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In electronics systems, the Weyl points can be classified into three types based on the geometry of the Fermi surface, and each type exhibits various unique and intriguing phenomena. While the type-I and type-II Weyl points have been achieved in both spinful and spinless systems, the realization of type-III Weyl points remains challenging, and has not been reported in artificial periodic systems. Here, we for the first time report the experimental observation of the type-III Weyl points in a sonic crystal. Remarkably, a single pair of type-III Weyl points are observed as the only band crossings in a frequency range, experimentally disproving a common belief in the field, namely, the minimal number of Weyl points in nonmagnetic systems is four. The consistency between experimental results and theoretical predictions confirms the existence of type-III Weyl points, noncontractible Fermi arc surface states, and chiral edge states. Our work not only fill the gap of the type-III Weyl point in sonic crystal but also stimulate related researches in other systems, such as photonic, mechanical and cold atom systems.

Topological insulators (TIs) hold promise as a platform for spintronics applications due to the fascinating spin-momentum locking (SML) of the surface states. One particular interest lies in using TIs as spin-polarized sources for spintronics structures. Here, we propose the band tilt induced nonlinear Nernst effect (NLNE) in TIs as a clean and efficient route to generate high-performance spin polarization (SP). We show that in the presence of SML and hexagonal warping effect a finite band tilt can induce an imbalance of two spin carriers and effectively give rise to spin-polarized NLNE current in TIs. In our scheme, both the spin current and charge current regime can be achieved under the thermal drive. The obtainable SP can be efficiently tuned in either a smooth or rapid way, exhibiting highly flexible tunability. In addition, near-unity SP can be generated within a wide range of tunable parameters, which is also found to be robust against temperature. Therefore, our work provides a mechanism to realize controllable room-temperature high-degree SP based on TIs, which is of essential importance for future spintronics applications., The accepted version in PRB Letter

The research on emergent particles in condensed matters has been attracting tremendous interest, and recently it is extended to magnetic systems. Here, we study the emergent particles stabilized by the symmetries of type-IV magnetic space groups (MSGs). Type-IV MSGs feature a special time reversal symmetry $\{\mathcal{T}|\boldsymbol{t}_0\}$, namely, the time reversal operation followed by a half lattice translation, which significantly alters the symmetry conditions for stabilizing the band degeneracies. In this work, based on symmetry analysis and modeling, we present a complete classification of emergent particles in type-IV MSGs by studying all possible (spinless and spinful, essential and accidental) particles in each of the 517 type-IV MSGs. Particularly, the detailed correspondence between the emergent particles and the type-IV MSGs that can host them are given in easily accessed interactive tables, where the basic information of the emergent particles, including the symmetry conditions, the effective Hamiltonian, the band dispersion and the topological characters can be found. According to the established encyclopedia, we find that several emergent particles that are previously believed to exist only in spinless systems will occur in spinful systems here, and vice versa, due to the $\{{\cal T}|\boldsymbol{t}_{0}\}$ symmetry. Our work not only deepens the understanding of the symmetry conditions for realizing emergent particles but also provides specific guidance for searching and designing materials with target particles., Comment: 7pages, 1 figures, 3 tables

In three-dimensional (3D) crystals, emergent particles arise when two or multiple bands contact and form degeneracy (band crossing) in the Brillouin zone. Recently a complete classification of emergent particles in 3D nonmagnetic crystals, which described by the type-II magnetic space groups (MSGs), has been established. However, a systematic investigation of emergent particles in magnetic crystals has not yet been performed, due to the complexity of the symmetries of magnetically ordered structures. Here, we address this challenging task by exploring the possibilities of the emergent particles in the 674 type-III MSGs. Based on effective k.p Hamiltonian and our classification of emergent particles [Yu et al., Sci. Bull. 67, 375 (2022) DOI:10.1016/j.scib.2021.10.023], we identify all possible emergent particles, including spinful and spinless, essential and accidental particles in the type-III MSGs. We find that all emergent particles in type-III MSGs also exist in type-II MSGs, with only one exception, i.e. the combined quadratic nodal line and nodal surface (QNL/NS). Moreover, tabulations of the emergent particles in each of the 674 type-III MSGs, together with the symmetry operations, the small corepresentations, the effective k.p Hamiltonians, and the topological character of these particles, are explicitly presented. Remarkably, combining this work and our homemade SpaceGroupIrep and MSGCorep packages will provide an effcient way to search topological magnetic materials with novel quasiparticles., 8 pages, 1 figure, 3 tables

