Weyl Particles Research Articles

Descripting e + and Weyl fermion as beam/current for pump/injection semiconductor devices

Based on the demand for an improvement in various corpuscle types of current injection, the objective of this technique is to provide a new concept of carrier generators for optoelectronic pump and injection devices. This investigation is conducted to improve current injection by using a particle other than the electron. The idea was conceived from condensed matter physics for a technique to implement positron for carrier transport in semiconductors with the source based on localized emissions. A radioactive source such as 22Na is incident on a tungsten vane moderator, thus having positive electrons flowing and tunneling as well as a laser-driven high-quality positron into semiconductor-based devices. In addition, tantalum arsenide (TaAs) hosting Weyl particles has been discovered to hold significant potential for cutting-edge technological uses. Injection of different carriers and their behavior in semiconductors will lead to the emergence of solid state optoelectronics with different carrier injections that possesses a high energy (100–500 keV) and the possibility of maximum energy that is approximately several tens of megaelectron volts. Significantly, these various carrier sources have a larger range of operational settings and output characteristics due to their various underlying emission principles, thus obtaining a greater kinetic energy for a positron. The transformation to Weyl fermions carries electric charge via a device far more quickly than ordinary electrons, therefore unlocking the potential of new materials with unusual transport properties.

A General Method for Obtaining Degenerate Solutions to the Dirac and Weyl Equations and a Discussion on the Experimental Detection of Degenerate States

AbstractIn this work, a general method is described for obtaining degenerate solutions of the Dirac equation, corresponding to an infinite number of electromagnetic 4‐potentials and fields, which are explicitly calculated. More specifically, using four arbitrary real functions, one can automatically construct a spinor that satisfies the Dirac equation for an infinite number of electromagnetic 4‐potentials, defined by those functions. An interesting characteristic of these solutions is that, in the case of Dirac particles with nonzero mass, the degenerate spinors should be localized, both in space and time. The method is also extended to the cases of massless Dirac and Weyl particles, where the localization of the spinors is no longer required. Finally, two experimental methods are proposed for detecting the presence of degenerate states.

Discovery of a maximally charged Weyl point

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 quantised 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.

On the Localization Properties of Weyl Particles

AbstractIn this work, it is shown that Weyl particles can exist at different states in zero electromagnetic field, either as free particles or at localized states. In addition, it is shown that the localization, as well as the energy, of the particles can be fully controlled using simple electric fields, which can easily be realized in practice. These results are particularly important regarding possible practical applications of Weyl particles, both considering solid‐state physics in materials supporting these particles and laser physics using ions trapped by laser beams, which can simulate the behavior of Weyl particles.

Degenerate solutions to the massless Dirac and Weyl equations and a proposed method for controlling the quantum state of Weyl particles

In a recent work, we have shown that all solutions to the Weyl equation and a special class of solutions to the Dirac equation are degenerate in the sense that they remain unaltered under the influence of a wide variety of different electromagnetic fields. In this study, our previous work is significantly extended, providing a wide class of degenerate solutions to the Dirac equation for massless particles. The electromagnetic fields corresponding to these solutions are calculated and examples regarding both spatially constant electromagnetic fields and electromagnetic waves are also provided. Furthermore, some general solutions to the Weyl equation are presented, and the corresponding electromagnetic fields are calculated. Based on these results, a method for fully controlling the quantum state of Weyl particles using appropriate electromagnetic fields is proposed. Finally, the transition from degenerate to non-degenerate solutions as the particles acquire mass is discussed.

Chiral transport of pseudospinors induced by synthetic gravitational field in photonic Weyl metamaterials

Weyl particles exhibit chiral transport property under external curved space-time geometry. This effect is called the chiral gravitational effect, which plays an important role in quantum field theory. However, the absence of real Weyl particles in nature hinders the observation of such interesting phenomena. In this paper, we show that chiral gravitational effect can be manifested in Weyl metamaterials with spatially controlled nonlocality. This inhomogeneous modulation results in a spatially dependent group velocity in the Weyl cone dispersion, which is equivalent to introducing a curved background space-time (or gravitational field) for Weyl pseudospinors according to the Clifford algebra. The Weyl equation describing the Weyl cone dispersions is reformulated into the covariant form that contains a nonAbelian gauge field by considering the parallel transport of pseudospinors. The synthetic gravitational field (i.e., space-time curvature) leads to the quantization of energy levels, including chiral zeroth order energy modes (or simply chiral zero modes) that determine the chiral transport property of pseudospinors. Inhomogeneous Weyl metamaterials can potentially become a platform for investigating the interaction between Weyl particles and gravitational fields in future table-top experiments. The theoretical model and the associated results may serve as a guidance for experimentalists to observe such chiral transport effects in photonic Weyl systems.

