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The bulk-boundary correspondence is a key concept in topological quantum materials. For instance, a quantum spin Hall insulator features a bulk insulating gap with gapless helical boundary states protected by the underlying Z2 topology. However, the bulk-boundary dichotomy and distinction are rarely explored in optical experiments, which can provide unique information about topological charge carriers beyond transport and electronic spectroscopy techniques. Here, we utilize mid-infrared absorption micro-spectroscopy and pump-probe micro-spectroscopy to elucidate the bulk-boundary optical responses of Bi4Br4, a recently discovered room-temperature quantum spin Hall insulator. Benefiting from the low energy of infrared photons and the high spatial resolution, we unambiguously resolve a strong absorption from the boundary states while the bulk absorption is suppressed by its insulating gap. Moreover, the boundary absorption exhibits a strong polarization anisotropy, consistent with the one-dimensional nature of the topological boundary states. Our infrared pump-probe microscopy further measures a substantially increased carrier lifetime for the boundary states, which reaches one nanosecond scale. The nanosecond lifetime is about one to two orders longer than that of most topological materials and can be attributed to the linear dispersion nature of the helical boundary states. Our findings demonstrate the optical bulk-boundary dichotomy in a topological material and provide a proof-of-principal methodology for studying topological optoelectronics., Comment: 26 pages, 4 figures

We present a systematic investigation of the magnetoresponses of the Weyl points (WPs) with a topological charge of n = 2, 3 and 4, and with both linear and quadratic energy tilt. The linear tilt always tends to squeeze the Landau levels (LLs) of both conduction and valence bands of all the WPs, and eventually leads to LL collapse in the type-II phase. However, the quadratic energy tilt has more complex influences on the LLs of the unconventional WPs. For charge-n (n = 2, 4) WP, the influence of the quadratic tilt on the LLs of conduction and valence bands are opposite, i.e. if the LLs of conduction (valence) bands are squeezed, then that of the valence (conduction) bands are broadened, and the squeezed LL spectrum will be collapsed in type-III phase. But, the LL collapse generally can not be found in the type-III charge-3 WP. Moreover, for charge-n (n = 2, 3) WP, the quadratic tilt breaks the degeneracy of the chiral LLs regardless of the direction of the magnetic field, leading to additional optical transitions and magneto-optical conductivity peaks at low frequencies. Interestingly, the four chiral LLs in charge-4 WP are always not degenerate. Hence, there inevitably exist magneto-optical conductivity peaks at low frequencies for charge-4 WP. Since the density of state of the LL spectrum is very large, one can expect that the low-frequency magneto-optical responses in unconventional WPs would be significant and may be used for developing efficient terahertz photodetectors.

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.

We present a systematic classification of emergent particles in all 528 magnetic layer groups and 394 magnetic rod groups, which describe two-dimensional and one-dimensional crystals respectively. Our approach is via constructing a correspondence between a given magnetic layer/rod group and one of the magnetic space group, such that all irreducible representations of the layer/rod group can be derived from those of the corresponding space group. Based on these group representations, we explicitly construct the effective models for possible band degeneracies and identify all emergent particles, including both spinless and spinful cases. We find that there are six kinds of particles protected by magnetic layer groups and three kinds by magnetic rod groups. Our work provides a useful reference for the search and design of emergent particles in lower dimensional crystals., 7 pages, 2 figures, 4 tables

Unconventional Weyl point with nonlinear dispersion features higher topological charge $|{\cal{C}}|>1$ and multiple topologically protected Fermi arc states at its boundary. As a novel topological state, it has been attracting widespread attention. However, the unconventional Weyl point with $|{\cal{C}}|=3$ has not yet been reported in realistic materials, even though it has been theoretically proposed for more than a decade. In this work, based on first-principles calculations and theoretical analysis, we predict the existing material, $\rm\alpha$-LiIO$_3$ as the first realistic example with this unconventional Weyl point. Particularly, in the phonon spectra of $\rm\alpha$-LiIO$_3$, two Weyl points with ${\cal{C}}=-3$, connected by time-reversal symmetry, appear at the neck crossing-point of a hourglass-type band, leading to two hourglass charge-3 Weyl phonons. The symmetry protection and the associated novel triple- and sextuple-helicoid surface arc states of the hourglass charge-3 Weyl phonons are revealed. Our results uncover a hidden topological character of $\rm\alpha$-LiIO$_3$ and also show that the phonon spectra is a great platform for exploring unconventional topological states.

