Chirality-Dependent Hall Effect in Weyl Semimetals

Multiple Weyl fermions and topological phase transition in two-dimensional ferromagnetic CrS2

The study of topological states in two-dimensional (2D) systems, especially with magnetic properties, has recently gained significant attention owing to their potential in spintronics and nanotechnology. Here, we propose a 2D ferromagnetic (FM) material, CrS2, which hosts multiple Weyl points (WPs) and can undergo a topological phase transition by rotating the magnetization direction. Based on first-principles calculations, we identify distinct Weyl points around the Fermi level: W1, W2, and W3. These points appear in both spin channels and include various types: type-I, type-II and type-III WPs. Corresponding Fermi arcs are clearly observed at the material edges. CrS2 displays a FM ground state with the easy magnetization direction along the c-axis. When the magnetization direction is rotated in the x-y plane, the W1 and W3 points open gaps, with the gap values remaining the same in all magnetization directions. The W2 can maintain a crossing at specific in-plane magnetization directions, indicating that the material retains its Weyl state. Additionally, we examine the effects of biaxial and uniaxial strains on electronic properties. Weyl points remain stable under biaxial strain of less than ±5%, but they disappear under uniaxial strain. In summary, our work proposes a 2D FM material with multiple coexisting Weyl fermions, where the topological states can be tuned by an external magnetic field.

Geometry, anomaly, topology, and transport in Weyl fermions

Weyl fermions are one of the simplest objects that link ideas in geometry and topology to high-energy physics and condensed matter physics. Although the existence of Weyl fermions as elementary particles remains dubious, there is mounting evidence of their existence as quasiparticles in certain condensed matter systems. Such systems are termed Weyl semimetals (WSMs). Needless to say, WSMs have emerged as a fascinating class of materials with unique electronic properties, offering a rich playground for both fundamental research and potential technological applications. This review examines recent advancements in understanding electron transport in WSMs. We begin with a pedagogical introduction to the geometric and topological concepts critical to understanding quantum transport in Weyl fermions. We then explore chiral anomaly, a defining feature of WSMs, and its impact on transport phenomena such as longitudinal magnetoconductance and planar Hall effect. The Maxwell-Boltzmann transport theory extended beyond the standard relaxation-time approximation is then discussed in the context of Weyl fermions, which is used to evaluate various transport properties. Attention is also given to the effects of strain-induced gauge fields and external magnetic fields in both time-reversal broken and inversion asymmetric inhomogeneous WSMs. The review synthesizes theoretical insights, experimental observations, and numerical simulations to provide a comprehensive understanding of the complex transport behaviors in WSMs, aiming to bridge the gap between theoretical predictions and experimental verification.

Half-quantized helical hinge currents in axion insulators

Fractional quantization can emerge in noncorrelated systems due to the parity anomaly, while its condensed matter realization is a challenging problem. We propose that in axion insulators (AIs), parity anomaly manifests a unique fractional boundary excitation: the half-quantized helical hinge currents. These helical hinge currents microscopically originate from the lateral Goos-Hänchen (GH) shift of massless side-surface Dirac electrons that are totally reflected from the hinges. Meanwhile, due to the presence of the massive top and bottom surfaces of the AI, the helical current induced by the GH shift is half-quantized. The semiclassical wave packet analysis uncovers that the hinge current has a topological origin and its half quantization is robust to parameter variations. Lastly, we propose an experimentally feasible six-terminal device to identify the half-quantized hinge channels by measuring the nonreciprocal conductances. Our results advance the realization of the half-quantization and topological magnetoelectric responses in AIs.

Fast nuclear spin relaxation rates in tilted cone Weyl semimetals: redshift factors from Korringa relation

Spin lattice relaxation rate is investigated for 3D tilted cone Weyl semimetals (TCWSMs). The nuclear spin relaxation rate is presented as a function of temperature and tilt parameter. We find that the relaxation rate behaves as(1-ζ2)-αwithα≈ 9 where 0 ⩽ζ< 1 is the tilt parameter. We demonstrate that such a strong enhancement forζ≲ 1 that gives rise to very fast relaxation rates, is contributed by a new hyperfine interactions arising from the tilt itself. This can be attributed to the combination of anisotropy of the Fermi surface and an additional part related to the structure of the spacetime: extracting an effective density of states (DOS)ρ̃from the Korringa relation, we show that it is related to the DOSρof the tilted cone dispersion by the 'redshift factor' asρ̃=ρ/1-ζ2. We interpret this relation as NMR manifestation of an emergent underlying spacetime structure in TCWSMs.