Topological Weyl semimetals have been attracting broad interest. Recently, a new type of Weyl point with topological charge of four, termed as charge-four Weyl point (C-4 WP), was proposed in spinless systems. Here, we show the minimum symmetry requirement for C-4 WP is point-group $T$ together with $\mathcal{T}$ symmetry or point-group $O$. We establish a minimum tight-binding model for C-4 WP on a cubic lattice with time-reversal symmetry and without spin-orbit coupling effect. This lattice model is a two-band one, containing only one pair of C-4 WPs with opposite chirality around the Fermi level. Based on both the low-energy effective Hamiltonian and the minimum lattice model, we investigate the electronic, optical, and magnetic properties of C-4 WP. Several chirality-dependent properties are revealed, such as chiral Landau bands, quantized circular photogalvanic effect and quadruple-helicoid surface arc states. Furthermore, we predict that under symmetry breaking, various exotic topological phases can evolve out of C-4 WPs. Our paper not only reveals several interesting phenomena associate with C-4 WPs, but also provides a simple and ideal lattice model of C-4 WP, which will be helpful for the subsequent study on C-4 WPs.

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 phonon candidates have been predicted in solid-state materials. In this study, based on symmetry analysis and first-principles calculation, we prove that zero-, one-, and two-dimensional degeneracy co-exist in the phonon dispersion of one single realistic solid-state material ${\mathrm{SnO}}_{2}$ with $P{4}_{2}/mnm$ structure. In contrast to the previously reported electronic systems, the topological phonons observed in ${\mathrm{SnO}}_{2}$ are not restricted by the Pauli exclusion principle, and they experience negligible spin-orbit coupling effect. Hence, ${\mathrm{SnO}}_{2}$ 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.

Topological physics of phonon spectra has been attracting great interest. Using the first-principle calculations and symmetry analysis, we show the realistic ternary nitride ${\mathrm{Li}}_{6}{\mathrm{WN}}_{4}$ 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 and can thereby be distinguished from those of conventional nodal-line and nodal-surface semimetals. Our study enriches the classification of topological quantum phases, predicts ideal material candidates, and provides a good platform for investigating the interaction between nodal-line and nodal-surface phonons.

Experimentally synthesized perovskite-type ${\mathrm{YRh}}_{3}\mathrm{B}$ with a $Pm\overline{3}m$ type structure was proposed as a topological material (TM) via first-principles calculations and the low-energy $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ effective Hamiltonian, which has a quadratic contact triple point (QCTP) at point $\mathrm{\ensuremath{\Gamma}}$ and six pairs of open nodal lines (NLs) of the hybrid type. Clear surface states observed in the surface spectrum confirmed the topological states. When spin-orbit coupling was considered, the QCTP at $\mathrm{\ensuremath{\Gamma}}$ transferred to the quadratic-type Dirac nodal point (NP). Under $1%$ tetragonal strained lattice constants, ${\mathrm{YRh}}_{3}\mathrm{B}$ hosted richer topological states, including a quadratic-type twofold degenerate NP, six pairs of open NLs of the hybrid type, and two closed NLs of type I and hybrid type. Moreover, it was proved that the NLs of ${\mathrm{YRh}}_{3}\mathrm{B}$ at its strained lattice constants contain all types of band-crossing points (BCPs) (i.e., type I, type II, and critical type). Such rich types of NP and NL states in one compound make it potentially applicable for multifunctional electronic devices as well as an appropriate platform to study entanglement among topological states.

Magnetic materials with high mobilities are intriguing subjects of research from both fundamental and application perspectives. Based on first-principle calculations, we investigate the physical properties of the already synthesized $A\mathrm{MnBi}$ ($A=\mathrm{K}$, Rb, Cs) family materials. We show that these materials are antiferromagnetic (AFM), with N\'eel temperatures above 300 K. They contain AFM ordered Mn layers, while the interlayer coupling changes from ferromagnetic (FM) for KMnBi to AFM for RbMnBi and CsMnBi. We find that these materials are narrow gap semiconductors. Owing to the small effective mass, the electron carrier mobility can be very high, reaching up to ${10}^{5}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{2}/(\mathrm{V}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$) for KMnBi. In contrast, the hole mobility is much suppressed, typically lower by two orders of magnitude. We further study their two-dimensional (2D) single layer structures, which are found to be AFM with fairly high mobility $\ensuremath{\sim}{10}^{3}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{2}/(\mathrm{V}\phantom{\rule{0.16em}{0ex}}\mathrm{s})$. Their N\'eel temperatures can still reach room temperature. Interesting, we find that the magnetic phase transition is also accompanied by a metal-insulator phase transition, with the paramagnetic metal phase possessing a pair of nonsymmorphic-symmetry-protected 2D spin-orbit Dirac points. Furthermore, the magnetism can be effectively controlled by the applied strain. When the magnetic ordering is turned into FM, the system can become a quantum anomalous Hall insulator with gapless chiral edge states.