The hypothetical Weyl particles in high-energy physics have been discovered in three-dimensional crystals as collective quasiparticle excitations near two-fold degenerate Weyl points. Such momentum-space Weyl particles carry quantized chiral charges, which can be measured by counting the number of Fermi arcs emanating from the corresponding Weyl points. It is known that merging unit-charged Weyl particles can create new ones with more charges. However, only very recently has it been realised that there is an upper limit - the maximal charge number that a two-fold Weyl point can host is four - achievable only in crystals without spin-orbit coupling. Here, we report the experimental realisation of such a maximally charged Weyl point in a three-dimensional photonic crystal. The four charges support quadruple-helicoid Fermi arcs, forming an unprecedented topology of two non-contractible loops in the surface Brillouin zone. The helicoid Fermi arcs also exhibit the long-pursued type-II van Hove singularities that can reside at arbitrary momenta. This discovery reveals a type of maximally charged Weyl particles beyond conventional topological particles in crystals., Comment: 17 pages, 4 figures

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

Motivated by easy access to complete corepresentation (corep) data of all the 1651 magnetic space groups (MSGs) in three-dimensional space, we have developed a Mathematica package MSGCorep to provide an offline database of coreps and various functions to manipulate them, based on our previous package SpaceGroupIrep. One can use the package MSGCorep to obtain the elements of any MSG and magnetic little group, to calculate the multiplication of group elements, to obtain the small coreps at any k-point and full coreps of any magnetic k-star for any MSG and show them in a user-friendly table form, to calculate and show the decomposition of direct products of full coreps between any two specified magnetic k-stars, and to determine the small coreps of energy bands. Both single-valued and double-valued coreps are supported. In addition, the 122 magnetic point groups (MPGs) and their coreps are also supported by this package. To the best of our knowledge, MSGCorep is the first package that is able to calculate the direct product of full coreps for any MSG and able to determine small coreps of energy bands for general purpose. In a word, the MSGCorep package is an offline database and tool set for MSGs, MPGs, and their coreps, and it is very useful to study the symmetries in magnetic and nonmagnetic materials., Comment: 23 pages, 7 figures, 6 tables, 1 supplementary material

The past decade has witnessed a surge of interest in exploring emergent particles in condensed matter systems. Novel particles, emerged as excitations around exotic band degeneracy points, continue to be reported in real materials and artificially engineered systems, but so far, we do not have a complete picture on all possible types of particles that can be achieved. Here, via systematic symmetry analysis and modeling, we accomplish a complete list of all possible particles in time reversal-invariant systems. This includes both spinful particles such as electron quasiparticles in solids, and spinless particles such as phonons or even excitations in electric-circuit and mechanical networks. We establish detailed correspondence between the particle, the symmetry condition, the effective model, and the topological character. This obtained encyclopedia concludes the search for novel emergent particles and provides concrete guidance to achieve them in physical systems., Comment: 5 pages for main text and 1228 pages for SM; comments are welcome, please contact Z. M. Yu, Y. Yao, or S. A. Yang

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.

We present a Mathematica program package MagneticTB, which can generate the tight-binding model for arbitrary magnetic space group. The only input parameters in MagneticTB are the (magnetic) space group number and the orbital information in each Wyckoff positions. Some useful functions including getting the matrix expression for symmetry operators, manipulating the energy band structure by parameters and interfacing with other software are also developed. MagneticTB can help to investigate the physical properties in both magnetic and non-magnetic system, especially for topological properties., 16 pages, 3 figures

We propose a new concept of two-dimensional (2D) Dirac semiconductor which is characterized by the emergence of fourfold degenerate band crossings near the band edge and provide a generic approach to realize this novel semiconductor in the community of material science. Based on the first-principle calculations and symmetry analysis, we discover recently synthesised triple-layer (TL)-BiOS2 is such Dirac semiconductor that features Dirac cone at X/Y point, protected by nonsymmorphic symmetry. Due to sandwich-like structure, each Dirac fermion in TL-BiOS2 can be regarded as a combination of two Weyl fermions with opposite chiralities, degenerate in momentum-energy space but separated in real space. Such Dirac semiconductor carries layer-dependent helical spin textures that never been reported before. Moreover, novel topological phase transitions are flexibly achieved in TL-BiOS2: (i) an vertical electric field can drive it into Weyl semiconductor with switchable spin polarization direction, (ii) an extensive strain is able to generate ferroelectric polarization and actuate it into Weyl nodal ring around X point and into another type of four-fold degenerate point at Y point. Our work extends the Dirac fermion into semiconductor systems and provides a promising avenue to integrate spintronics and optoelectronics in topological materials., 5 pages, 3 figures

We propose a class of nodal line semimetals that host an eightfold-degenerate double Dirac nodal line (DDNL) with negligible spin-orbit coupling. We find only 5 of the 230 space groups host the DDNL. The DDNL can be considered as a combination of two Dirac nodal lines, and has a trivial Berry phase. This leads to two possible but completely different surface states, namely, a torus surface state covering the whole surface Brillouin zone and no surface state at all. Based on first-principles calculations, we predict that the hydrogen storage material LiBH is an ideal DDNL semimetal, where the line resides at Fermi level, is relatively flat in energy, and exhibits a large linear energy range. Interestingly, both the two novel surface states of DDNL can be realized in LiBH. Further, we predict that with a magnetic field parallel to DDNL, the Landau levels of DDNL are doubly degenerate due to a Kramers-like degeneracy and have a doubly degenerate zero mode.