Fully spin-polarized Weyl fermions and in/out-of-plane quantum anomalous Hall effects in a two-dimensional d0 ferromagnet

The quantum anomalous Hall effect (QAHE) in intrinsic ferromagnets has attracted considerable attention recently. Previously, studies of the QAHE have mostly focused on the default assumption of out-of-plane magnetization. In fact, the QAHE can also be achieved via in-plane magnetization, but such candidate materials are very scarce. Here, we find that two-dimensional (2D) YN₂ not only possesses the previously reported out-of-plane QAHE, but it also possesses a tunable in-plane QAHE. More importantly, unlike the previously reported in-plane QAHE in d/f-type ferromagnets, here we report the effect in a 2D d⁰ ferromagnet, namely YN₂, for the first time. In the ground state, a YN₂ monolayer has a half-metal band structure, and manifests six pairs of fully spin-polarized Weyl points at the Fermi level. When spin-orbit coupling is included, the YN₂ monolayer can realize multiple topological phases, determined based on the magnetization direction. Under in-plane magnetization, the YN₂ monolayer shows either the Weyl state or in-plane QAHE state. Remarkably, the Chern number (±1) and the propagating direction of QAHE edge channels can be continuously switched via shifting the direction of the in-plane magnetic field. When magnetization is applied out-of-plane, the YN₂ monolayer realizes an out-of-plane QAHE phase with a high Chern number of 3. The nontrivial edge states for all the topological phases in the YN₂ monolayer have been clearly identified. This work suggests that 2D YN₂ is an excellent candidate for investigating in-plane QAHE phases in d⁰ ferromagnets.

Quantized Circulation of Anomalous Shift in Interface Reflection

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 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.

Recent advances in 2D, 3D and higher-order topological photonics

Over the past decade, topology has emerged as a major branch in broad areas of physics, from atomic lattices to condensed matter. In particular, topology has received significant attention in photonics because light waves can serve as a platform to investigate nontrivial bulk and edge physics with the aid of carefully engineered photonic crystals and metamaterials. Simultaneously, photonics provides enriched physics that arises from spin-1 vectorial electromagnetic fields. Here, we review recent progress in the growing field of topological photonics in three parts. The first part is dedicated to the basics of topological band theory and introduces various two-dimensional topological phases. The second part reviews three-dimensional topological phases and numerous approaches to achieve them in photonics. Last, we present recently emerging fields in topological photonics that have not yet been reviewed. This part includes topological degeneracies in nonzero dimensions, unidirectional Maxwellian spin waves, higher-order photonic topological phases, and stacking of photonic crystals to attain layer pseudospin. In addition to the various approaches for realizing photonic topological phases, we also discuss the interaction between light and topological matter and the efforts towards practical applications of topological photonics.

Topological Amorphous Metals

We study amorphous systems with completely random sites and find that, through constructing and exploring a concrete model Hamiltonian, such a system can host an exotic phase of topological amorphous metal in three dimensions. In contrast to the traditional Weyl semimetals, topological amorphous metals break translational symmetry, and thus they cannot be characterized by the first Chern number defined based on the momentum space band structures. Instead, their topological properties will manifest in the Bott index and the Hall conductivity as well as the surface states. By studying the energy band and quantum transport properties, we find that topological amorphous metals exhibit a diffusive metal behavior. We further introduce a practical experimental proposal with electric circuits where the predicted phenomena can be observed using state-of-the-art technologies. Our results open the door to exploring topological gapless phenomena in amorphous systems.

Topological valley pumping in Weyl semimetals

We investigate the topological pumping effect in Weyl semimetals, subject to the modulation of two ac electric fields along y  and z directions, respectively. We show that the pumping effect originates from the anomalous velocity related to the Berry curvature. The direction of the pumping current is dependent on the chirality of the Weyl fermions. While the total particle current is vanishing because the Weyl points of opposite chirality always come in pairs in Weyl semimetals, the pump gives rise to a net chirality current or valley current. The noiseless valley current generated can be useful in valleytronic applications.

Electrically tunable valley polarization in Weyl semimetals with tilted energy dispersion

Tunneling transport across electrical potential barriers in Weyl semimetals with tilted energy dispersion is investigated. We report that the electrons around different valleys experience opposite direction refractions at the barrier interface when the energy dispersion is tilted along one of the transverse directions. Chirality dependent refractions at the barrier interface polarize the Weyl fermions in angle-space according to their valley index. A real magnetic barrier configuration is used to select allowed transmission angles, which results in electrically controllable and switchable valley polarization. Our findings may pave the way for experimental investigation of valley polarization, as well as valleytronic and electron optic applications in Weyl semimetals.

Fermi-Arc-Induced Vortex Structure in Weyl Beam Shifts

In periodic media, despite the close relationship between geometrical effects in the bulk and topological surface states, the two are typically probed separately. We show that when beams in a Weyl medium reflect off an interface with a gapped medium, the trajectory is influenced by both bulk geometrical effects and the Fermi arc surface states. The reflected beam experiences a displacement, analogous to the Goos-Hänchen or Imbert-Fedorov shifts, that forms a half-vortex in the two-dimensional surface momentum space. The half-vortex is centered where the Fermi arc of the reflecting surface touches the Weyl cone, with the magnitude of the shift scaling as an inverse square root away from the touching point, and diverging at the touching point. This striking feature provides a way to use bulk transport to probe the topological characteristics of a Weyl medium.

