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Nodal monoloop, enjoying the cleanest scenario with a single loop, is recognized as the basic building block of intricate linked loops including chains, nets, and knots. Here, we explore the interplay of magnetic ordering and band topology in one system by introducing a brand-new quantum state, referred to as Weyl monoloop semi-half-metal, which is characterized by a single loop at the Fermi level stemming from the same spin channel. Such a nodal line Fermion, yielding 100% spin polarization, is protected by mirror ( M z ) symmetry. As a prominent example, a realistic rutile-type metal fluorides LiV2F6 achieves the hitherto unmaterialized state, featuring fully spin-polarized ultraflat surface states. More interestingly, LiV2F6 has a "soft" ferromagnetic property, which is one of the desired systems to control the anomalous Hall effect by rotating the magnetization direction. Our findings offer a promising candidate for exploring the topology and magnetism with intriguing effects.

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

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.

In the presence of hexagonal warping (HW), the surface state of a topological insulator (TI) exhibits anisotropic Fermi surface. In this work, we study the transport properties of a TI junction, focusing on the influence of anisotropic effect of the HW on the scattering. We establish general expressions for calculating the scattering with an arbitrary angle between TI surface and the potential step, and find that the transmission probability and conductance have a strong dependence on the angle, making TI junction a promising platform for studying the geometry control of transport. Remarkably, we find that the HW can induce a triple refraction, which gives rise to an anomalous increase of the transmission probability when varying incident angle. The general expressions here for calculating the scattering can be extended to the systems with trigonally and tetragonally warped Fermi surface.

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.

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 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

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

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.

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

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.

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

Recently, a type-II Weyl fermion was theoretically predicted to appear at the contact of electron and hole Fermi surface pockets. A distinguishing feature of the surfaces of type-II Weyl semimetals is the existence of topological surface states, so-called Fermi arcs. Although WTe2 was the first material suggested as a type-II Weyl semimetal, the direct observation of its tilting Weyl cone and Fermi arc has not yet been successful. Here, we show strong evidence that WTe2 is a type-II Weyl semimetal by observing two unique transport properties simultaneously in one WTe2 nanoribbon. The negative magnetoresistance induced by a chiral anomaly is quite anisotropic in WTe2 nanoribbons, which is present in b-axis ribbon, but is absent in a-axis ribbon. An extra-quantum oscillation, arising from a Weyl orbit formed by the Fermi arc and bulk Landau levels, displays a two dimensional feature and decays as the thickness increases in WTe2 nanoribbon., Exotic transport properties of type-II Weyl semimetals have been predicted but are yet to be experimentally evidenced. Here, Li et al. report evidences of an anisotropy of negative magnetoresistance and a quantum oscillation arising from the predicted Weyl orbit in the type-II Weyl semimetal WTe2.

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.

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.

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

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.

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

Spin–orbit torque has recently been intensively investigated for the purposes of manipulating the magnetization in magnetic nano-devices and understanding fundamental physics. Therefore, the search for novel materials or material combinations that exhibit a strong enough spin-torque effect has become one of the top priorities in this field of spintronics. Weyl semimetal, a new topological material that features open Fermi arc with strong spin–orbit coupling and spin–momentum locking effect, is naturally expected to exhibit an enhanced spin-torque effect in magnetic nano-devices. Here we observe a significantly enhanced spin conductivity, which is associated with the field-like torque at low temperatures. The enhancement is obtained in the b-axis WTe2/Py bilayers of nano-devices but not observed in the a-axis of WTe2/Py nano-devices, which can be ascribed to the enhanced spin accumulation by the spin–momentum locking effect of the Fermi arcs of the Weyl semimetal WTe2., The Fermi arcs, topological surface states of Weyl semimetals can enable the intriguing spin control and facilitate topological spintronics. Here the authors report the spin-orbit torque at the interface of WTe2/Py and attribute it to the enhanced spin accumulation by the spin-momentum locking effect of the Fermi arcs of WTe2.

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

At a normal-metal/superconductor interface, an incident electron from the normal-metal (N) side can be normally reflected as an electron or Andreev reflected as a hole. We show that pronounced lateral shifts along the interface between the incident and the reflected quasiparticles can happen in both reflection processes, which are analogous to the Goos-H\"anchen effect in optics. Two concrete model systems are considered. For the simplest model in which the N side is of the two-dimensional electron gas, we find that while the shift in Andreev reflection stays positive, the shift in normal reflection can be made either positive or negative, depending on the excitation energy. For the second model with the N side taken by graphene, the shift in Andreev reflection can also be made negative, and the shifts have rich behavior due to the additional sublattice pseudospin degree of freedom. We show that the shift strongly modifies the dispersion for the confined waveguide modes in an SNS structure. We also suggest a possible experimental setup for detecting the shift., Comment: 9 pages, 8 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

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.