We study the possibility of triply-degenerate points (TPs) that can be stabilized in spinless crystalline systems. Based on an exhaustive search over all 230 space groups, we find that the spinless TPs can exist at both high-symmetry points and high-symmetry paths, and they may have either linear or quadratic dispersions. For TPs located at high-symmetry points, they all share a common minimal set of symmetries, which is the point group $T$. The TP protected solely by the $T$ group is chiral and has a Chern number of $\pm2$. By incorporating additional symmetries, this TP can evolve into chiral pseudospin-1 point, linear TP without chirality, or quadratic contact TP. For accidental TPs residing on a high-symmetry path, they are not chiral but can have either linear or quadratic dispersions in the plane normal to the path. We further construct effective $k\cdot p$ models and minimal lattice models for characterizing these TPs. Distinguished phenomena for the chiral TPs are discussed, including the extensive surface Fermi arcs and the chiral Landau bands., Comment: 9 pages, 5 figures

The surface Fermi arc, as a hallmark of Weyl semimetals (WSMs), has been well known in current research, but it remains a challenge to unveil novel phenomena associated with the Fermi arc. Here, we predict a heretofore unrecognized process in WSMs, namely, the photoinduced transition between the bulk states and the Fermi arc. We find this process is significant and can lead to a large effective three-dimensional shift current on the boundaries with the Fermi arc in wide terahertz range. Moreover, due to the low symmetry of the boundaries, the surface photogalvanic effect predicted here can appear in a large class of WSMs that do not have bulk shift current. Hence, our work not only unveils a hidden photogalvanic effect in WSMs but also suggests that all the WSMs are promising material candidates for developing efficient terahertz photodetectors., Comment: 6 pages, 5 figures

The interplay between magnetism and band topology is a focus of current research on magnetic topological systems. Based on first-principle calculations and symmetry analysis, we reveal multiple intriguing topological states can be realized in a single system EuAgAs, controlled by the magnetic ordering. The material is Dirac semimetal in the paramagnetic state, with a pair of accidental Dirac points. Under different magnetic configurations, the Dirac points can evolve into magnetic triply-degenerate points, magnetic linear and double Weyl points, or being gapped out and making the system a topological mirror semimetal characterized by mirror Chern numbers. The change in bulk topology is also manifested in the surface states, including the surface Fermi arcs and surface Dirac cones. In addition, the antiferromagnetic states also feature a nontrivial Z4 index, implying a higher order topology. These results deepen our understanding of magnetic topological states and provide new perspectives for spintronic applications., Comment: 8 pages,7 figures

Nodal lines, as one-dimensional band degeneracies in momentum space, usually feature a linear energy splitting. Here, we propose the concept of magnetic higher-order nodal lines, which are nodal lines with higher-order energy splitting and realized in magnetic systems with broken time reversal symmetry. We provide sufficient symmetry conditions for stabilizing magnetic quadratic and cubic nodal lines, based on which concrete lattice models are constructed to demonstrate their existence. Unlike its counterpart in nonmagnetic systems, the magnetic quadratic nodal line can exist as the only band degeneracy at the Fermi level. We show that these nodal lines can be accompanied by torus surface states, which form a surface band that span over the whole surface Brillouin zone. Under symmetry breaking, these magnetic nodal lines can be transformed into a variety of interesting topological states, such as three-dimensional quantum anomalous Hall insulator, multiple linear nodal lines, and magnetic triple-Weyl semimetal. The three-dimensional quantum anomalous Hall insulator features a Hall conductivity $\sigma_{xy}$ quantized in unit of $e^2/(hd)$ where $d$ is the lattice constant normal to the $x$-$y$ plane. Our work reveals previously unknown topological states, and offers guidance to search for them in realistic material systems., Comment: 10 pages, 8 figures

Topological band insulators and (semi-) metals can arise out of atomic insulators when the hopping strength between electrons increases. Such topological phases are separated from the atomic insulator by a bulk gap closing. In this work, we show that in many (magnetic) space groups, the crystals with certain Wyckoff positions and orbitals being occupied must be semimetal or metals in the atomic limit, e.g. the hopping strength between electrons is infinite weak but not vanishing, which then are termed atomic (semi-)metals (ASMs). We derive a sufficient condition for realizing ASMs in spinless and spinful systems. Remarkably, we find that increasing the hopping strength between electrons may transform an ASM into an insulator with both symmetries and electron fillings of crystal are preserved. The induced insulators inevitably are topologically non-trivial and at least are obstructed atomic insulators (OAIs) that are labeled as trivial insulator in topological quantum chemistry website. Particularly, using silicon as an example, we show ASM criterion can discover the OAIs missed by the recently proposed criterion of filling enforced OAI. Our work not only establishes an efficient way to identify and design non-trivial insulators but also predicts that the group-IV elemental semiconductors are ideal candidate materials for OAI., Comment: 4 pages, 4 figures