Novel fermions with a pseudospin-1 structure can be realized as emergent quasiparticles in condensed matter systems. Here, we investigate its unusual properties during the Andreev reflection at a normal-metal/ superconductor (NS) interface. We show that distinct from the previously studied pseudospin-1/2 and two-dimensional electron gas models, the pseudospin-1 fermions exhibit a strongly enhanced Andreev reflection probability, and remarkably, can be further tuned to approach perfect Andreev reflection with unit efficiency for all incident angles, exhibiting a previously unknown super-Andreev reflection effect. The super-Andreev reflection leads to perfect transparency of the NS interface that strongly promotes charge injection into the superconductor and directly manifests as a differential conductance peak which can be readily probed in experiment. Additionally, we find that sizable longitudinal shifts exist in the normal and Andreev reflections of pseudospin-1 fermions. Distinct from the pseudospin-1/2 case, the shift is always in the forward direction in the subgap regime, regardless of whether the reflection is of retro- or specular type.

Weak topological insulators and Dirac semimetals are gapped and nodal phases with distinct topological properties, respectively. Here, we propose a novel topological phase that exhibits features of both and is dubbed composite Dirac semimetal (CDSM). In its bulk, the CDSM has a pair of Dirac points and a pair of bands inverted along a high-symmetry path. At side surfaces, a pair of Fermi arcs connecting the projected Dirac points coexist with a pair of Fermi loops traversing the surface Brillouin zone. A nonsymmorphic symmetry dictates degeneracies between the Fermi arcs and the Fermi loops. We characterize the CDSM by multiple topological invariants and show that, under a transition without breaking any symmetry, it deforms into a topological crystalline insulator hosting two pairs of surface Fermi loops. We demonstrate the CDSM in two models and predict its realization in the KAuTe-family materials.

Despite a rapidly expanding inventory of possible crystalline Weyl semimetals, all of them are constrained by the Nielsen-Ninomiya no-go theorem, namely, that left- and right-handed Weyl points appear in pairs. With time-reversal ($\mathcal{T}$) symmetry, an even stronger version holds for the semimetals, i.e., all eight time-reversal-invariant points in the Brillouin zone (BZ) simultaneously host Weyl points or not. Accompanying the no-go theorem, the surface of the system features Fermi arc states, which connect pairs of surface projected Weyl points. Here, we explicitly construct a topological phase of a $\mathcal{T}$-invariant crystalline metal, which features a single Weyl point residing at the center of the BZ, surrounded by nodal walls spreading over the entire BZ boundary. In other words, a single Weyl point is realized with the no-go theorem being circumvented. Moreover, the surface Fermi arcs, considered as a hallmark of Weyl semimetals, do not appear for this composite topological phase. We show that this phase can be realized for space groups No. 19 and No. 92, with and without spin-orbit coupling, respectively.

Nodal-line metals and semimetals, as interesting topological states of matter, have been mostly studied in nonmagnetic materials. Here, based on first-principles calculations and symmetry analysis, we predict that fully spin polarized Weyl loops can be realized in the half-metal state of the three-dimensional material ${\mathrm{Li}}_{3}{({\mathrm{FeO}}_{3})}_{2}$. We show that this material has a ferromagnetic ground state, and it is a half-metal with only a single spin channel present near the Fermi level. The spin-up bands form two separate Weyl loops close to the Fermi level, which arise from band inversions and are protected by the glide mirror symmetry. One loop is type I, whereas the other loop is the hybrid type. Corresponding to these two loops in the bulk, on the (100) surface, there exist two fully spin polarized drumheads of surface states within the surface projections of the loops. The effects of the electron correlation and the spin-orbit coupling, as well as the possible hourglass Weyl chains in the nonmagnetic state, are discussed. The realization of fully spin polarized Weyl-loop fermions in the bulk and drumhead fermions on the surface for a half-metal may generate promising applications in spintronics.