Chirality Josephson Current Due to a Novel Quantum Anomaly in Inversion-Asymmetric Weyl Semimetals

We study Josephson junctions based on inversion-asymmetric but time-reversal symmetric Weyl semimetals under the influence of Zeeman fields. We find that, due to distinct spin textures, the Weyl nodes of opposite chirality respond differently to an external magnetic field. Remarkably, a Zeeman field perpendicular to the junction direction results in a phase shift of opposite sign in the current-phase relations of opposite chirality. This leads to a finite chirality Josephson current (CJC) even in the absence of a phase difference across the junction. This feature could allow for applications in chiralitytronics. In the long junction and zero temperature limit, the CJC embodies a novel quantum anomaly of Goldstone bosons at π phase difference which is associated with a Z_{2} symmetry at low energies. It can be detected experimentally via an anomalous Fraunhofer pattern.

Controlling electron flow in anisotropic Dirac materials heterojunctions: a super-diverging lens

Ballistic heterojunctions of Dirac materials offer the opportunity of exploring optics-like phenomena in electronic systems. In this paper, a new perfect lens through special positive refraction is predicted with omnidirectional Klein tunneling of massless Dirac fermions. The novel optics component called a super-diverging lens (SDL) is the counterpart of a Veselago lens (VL). The use of SDL and VL creates a device that simulates the ocular vision. This atypical refraction is due to electrons obeying different Snell's laws of pseudo-spin and group velocity in heterojunctions with elliptical Dirac cones. These findings pave the way for an electron elliptical Dirac optics and open up new possibilities for the guiding of electrons.

Unconventional Pairing Induced Anomalous Transverse Shift in Andreev Reflection

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-metal-superconductor interface-offers a basic mechanism to probe the pairing. 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 with a novel underlying mechanism, but also suggests a possible new technique capable of probing the structure of unconventional pairings.

Electronic scattering, focusing, and resonance by a spherical barrier in Weyl semimetals

We solve the Weyl electron scattered by a spherical step potential barrier. Tuning the incident energy and the potential radius, one can enter both quasiclassical and quantum regimes. Transport features related to far-field currents and integrated cross sections are studied to reveal the preferred forward scattering. In the quasiclassical regime, a strong focusing effect along the incident spherical axis is found in addition to optical caustic patterns. In the quantum regime, at energies of successive angular momentum resonances, a polar aggregation of electron density is found inside the potential. The findings will be useful in transport studies and electronic lens applications in Weyl systems.

Shot noise and electronic properties in the inversion-symmetric Weyl semimetal resonant structure

Using the transfer matrix method, the authors combine the analytical formula with numerical calculation to explore the shot noise and conductance of massless Weyl fermions in the Weyl semimetal resonant junction. By varying the barrier strength, the structure of the junction, the Fermi energy, and the crystallographic angle, the shot noise and conductance can be tuned efficiently. For a quasiperiodic superlattice, in complete contrast to the conventional junction case, the effect of the disorder strength on the shot noise and conductance depends on the competition of classical tunneling and Klein tunneling. Moreover, the delta barrier structure is also vital in determining the shot noise and conductance. In particular, a universal Fano factor has been found in a single delta potential case, whereas the resonant structure of the Fano factor perfectly matches with the number of barriers in a delta potential superlattice. These results are crucial for engineering nanoelectronic devices based on this topological semimetal material.

Three-dimensional Pentagon Carbon with a genesis of emergent fermions

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.

Resonance states and beating pattern induced by quantum impurity scattering in Weyl/Dirac semimetals

Currently, Weyl semimetals (WSMs) are drawing great interest as a new topological nontrivial phase. When most of the studies concentrated on the clean host WSMs, it is expected that the dirty WSM system would present rich physics due to the interplay between the WSM states and the impurities embedded inside these materials. We investigate theoretically the change of local density of states in three-dimensional Dirac and Weyl bulk states scattered off a quantum impurity. It is found that the quantum impurity scattering can create nodal resonance and Kondo peak/dip in the host bulk states, remarkably modifying the pristine spectrum structure. Moreover, the joint effect of the separation of Weyl nodes and the Friedel interference oscillation causes the unique battering feature. We in detail an- alyze the different contribution from the intra- and inter-node scattering processes and present various scenarios as a consequence of competition between them. Importantly, these behaviors are sensitive significantly to the displacement of Weyl nodes in energy or momentum, from which the distinctive fingerprints can be extracted to identify various semimetal materials experimentally by employing the scanning tunneling microscope.

Predicted Unusual Magnetoresponse in Type-II Weyl Semimetals

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.