Higher-order topological phases and real topological phases are two emerging topics in topological states of matter, which have been attracting considerable research interest. However, it remains a challenge to find realistic materials that can realize these exotic phases. Here, based on first-principles calculations and theoretical analysis, we identify graphyne, the representative of the graphyne-family carbon allotropes, as a two-dimensional (2D) second-order topological insulator and a real Chern insulator. We show that graphyne has a direct bulk band gap at the three $M$ points, forming three valleys. The bulk bands feature a double band inversion, which is characterized by the nontrivial real Chern number enabled by the spacetime-inversion symmetry. The real Chern number is explicitly evaluated by both the Wilson-loop method and the parity approach, and we show that it dictates the existence of Dirac type edge bands and the topological corner states. Furthermore, we find that the topological phase transition in graphyne from the second-order topological insulator to a trivial insulator is mediated by a 2D Weyl semimetal phase. The robustness of the corner states against symmetry breaking and possible experimental detection methods are discussed., 9 pages, 5+3 figures

As a novel type of fermionic state, hybrid nodal loop with the coexistence of both type-I and type- II band crossings has attracted intense research interest. However, it remains a challenge to realize hybrid nodal loop in both two-dimensional (2D) materials and in ferromagnetic (FM) materials. Here, we propose the first FM hybrid nodal loop in 2D CrN monolayer. We show that the material has a high Curie temperature (> 600 K) FM ground state, with the out-of-plane [001] magnetization. It shows a half-metallic band structure with two bands in the spin-up channel crossing each other near the Fermi level. These bands produce both type-I and type-II band crossings, which form a fully spin-polarized hybrid nodal loop. We find the nodal loop is protected by the mirror symmetry and robust against spin-orbit coupling (SOC). An effective Hamiltonian characterizing the hybrid nodal loop is established. We further find the configuration of nodal loop can be shifted under external perturbations such as strain. Most remarkably, we demonstrate that both type-I and type-II Weyl nodes can be realized from such FM hybrid nodal loop by simply shifting the magnetization from out-of-plane to in-plane. Our work provides an excellent candidate to realize FM hybrid nodal loop and Weyl fermions in 2D material, and is also promising for related topological applications with their intriguing properties., 8 pages,7 figures

Topological metals (TMs) are a kind of special metallic materials, which feature nontrivial band crossings near the Fermi energy, giving rise to peculiar quasiparticle excitations. TMs can be classified based on the characteristics of these band crossings. For example, according to the dimensionality of the crossing, TMs can be classified into nodal-point, nodal-line, and nodal-surface metals. Another important property is the type of dispersion. According to degree of the tilt of the local dispersion around the crossing, we have type-I and type-II dispersions. This leads to significant distinctions in the physical properties of the materials, owing to their contrasting Fermi surface topologies. In this article, we briefly review the recent advances in this research direction, focusing on the concepts, the physical properties, and the material realizations of the type-II nodal-point and nodal-line TMs., 12 pages, 16 figures

A particle beam may undergo an anomalous spatial shift when it is reflected at an interface. The shift forms a vector field defined in the two-dimensional interface momentum space. We show that, although the shift vector at individual momentum is typically sensitive to the system details, its integral along a close loop, i.e., its circulation, could yield a robust quantized number under certain symmetry conditions of interest. Particularly, this is the case when the beam is incident from a trivial medium, then the quantized circulation of anomalous shift (CAS) directly manifests the topological character of the other medium. We demonstrate that the topological charge of a Weyl medium as well as the unconventional pair potentials of a superconductor can be captured and distinguished by CAS. Our work unveils a hidden quantized feature in a ubiquitous physical process, which may also offer a new approach for probing topological media., Comment: 5 pages, 4 figures

We introduce the concept of valley-layer coupling (VLC) in two-dimensional materials, where the low-energy electronic states in the emergent valleys have valley-contrasted layer polarization such that each state is spatially localized on the top or bottom super-layer. The VLC enables a direct coupling between valley and gate electric field, opening a new route towards electrically controlled valleytronics. We analyze the symmetry requirements for the system to host VLC, demonstrate our idea via first-principles calculations and model analysis of a concrete 2D material example, and show that an electric, continuous, wide-range, and switchable control of valley polarization can be achieved by VLC. Furthermore, we find that systems with VLC can exhibit other interesting physics, such as valley-contrasting linear dichroism and optical selection of the electric polarization of interlayer excitons., 6 pages, 4 figures

We have developed a Mathematica program package SpaceGroupIrep which is a database and tool set for irreducible representations (IRs) of space group in BC convention, i.e. the convention used in the famous book "The mathematical theory of symmetry in solids" by C. J. Bradley & A. P. Cracknell. Using this package, elements of any space group, little group, Herring little group, or central extension of little co-group can be easily obtained. This package can give not only little-group (LG) IRs for any k-point but also space-group (SG) IRs for any k-stars in intuitive table form, and both single-valued and double-valued IRs are supported. This package can calculate the decomposition of the direct product of SG IRs for any two k-stars. This package can determine the LG IRs of Bloch states in energy bands in BC convention and this works for any input primitive cell thanks to its ability to convert any input cell to a cell in BC convention. This package can also provide the correspondence of k-points and LG IR labels between BCS (Bilbao Crystallographic Server) and BC conventions. In a word, the package SpaceGroupIrep is very useful for both study and research, e.g. for analyzing band topology or determining selection rules., 28 pages (main text) + 629 pages (supplementary material), 11 figures