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

The recent proposal of the type-II Weyl semimetal state has attracted significant interest. In this Letter, we propose the concept of the three-dimensional type-II Dirac fermion and theoretically identify this new symmetry-protected topological state in the large family of transition-metal icosagenides, MA_{3} (M=V, Nb, Ta; A=Al, Ga, In). We show that the VAl_{3} family features a pair of strongly Lorentz-violating type-II Dirac nodes and that each Dirac node can be split into four type-II Weyl nodes with chiral charge ±1 via symmetry breaking. Furthermore, we predict that the Landau level spectrum arising from the type-II Dirac fermions in VAl_{3} is distinct from that of known Dirac or Weyl semimetals. We also demonstrate a topological phase transition from a type-II Dirac semimetal to a quadratic Weyl semimetal or a topological crystalline insulator via crystalline distortions.

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

Topological metals and semimetals (TMs) have recently drawn significant interest. These materials give rise to condensed matter realizations of many important concepts in high-energy physics, leading to wide-ranging protected properties in transport and spectroscopic experiments. The most studied TMs, i.e., Weyl and Dirac semimetals, feature quasiparticles that are direct analogues of the textbook elementary particles. Moreover, the TMs known so far can be characterized based on the dimensionality of the band crossing. While Weyl and Dirac semimetals feature zero-dimensional points, the band crossing of nodal-line semimetals forms a one-dimensional closed loop. In this paper, we identify a TM which breaks the above paradigms. Firstly, the TM features triply-degenerate band crossing in a symmorphic lattice, hence realizing emergent fermionic quasiparticles not present in quantum field theory. Secondly, the band crossing is neither 0D nor 1D. Instead, it consists of two isolated triply-degenerate nodes interconnected by multi-segments of lines with two-fold degeneracy. We present materials candidates. We further show that triplydegenerate band crossings in symmorphic crystals give rise to a Landau level spectrum distinct from the known TMs, suggesting novel magneto-transport responses. Our results open the door for realizing new topological phenomena and fermions including transport anomalies and spectroscopic responses in metallic crystals with nontrivial topology beyond the Weyl/Dirac paradigm.

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.

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

The gapless edge states with a certain spin and valley index can be realized in topological domain walls. In this work, we study various domain walls thoroughly and demonstrate that multiple kinds of edge states with a perfect spin and valley polarization can be realized, leading to the possibility of establishing spin filters and spin-valley filters. Moreover, the spin and valley index of the edge states can be tuned by an external electric field. We also investigate the finite size effect on the edge states when two domain walls approach each other. Generally, the finite size effect can cause the hybridization of edge states and open a gap. We find that the evolutions of the bandgap as a function of the finite size effect for different topological edge states are distinguished from each other.

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.

Effective gravity and gauge fields are emergent properties intrinsic for low-energy quasiparticles in topological semimetals. Here, taking two Dirac semimetals as examples, we demonstrate that applied lattice strain can generate warped spacetime, with fascinating analogues in astrophysics. Particularly, we study the possibility of simulating black-hole/white-hole event horizons and gravitational lensing effect. Furthermore, we discover strain-induced topological phase transitions, both in the bulk materials and in their thin films. Especially in thin films, the transition between the quantum spin Hall and the trivial insulating phases can be achieved by a small strain, naturally leading to the proposition of a novel piezo-topological transistor device. Possible experimental realizations and analogue of Hawking radiation effect are discussed. Our result bridges multiple disciplines, revealing topological semimetals as a unique table-top platform for exploring interesting phenomena in astrophysics and general relativity; it also suggests realistic materials and methods to achieve controlled topological phase transitions with great potential for device applications., 29 pages,6 figures

We show several distinct signatures in the magnetoresponse of type-II Weyl semimetals. The energy tilt tends to squeeze the Landau levels (LLs), and, for a type-II Weyl node, there always exists a critical angle between the B field and the tilt, at which the LL spectrum collapses, regardless of the field strength. Before the collapse, signatures also appear in the magneto-optical spectrum, including the invariable presence of intraband peaks, the absence of absorption tails, and the special anisotropic field dependence.