nodal surface state (in kz = π plane). The nontrivial topological states are explained based on the orbital-resolved band structures. Our study shows that pure Zr can serve as a new platform to investigate the interplay between the nodal line state and the nodal surface state.