We reveal a class of three-dimensional $d$-orbital topological materials in the antifluorite Cu$_2$S family. Derived from the unique properties of low-energy $t_{2g}$ states, their phases are solely determined by the sign of spin-orbit coupling (SOC): topological insulator for negative SOC, whereas topological semimetal for positive SOC; both having Dirac-cone surface states but with contrasting helicities. With broken inversion symmetry, the semimetal becomes one with a nodal box consisting of butterfly-shaped nodal lines that are robust against SOC. Further breaking the tetrahedral symmetry by strain leads to an ideal Weyl semimetal with four pairs of Weyl points. Interestingly, the Fermi arcs coexist with a surface Dirac cone on the (010) surface, as required by a $Z_2$-invariant., 5+2 pages, 5+2 figures

Dirac point in two-dimensional (2D) materials has been a fascinating subject of research. Recently, it has been theoretically predicted that Dirac point may also be stabilized in 2D magnetic systems. However, it remains a challenge to identify concrete 2D materials which host such magnetic Dirac point. Here, based on first-principles calculations and theoretical analysis, we propose a stable 2D material, the monolayers TaCoTe$_2$, as an antiferromagnetic (AFM) 2D Dirac material. We show that it has an AFM ground state with an out-of-plane N\'{e}el vector. It hosts a pair of 2D AFM Dirac points on the Fermi level in the absence of spin-orbit coupling (SOC). When the SOC is considered, a small gap is opened at the original Dirac points. Meanwhile, another pair of Dirac points appear on the Brillouin zone boundary below the Fermi level, which are robust under SOC and have a type-II dispersion. Such a type-II AFM Dirac point has not been observed before. We further show that the location of this Dirac point as well as its dispersion type can be controlled by tuning the N\'{e}el vector orientation., Comment: 8 pages, 8 figures

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.

Two-dimensional (2D) materials with nodal-loop band crossing have been attracting great research interest. However, it remains a challenge to find 2D nodal loops that are robust against spin-orbit coupling (SOC) and realized in magnetic states. Here, based on first-principles calculations and theoretical analysis, we predict that monolayer MnN is a 2D nodal-loop half metal with fully spin polarized nodal loops. We show that monolayer MnN has a ferromagnetic ground state with out-of-plane magnetization. Its band structure shows half metallicity with three low-energy bands belonging to the same spin channel. The crossing between these bands forms two concentric nodal loops centered around the $\Gamma$ point near the Fermi level. Remarkably, the nodal loops and their spin polarization are robust under SOC, due to the protection of a mirror symmetry. We construct an effective model to characterize the fully polarized emergent nodal-loop fermions. We also find that a uniaxial strain can induce a loop transformation from a localized single loop circling around $\Gamma$ to a pair of extended loops penetrating the Brillouin zone., Comment: 7 pages, 5 fgures

Nodal loops in two-dimensional (2D) systems are typically vulnerable against spin-orbit coupling (SOC). Here, we explore 2D systems with a type of doubly degenerate nodal loops that are robust under SOC and feature an hourglass type dispersion. We present symmetry conditions for realizing such hourglass Weyl loops, which involve nonsymmorphic lattice symmetries. Depending on the symmetry, the loops may exhibit different patterns in the Brillouin zone. Based on first-principles calculations, we identify the monolayer GaTeI-family materials as a realistic material platform to realize such loops. These materials host a single hourglass Weyl loop circling around a high-symmetry point. Interestingly, there is also a spin-orbit Dirac point enabled by an additional screw axis. We show that the hourglass Weyl loop and the Dirac point are robust under a variety of applied strains. By breaking the screw axis, the Dirac point can be transformed into a second Weyl loop. Furthermore, by breaking the glide mirror, the hourglass Weyl loop and the spin-orbit Dirac point can both be transformed into a pair of spin-orbit Weyl points. Our work offers guidance and realistic material candidates for exploring fascinating physics of several novel 2D emergent fermions., Comment: 11 pages, 10 figures

A second-order topological insulator (SOTI) in $d$ spatial dimensions features topologically protected gapless states at its $(d-2)$-dimensional boundary at the intersection of two crystal faces, but is gapped otherwise. As a novel topological state, it has been attracting great interest, but it remains a challenge to identify a realistic SOTI material in two dimensions (2D). Here, based on combined first-principles calculations and theoretical analysis, we reveal the already experimentally synthesized 2D material graphdiyne as the first realistic example of a 2D SOTI, with topologically protected 0D corner states. The role of crystalline symmetry, the robustness against symmetry-breaking, and the possible experimental characterization are discussed. Our results uncover a hidden topological character of graphdiyne and promote it as a concrete material platform for exploring the intriguing physics of higher-order topological phases., 6 pages, 5 figures