Topological metals and semimetals are new states of matter which are attracting great interest in current research. Here, based on first-principles calculations and symmetry analysis, we propose that the family of titanium-based compounds ${\mathrm{Ti}}_{3}X$ ($X$=Al, Ga, Sn, Pb) are unexplored topological metals. These materials feature the coexistence of a nodal loop and a nodal surface in their low-energy band structures. Taking ${\mathrm{Ti}}_{3}\mathrm{Al}$ as an example, we show that the material has an almost ideal nodal loop in the sense that the loop is close to the Fermi level and it is nearly flat in energy with energy variation $l0.25$ meV. The loop is protected by one of the two independent symmetries: the combined space-time-inversion symmetry and the mirror-reflection symmetry. The nodal surface at the ${k}_{z}=\ensuremath{\pi}$ plane is guaranteed by the nonsymmorphic screw rotational symmetry and the time-reversal symmetry. We discuss the effect of spin-orbit coupling and construct an effective model for describing the nodal loop. Our findings indicate that the ${\mathrm{Ti}}_{3}\mathrm{Al}$ family of compounds can serve as an excellent material platform for studying new topological phases and, particularly, the interplay between nodal loop and nodal surface fermions.

We theoretically study the three-dimensional topological semimetals with nodal surfaces protected by crystalline symmetries. Different from the well-known nodal-point and nodal-line semimetals, in these materials, the conduction and valence bands cross on closed nodal surfaces in the Brillouin zone. We propose different classes of nodal surfaces, both in the absence and in the presence of spin-orbit coupling (SOC). In the absence of SOC, a class of nodal surfaces can be protected by spacetime inversion symmetry and sublattice symmetry and characterized by a $\mathbb{Z}_2$ index, while another class of nodal surfaces are guaranteed by a combination of nonsymmorphic two-fold screw-rotational symmetry and time-reversal symmetry. We show that the inclusion of SOC will destroy the former class of nodal surfaces but may preserve the latter provided that the inversion symmetry is broken. We further generalize the result to magnetically ordered systems and show that protected nodal surfaces can also exist in magnetic materials without and with SOC, given that certain magnetic group symmetry requirements are satisfied. Several concrete nodal-surface material examples are predicted via the first-principles calculations. The possibility of multi-nodal-surface materials is discussed., 13 pages, 12 figures

We study the electronic and scattering properties of graphene with moderate Rashba spin-orbit coupling (SOC). The Rashba SOC in graphene tends to distort the band structure and gives rise to a trigonally warped Fermi surface. For electrons at a pronouncedly warped Fermi surface, the spin direction exhibits a staircase profile as a function of the momentum, making an unusual spin texture. We also study the spin-resolved scattering on a Rashba barrier and find that the trigonal warping is essential for producing spin polarization of the transmitted current. Particularly, both the direction and strength of the spin polarization can be controlled by kinds of electric methods. Our work unveils that not only SOC but also the geometry of the Fermi surface is important for generating spin polarization.

Nonsymmorphic space group symmetries can generate exotic band crossings in topological metals and semimetals. Here, based on symmetry analysis and first-principles calculations, we reveal rich band-crossing features in the existing layered compounds ${\mathrm{Ta}}_{3}{\mathrm{SiTe}}_{6}$ and ${\mathrm{Nb}}_{3}{\mathrm{SiTe}}_{6}$, enabled by nonsymmorphic symmetries. We show that in the absence of spin-orbit coupling (SOC), these three-dimensional (3D) bulk materials possess accidental Dirac loops and essential fourfold nodal lines. In the presence of SOC, there emerges an hourglass Dirac loop---a fourfold degenerate nodal loop, on which each point is a neck point of an hourglass-type dispersion. We show that this interesting type of band crossing is protected and dictated by the nonsymmorphic space group symmetries and it gives rise to drumheadlike surface states. Furthermore, we also investigate these materials in the monolayer form. We show that these two-dimensional (2D) monolayers host nodal lines in the absence of SOC and the nodal lines transform to essential spin-orbit Dirac points when SOC is included. Our work suggests a realistic material platform for exploring the fascinating physics associated with nonsymmorphic band crossings in both 3D and 2D systems.

We study the transport properties at the surface state of a topological insulator (TI) with a potential barrier. Due to the hexagonal warping effect, the number of reflected propagating states for an incident electron beam can change from one to two, corresponding to a change from normal reflection to double reflection, by controlling the Fermi energy. Associated with the change, the corresponding reflection probability shows a significant jump in its derivative. In addition, for a junction with potential barrier, the reflection probability should oscillate as the potential energy varies due to the tunneling resonance. The oscillation period in the TI junction is closely related to the warping strength. Thus, these two proposals both can be used to identify the hexagonal warping strength of the TI surface state and especially, are robust against the influence of lateral surfaces, as they depend on the relative variation of transport. Remarkably, the latter proposal is compatible with the recent experiment in graphene [Chen et al., Science 353, 1522 (2016)], which achieves a direct measurement of the angle-resolved scattering probability.