On 3D and 1D Weyl particles in a 1D box

We construct the most general families of self-adjoint boundary conditions for three (equivalent) Weyl Hamiltonian operators, each describing a three-dimensional Weyl particle in a one-dimensional box situated along a Cartesian axis. These results are essentially obtained by using the most general family of self-adjoint boundary conditions for a Dirac Hamiltonian operator that describes a one-dimensional Dirac particle in a box, in the Weyl representation, and by applying simple changes of representation to this operator. Likewise, we present the most general family of self-adjoint boundary conditions for a Weyl Hamiltonian operator that describes a one-dimensional Weyl particle in a one-dimensional box. We also obtain and discuss throughout the article distinct results related to the Weyl equations in (3+1) and (1+1) dimensions, in addition to their respective wave functions, and present certain key results related to representations for the Dirac equation in (1+1) dimensions.

On the Boundary Conditions for the 1D Weyl–Majorana Particle in a Box

In (1+1) space-time dimensions, we can have two particles that are Weyl and Majorana particles at the same time---1D Weyl-Majorana particles. That is, the right-chiral and left-chiral parts of the two-component Dirac wave function that satisfies the Majorana condition, in the Weyl representation, describe these particles, and each satisfies their own Majorana condition. Naturally, the nonzero component of each of these two two-component wave functions satisfies a Weyl equation. We investigate and discuss this issue and demonstrate that for a 1D Weyl-Majorana particle in a box, the nonzero components, and therefore the chiral wave functions, only admit the periodic and antiperiodic boundary conditions. From the latter two boundary conditions, we can only construct four boundary conditions for the entire Dirac wave function. Then, we demonstrate that these four boundary conditions are also included within the most general set of self-adjoint boundary conditions for a 1D Majorana particle in a box.

Probing the Berry curvature and Fermi arcs of a Weyl circuit

The Weyl particle is the massless fermionic cousin of the photon. While no fundamental Weyl particles have been identified, they arise in condensed matter and meta-material systems, where their spinor nature imposes topological constraints on low-energy dispersion and surface properties. Here we demonstrate a topological circuit with Weyl dispersion at low-momentum, realizing a 3D lattice that behaves as a half-flux Hofstadter model in all principal planes. The circuit platform provides access to the complete complex-valued spin-texture of all bulk- and surface- states, thereby revealing not only the presence of Weyl points and the Fermi arcs that connect their surface-projections, but also, for the first time, the Berry curvature distribution through the Brillouin zone and the associated quantized Chiral charge of the Weyl points. This work opens a path to exploration of interacting Weyl physics in superconducting circuits, as well as studies of how manifold topology impacts band topology in three dimensions.

Dirac, Majorana and Weyl particles in the oscillating de Sitter universe, reflection from the cosmological barrier

It is known that the geometry of the Lobachevsky space acts on the fields of particles with spins 0, 1/2, 1 as an ideal mirror distributed in space. The depth of penetration of the field in such a medium increases with increasing field energy. Since the Lobachevsky model is a constituent element in some cosmological models, this property means that in such models it is necessary to take into account the effect of the presence of a “cosmological mirror”; it must lead to a redistribution of the particle density in space. The earlier analysis assumed the static nature of the space-time geometry. In this article, we generalize the research of the spin 1/2 field in the case of the oscillating model of the de Sitter universe. The Dirac equation is solved in the non-static quasi-Cartesian coordinates. At this, we substantially use the diagonalization of a generalized helicity operator. The wave functions of the particle are nontrivially time-dependent; however the effect of a complete reflection of the particles from an effective potential barrier is preserved. For the real Majorana 4-spinor field, the similar results are valid. For the solutions describing the reflection effect to be constructed, we must use linear combinations of solutions with opposite helicities. Such combinations are forbidden for 2-component Weyl particles, for this reason such particles cannot be reflected by the cosmological barrier.

Discovery of Lorentz-violating type II Weyl fermions in LaAlGe.

In quantum field theory, Weyl fermions are relativistic particles that travel at the speed of light and strictly obey the celebrated Lorentz symmetry. Their low-energy condensed matter analogs are Weyl semimetals, which are conductors whose electronic excitations mimic the Weyl fermion equation of motion. Although the traditional (type I) emergent Weyl fermions observed in TaAs still approximately respect Lorentz symmetry, recently, the so-called type II Weyl semimetal has been proposed, where the emergent Weyl quasiparticles break the Lorentz symmetry so strongly that they cannot be smoothly connected to Lorentz symmetric Weyl particles. Despite some evidence of nontrivial surface states, the direct observation of the type II bulk Weyl fermions remains elusive. We present the direct observation of the type II Weyl fermions in crystalline solid lanthanum aluminum germanide (LaAlGe) based on our photoemission data alone, without reliance on band structure calculations. Moreover, our systematic data agree with the theoretical calculations, providing further support on our experimental results.

Semiclassical dynamics of Dirac and Weyl particles in rotating coordinates

The semiclassical kinetic theory of Dirac particles in the presence of external electromagnetic fields and global rotation is established. To provide the Hamiltonian formulation of Dirac particles a symplectic two-form which is a matrix in spin indices is proposed. The particle number and current densities for the Dirac particles are acquired in the helicity basis. Following a similar procedure, semiclassical kinetic theory of the Weyl particles is accomplished. It is shown that the phase-space dynamics of the Weyl and Dirac particles is directly linked. The anomalous chiral effects due to the external electromagnetic fields and angular velocity of the frame are calculated.