Relativistic massless Weyl and Dirac fermions exhibit the isotropic and linear dispersion relations to preserve the Poincar\'{e} symmetry, the most fundamental symmetry in high energy physics. In solids, the counterparts of the Poincar\'{e} symmetry are crystallographic symmetries, and hence, it is natural to explore generalizations of Dirac and Weyl fermions compatible with their crystallographic symmetries and then the new physics coming along with them. Here, we study an important kind of generalization, namely massless Dirac fermions with higher-order dispersion relations protected by crystallographic symmetries in three-dimensional nonmagnetic systems. We perform a systematic search over all 230 space groups with time-reversal symmetry and spin-orbit coupling considered. We find that the order of dispersion cannot be higher than three, i.e., only the quadratic and cubic Dirac points (QDPs and CDPs) are possible. We discover previously unknown classes of higher-order Dirac points, including the chiral QDPs with Chern numbers of $\pm 4$ and the QDPs/CDPs without centrosymmetry. Especially the chiral QDPs feature four extensive surface Fermi arcs and four chiral Landau bands and hence leads to observable signatures in spectroscopic and transport experiments. We further show that these higher-order Dirac points represent parent phases for other exotic topological structures. Via controlled symmetry breaking, QDPs and CDPs can be transformed into double Weyl points, triple Weyl points, charge-2 Dirac points or Weyl loops. Using first-principles calculations, we also identify possible material candidates, including $\alpha$-TeO$_2$ and YRu$_4$B$_4$, which realize the predicted nodal structures., Comment: 23 pages, 7 figures

The anomalous spatial shifts at interface scattering, first studied in geometric optics, recently found their counterparts in the electronic context. It was shown that both longitudinal and transverse shifts, analogous to the Goos-Hanchen and Imbert-Fedorov effects in optics, can exist when electrons are scattered at a junction interface. More interestingly, the shifts are also discovered in the process of Andreev reflection at a normal/superconductor interface. Particularly, for the case with unconventional superconductors, it was discovered that the transverse shift can arise solely from the superconducting pair potential and exhibit characteristic features depending on the pairing. Here, we briefly review the recent works in this field, with an emphasis on the physical picture and theoretical understanding., Comment: 17 pages, an invited review contributing to a special issue "Recent Advances in Topological Materials" of Frontiers of Physics

Superconductors with unconventional pairings have been a fascinating subject of research, for which a central issue is to explore effects that can be used to characterize the pairing. The process of Andreev reflection--the reflection of an electron as a hole at a normal-mental-superconductor interface by transferring a Cooper pair into the superconductor--offers a basic mechanism to probe the pairing through transport. Here we predict that in Andreev reflection from unconventional superconductors, the reflected hole acquires an anomalous spatial shift normal to the plane of incidence, arising from the unconventional pairing. The transverse shift is sensitive to the superconducting gap structure, exhibiting characteristic features for each pairing type, and can be detected as voltage signals. Our work not only unveils a fundamentally new effect but also suggests a powerful new technique capable of probing the structure of unconventional pairings., 4 pages, 4 figures

Crossed Andreev reflection (CAR) is an intriguing effect that occurs in a normal-superconductor-normal junction. In CAR, an incoming electron from one terminal is coherently scattered as an outgoing hole into the other terminal. Here, we reveal that there exists a transverse spatial shift in CAR, i.e., the plane of CAR for the outgoing hole may have a sizable transverse shift from the plane of incidence for the incoming electron. We explicitly demonstrate the effect in a model system based on Weyl semimetals. We further show that the effect is quite general and exists when the terminals have sizable spin-orbit coupling. In addition, we find that the corresponding shift in the elastic cotunneling process shows different behaviors, and it vanishes when the two terminals are identical. Based on these findings, we suggest possible experimental setups for detecting the effect, which may also offer an alternative method for probing CAR., 10 pages, 6 figures

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.

Most electronic properties of metals are determined solely by the low-energy states around the Fermi level, and for topological metals/semimetals, these low-energy states become distinct because of their unusual energy dispersion and emergent pseudospin degree of freedom. Here, we propose a class of materials which are termed as quadratic contact point (QCP) semimetals. In these materials, the conduction and valence bands contact at isolated points in the Brillouin zone, around which the band dispersions are quadratic along all three directions. We show that in the absence/presence of spin-orbit coupling, there may exist triply/quadruply-degenerate QCPs that are protected by the crystalline symmetry. We construct effective models to characterize the low-energy fermions near these QCPs. Under strong magnetic field, unlike the usual 3D electron gas, there appear unconventional features in the Landau spectrum. The QCP semimetal phase is adjacent to a variety of topological phases. For example, by breaking symmetries via Zeeman field or lattice strain, it can be transformed into a Weyl semimetal with Weyl and double Weyl points, a Z2 topological insulator/metal, or a Dirac semimetal. Via first-principles calculations, we identify realistic materials Cu2Se and RhAs3 as candidates for QCP semimetals., 10 pages and 9 figures

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

A nodal loop is formed by band crossing along a one-dimensional closed manifold, with each point on the loop a linear nodal point in the transverse dimensions and can be classified as type-I or type-II depending on the band dispersion. Here, we propose a class of nodal loops composed of both type-I and type-II points, which are hence termed as hybrid nodal loops. Based on firstprinciples calculations, we predict the realization of such loops in the existing electride material Ca2As. For a hybrid loop, the Fermi surface consists of coexisting electron and hole pockets that touch at isolated points for an extended range of Fermi energies, without the need for fine-tuning. This leads to unconventional magnetic responses, including the zero-field magnetic breakdown and the momentum space Klein tunneling observable in the magnetic quantum oscillations, as well as the peculiar anisotropy in the cyclotron resonance., Comment: 5 pages, 4 figures

In electronic band structures, nodal lines may arise when two (or more) bands contact and form a one-dimensional manifold of degeneracy in the Brillouin zone. Around a nodal line, the dispersion for the energy difference between the bands is typically linear in any plane transverse to the line. Here, we perform an exhaustive search over all 230 space groups for nodal lines with higher-order dispersions that can be stabilized by crystalline symmetry in solid state systems with spin-orbit coupling and time reversal symmetry. We find that besides conventional linear nodal lines, only lines with quadratic or cubic dispersions are possible, for which the allowed degeneracy cannot be larger than two. We derive effective Hamiltonians to characterize the novel low-energy fermionic excitations for the quadratic and cubic nodal lines, and explicitly construct minimal lattice models to further demonstrate their existence. Their signatures can manifest in a variety of physical properties such as the (joint) density of states, magneto-response, transport behavior, and topological surface states. Using ab-initio calculations, we also identify possible material candidates that realize these exotic nodal lines., Comment: 15 pages, 7 figures

The conventional $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ method fails to capture the full and essential physics of many symmetry enriched multiple nodal line structures in the three-dimensional Brillouin zone. Here, we present a systematical method to construct the effective lattice model of mirror symmetry protected three-dimensional multiple nodal line semimetals, when the spin-orbit interaction is ignored. For systems with a given pair of perpendicular nodal rings, we obtain all the effective lattice models and 11 inequivalent nodal line Fermi surfaces together with their related constraints. By means of first-principles calculations, we first propose a family of real materials, the $\ensuremath{\beta}$ phase of ternary nitrides ${X}_{2}{\mathrm{GeN}}_{2}$ ($X=\mathrm{Ca},\mathrm{Sr},\mathrm{Ba}$), that support one kind of these Fermi surfaces. Therefore, our work deepens the understanding of the nodal line structures and promotes the experimental progress of topological nodal line semimetals.

Nodal-loop semimetals are materials in which the conduction and valence bands cross on a one-dimensional loop in the reciprocal space. For the nodal-loop character to manifest in physical properties, it is desired that the loop is close to the Fermi level, relatively flat in energy, simple in its shape, and not coexisting with other extraneous bands. Here, based on the first-principles calculations, we show that the monoclinic CuTeO$_3$ is a realistic nodal-loop semimetal that satisfies all these requirements. The material features only a single nodal loop around the Fermi level, protected by either of the two independent symmetries: the $\mathcal{PT}$ symmetry and the glide mirror symmetry. The size of the loop can be effectively tuned by strain, and the loop can even be annihilated under stain, making a topological phase transition to a trivial insulator phase. Including the spin-orbit coupling opens a tiny gap at the loop, and the system becomes a $\mathbb{Z}_2$ topological semimetal with a nontrivial bulk $\mathbb{Z}_2$ invariant but no global bandgap. The corresponding topological surface states have been identified. We also construct a low-energy effective model to describe the nodal loop and the effect of spin-orbit coupling., Comment: 8 pages, 10 figures

An incoming electron is reflected back as a hole at a normal-metal-superconductor interface, a process known as Andreev reflection. We predict that there exists a universal transverse shift in this process due to the effect of spin-orbit coupling in the normal metal. Particularly, using both the scattering approach and the argument of angular momentum conservation, we demonstrate that the shifts are pronounced for lightly-doped Weyl semimetals, and are opposite for incoming electrons with different chirality, generating a chirality-dependent Hall effect for the reflected holes. The predicted shift is not limited to Weyl systems, but exists for a general three-dimensional spin-orbit- coupled metal interfaced with a superconductor., 5 pages, 2 figures

A nodal loop appears when two bands, typically one electronlike and one holelike, are crossing each other linearly along a one-dimensional manifold in reciprocal space. Here, we propose a type of nodal loop which emerges from the crossing between two bands which are both electronlike (or holelike) along a certain direction. Close to any point on such a loop (dubbed as a type-II nodal loop), the linear spectrum is strongly tilted and tipped over along one transverse direction, leading to marked differences in magnetic, optical, and transport responses compared with conventional (type-I) nodal loops. We show that the compound ${\mathrm{K}}_{4}{\mathrm{P}}_{3}$ is an example that hosts a pair of type-II nodal loops close to the Fermi level. Each loop traverses the whole Brillouin zone, and hence can only be annihilated in a pair when symmetry is preserved. The symmetry and topological protections of the loops as well as the associated surface states are discussed.