Analog simulation of Weyl particles with cold atoms

In this letter we report on a novel approach to study the dynamics of harmonically confined Weyl particles using magnetically trapped fermionic atoms. We find that after a kick of its center of mass, the system relaxes towards a steady state even in the absence of interactions, in stark contrast with massive particles which would oscillate without damping. Remarkably, the equilibrium distribution is non-Boltzmann, exhibiting a strong anisotropy which we study both numerically and experimentally.

Weyl particles discovered

Weyl particles discovered

Berry phase, Lorentz covariance, and anomalous velocity for Dirac and Weyl particles

We consider the relation between spin and the Berry-phase contribution to the anomalous velocity of massive and massless Dirac particles. We extend the Berry connection that depends only on the spatial components of the particle momentum to one that depends on the the space and time components in a covariant manner. We show that this covariant Berry connection captures the Thomas-precession part of the Bargmann-Michel-Telegdi spin evolution, and contrast it with the traditional (unitary, but not naturally covariant) Berry connection that describes spin-orbit coupling. We then consider how the covariant connection enters the classical relativistic dynamics of spinning particles due to Mathisson, Papapetrou and Dixon. We discuss the problems that arise with Lorentz covariance in the massless case, and trace them mathematically to a failure of the Wigner-translation part of the massless-particle little group to be an exact gauge symmetry in the presence of interactions, and physically to the fact that the measured position of a massless spinning particle is necessarily observer dependent.

Canonical quantization, path integral representations, and pseudoclassical description of massive Weyl neutrinos in external backgrounds

We study massive $1/2$-spin particles in various external backgrounds, keeping in mind applications to neutrino physics. We are mainly interested in massive Majorana (Weyl) fields. However, massive neutral Dirac particles are also considered. We formulate classical Lagrangian theory of the massive Weyl field in terms of Grassmann-odd two-component spinors. Then, we construct the Hamiltonian formulation of such a theory, which turns out to be a theory with second-class constraints. Using this formulation, we canonically quantize the massive free Weyl field. We derive propagators of the Weyl field and relate them to the propagator of a massive Dirac particle. We also study the massive Weyl particles propagating in the background mater. We find the path integral representation for the propagator of such a field, as well as the corresponding pseudoclassical particle action. The massless limit of the Weyl field interacting with the matter is considered and compared with results of other works. Finally, the path integral representation for the propagator of the neutral massive Dirac particle with an anomalous magnetic moment moving in the background matter and external electromagnetic field, as well as the corresponding pseudoclassical particle action, are constructed.

The zig-zag road to reality

In the standard model of particle physics, all fermions are fundamentally massless and only acquire their effective bare mass when the Higgs field condenses. Therefore, in a fundamental de Broglie–Bohm pilot-wave quantum field theory (valid before and after the Higgs condensation), position beables should be attributed to massless fermions. In our endeavour to build a pilot-wave theory of massless fermions, which would be relevant for the study of quantum non-equilibrium in the early universe, we are naturally led to Weyl spinors and to particle trajectories which give meaning to the ‘zig-zag’ picture of the electron discussed recently by Penrose. We show that a positive-energy massive Dirac electron of given helicity can be thought of as a superposition of positive and negative energy Weyl particles of the same helicity and that a single massive Dirac electron can in principle move luminally at all times. This is however not true for the many body situation required by quantum field theory and we conclude that a more natural theory arises from attributing beable status to the positions of massless Dirac particles instead of to Weyl particles.

Pseudoclassical theories of majorana, weyl, and majorana-weyl particles

A pseudoclassical theories of Majorana, Weyl and Majorana--Weyl particles in the space--time dimensions $D=2n$ are constructed. The canonical quantization of these theories is carried out and as a result we obtain the quantum mechanical description of neutral particle in $D=2n$ , Weyl particle in $D=2n$ and neutral Weyl particle in $D=4n+2$. In $D=2,4({\rm mod}8)$ dimensional space--time the description of the neutral particle coincides with the field theoretical description of the Majorana particle in the Foldy--Wouthuysen representation. In $D=8k+2$ dimensions the neutral Weyl particle coincides with the Majorana--Weyl particle in the Foldy--Wouthuysen representation.

Pseudoclassical Description of Higher Spins in 2 + 1 Dimensions

A pseudoclassical supersymmetric model to describe massive particles with higher spins (integer and half-integer) in 2 + 1 dimensions is proposed. The quantization of the model leads to the minimal (with only one polarization state) quantum theory. In particular, the Bargmann–Wigner type equations for higher spins arise in the course of canonical quantization. The cases of spin one-half and one are considered in detail. Here one gets Dirac particles and Chern–Simons particles respectively. A relation with the field theory is discussed. On the basis of the model proposed, and using dimensional reduction considerations, a model to describe Weyl particles with higher spins in 3 + 1 dimensions is constructed.

Pseudoclassical description of massive Dirac particles in odd dimensions

A pseudoclassical model is proposed to describe massive Dirac (spin one-half) particles in arbitrary odd dimensions. The quantization of the model reproduces the minimal quantum theory of spinning particles in such dimensions. A dimensional duality between the model proposed and the pseudoclassical description of Weyl particles in even dimensions is discussed.