Nonsymmorphic symmetries, which involve fractional lattice translations, can generate exotic types of fermionic excitations in crystalline materials. Here we propose a topological phase arising from nonsymmorphic symmetries—the hourglass Dirac chain metal, and predict its realization in the rhenium dioxide. We show that ReO2 features hourglass-type dispersion in the bulk electronic structure dictated by its nonsymmorphic space group. Due to time reversal and inversion symmetries, each band has an additional two-fold degeneracy, making the neck crossing-point of the hourglass four-fold degenerate. Remarkably, close to the Fermi level, the neck crossing-point traces out a Dirac chain—a chain of connected four-fold-degenerate Dirac loops—in the momentum space. The symmetry protection, the transformation under symmetry-breaking, and the associated topological surface states of the Dirac chain are revealed. Our results open the door to an unknown class of topological matters, and provide a platform to explore their intriguing physics., Exotic topological particles are reported to arise from special types of symmetry protection. Here, Wang et al. predict an hourglass Dirac chain metal protected by nonsymmorphic symmetries and its possible realization in ReO2.

Using first-principles calculations, we report that ZrO is a topological material with the coexistence of three pairs of type-II triply degenerate nodal points (TNPs) and three nodal rings (NRs), when spin-orbit coupling (SOC) is ignored. Noticeably, the TNPs reside around Fermi energy with large linear energy range along tilt direction (> 1 eV) and the NRs are formed by three strongly entangled bands. Under symmetry-preserving strain, each NR would evolve into four droplet-shaped NRs before fading away, producing distinct evolution compared with that in usual two-band NR. When SOC is included, TNPs would transform into type-II Dirac points while all the NRs have gaped. Remarkably, the type-II Dirac points inherit the advantages of TNPs: residing around Fermi energy and exhibiting large linear energy range. Both features facilitate the observation of interesting phenomena induced by type-II dispersion. The symmetry protections and low-energy Hamiltonian for the nontrivial band crossings are discussed., 7 pages, 5 figures, J. Phys. Chem. Lett. 2017

Topological metals with protected band-crossing points have been attracting great interest. Here we report novel topological band features in a family of metal diboride materials. Using first- principles calculations, we show that these materials are metallic, and close to Fermi level, there appears coexistence of one pair of nodal rings and one pair of triply-degenerate nodal points (TNPs). The nodal ring here is distinct from the previously studied ones in that its formation requires four entangled bands, not just two as in previous cases, hence it is termed as a four-band nodal ring (FNR). Remarkably, we show that FNR features Dirac-cone-like surface states, in contrast to the usual drumhead surface states for two-band nodal rings. Due to the presence of inversion symmetry, the TNP here is also different from those discussed previously in inversion-asymmetric systems. Especially, when spin-orbit coupling is included, the TNP here transforms into a novel Dirac point that is close to the borderline between the type-I and type-II Dirac point categories. We discuss their respective symmetry protections, and construct effective models for their characterization. The large linear energy range (> 2 eV) in these materials should facilitate the experimental detection of the signatures of these nontrivial band crossings., 8 pages, 8 figures

Dirac points in two-dimensional (2D) materials have been a fascinating subject of research, with graphene as the most prominent example. However, the Dirac points in existing 2D materials, including graphene, are vulnerable against spin-orbit coupling (SOC). Here, based on first-principles calculations and theoretical analysis, we propose a new family of stable 2D materials, the HfGeTe-family monolayers, which represent the first example to host so-called spin-orbit Dirac points (SDPs) close to the Fermi level. These Dirac points are special in that they are formed only under significant SOC, hence they are intrinsically robust against SOC. We show that the existence of a pair of SDPs are dictated by the nonsymmorphic space group symmetry of the system, which are very robust under various types of lattice strains. The energy, the dispersion, and the valley occupation around the Dirac points can be effectively tuned by strain. We construct a low-energy effective model to characterize the Dirac fermions around the SDPs. Furthermore, we find that the material is simultaneously a 2D $\mathbb{Z}_2$ topological metal, which possesses nontrivial $\mathbb{Z}_2$ invariant in the bulk and spin-helical edge states on the boundary. From the calculated exfoliation energies and mechanical properties, we show that these materials can be readily obtained in experiment from the existing bulk materials. Our result reveals HfGeTe-family monolayers as a promising platform for exploring spin-orbit Dirac fermions and novel topological phases in two-dimensions., Comment: 9 pages, 6 figures

Carbon, the basic building block of our universe, enjoys a vast number of allotropic structures. Owing to its bonding characteristic, most carbon allotropes possess the motif of hexagonal rings. Here, with first-principles calculations, we discover a new metastable three-dimensional carbon allotrope entirely composed of pentagon rings. The unique structure of this Pentagon Carbon leads to extraordinary electronic properties, making it a cornucopia of emergent topological fermions. Under lattice strain, Pentagon Carbon exhibits topological phase transitions, generating a series of novel quasiparticles, from isospin-1 triplet fermions to triply degenerate fermions and further to Hopf-link Weyl-loop fermions. Its Landau level spectrum also exhibits distinct features, including a huge number of almost degenerate chiral Landau bands, implying pronounced magneto-transport signals. Our work not only discovers a remarkable carbon allotrope with highly rare structural motifs, it also reveals a fascinating hierarchical particle genesis with novel topological fermions beyond the Dirac and Weyl paradigm., Whether carbon allotropes may host emergent topological fermions is unknown. Here, Zhong et al. predict a three-dimensional carbon allotrope entirely composed of pentagon rings, showing a plethora of topological fermions under strain.