Massive Dirac Particles Research Articles

On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions

A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions Ψ(xph,xe1,xe2) where xph,xe1,xe2 are generic spacetime events of the photon and two electrons, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds {xph=xe1} and {xph=xe2} compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times.

Massive Dirac particles based on gapped graphene with Rosen–Morse potential in a uniform magnetic field

Abstract In this paper, we explore the gapped graphene structure in the two-dimensional plane in the presence of the Rosen-Morse potential and the external uniform magnetic field. In order to describe the corresponding structure, we consider the propagation of electrons in graphene as relativistic fermion quasi-particles, and analyze it by the wave functions of two-component spinors with pseudo-spin symmetry using the Dirac equation. Next, to solve and analyze the Dirac equation, we obtain the eigenvalues and eigenvectors using the Legendre differential equation. After that, we obtain the bounded states of energy depending on the coefficients of Rosen-Morse and magnetic potentials in terms of quantum numbers of principal n and spin-orbit k. Then, the values of the energy spectrum for the ground state and the first excited state are calculated, and the wave functions and the corresponding probabilities are plotted in terms of coordinates r. In what follows, we explore the band structure of gapped graphene by the modified dispersion relation and write it in terms of the two-dimensional wave vectors K x and K y . Finally, the energy bands are plotted in terms of the wave vectors K x and K y with and without the magnetic term.

Massive and massless two-dimensional Dirac particles in electric quantum dots

In this work we investigate the confining properties of charged particles of a Dirac material in the plane subject to an electrostatic potential well, that is, in an electric quantum dot. Our study focuses on the effect of mass and angular momenta on such confining properties. To have a global picture of confinement, both bound and resonance states are considered. The resonances will be examined by means of the Wigner time delay of the scattering states, as well as through the complex eigenvalues of outgoing states in order to show that they are physically meaningful. By tuning the potential intensity of the well, electron captures and atomic collapses are observed for critical values. In these processes, the bound states of the discrete spectrum become resonances of the continuous spectrum or vice versa. For massive charges, the atomic collapse phenomenon keeps the number of bound levels in the quantum dot below a maximum value. In the massless case, the bound states have zero energy and occur only for some discrete values of the potential depth, as is known. We also show that although the intensity of the resonances for massive particles is not significantly influenced by angular momenta, on the contrary, for massless particles they are quite sensitive to angular momenta, as it is the case of graphene.

Zero-range potentials for Dirac particles: Bound-state problems

A model in which a massive Dirac particle in R3 is bound by N⩾1 spatially distributed zero-range potentials is presented. Interactions between the particle and the potentials are modeled by subjecting a particle’s bispinor wave function to certain limiting conditions at the potential centers. Each of these conditions is parametrized by a 2 × 2 Hermitian matrix (or, equivalently, a real scalar and a real vector) and mixes the upper and the lower components of the wave function. The problem of determining particle’s bound-state eigenenergies is reduced to the problem of finding real zeros of a determinant of a certain 2N×2N matrix. As the lower component of the particle’s wave function is inverse-square singular at each of the potential centers, the wave function itself is not square-integrable. Nevertheless, one can define a scalar pseudo-product with the property that wave functions belonging to different eigenenergies are orthogonal with respect to it. The wave functions may then be normalized so that their self-pseudo-products are plus one, minus one or zero. An auxiliary set of Sturmian functions is constructed and used to derive an explicit representation of particle’s matrix Green’s function. For illustration purposes, two particular systems are studied in detail: (1) a particle bound in a field of a single zero-range potential, (2) a particle bound in a field of two identical zero-range potentials.

Динамика спинирующих частиц в вихревом гравитационном поле Рассматриваются эффекты взаимодействия спи

The effects of interaction of spinning particles having intrinsic angular momentum (spin) and describing by the Dirac equation with a vortex gravitational field are considered. It is shown that the interaction of massive Dirac particles with a vortex gravitational field leads to a precession of their spin around the axis of rotation of the vortex gravitational field. In the case of massless particles, their spin is set along the axis of rotation and remains constant, thus the spin of Dirac massless particles indicates the direction of the rotation axis of the vortex gravitational field.

Twisting D(2,1; α) Superspace

AbstractWe develop a three‐dimensionaltheory of rigid supersymmetry describing the dynamics of a set of hypermultipletson a curved AdS3worldvolume background, whose supersymmetry is captured by the supergroup. To unveil some remarkable features of this model, we perform two twists, involving the SLfactors of the theory. After the first twist, our spacetime Lagrangian exhibits a Chern‐Simons term associated with an odd one‐form field, together with a fermionic “gauge‐fixing”, in the spirit of the Rozansky‐Witten model. The second twist allows to interpret thesetup as a framework capable of describing massive Dirac particles. In particular, this yields a generalisation of the Alvarez‐Valenzuela‐Zanelli model of “unconventional supersymmetry”. We comment on specific values of the combination, which in our model is related to a sort of gauging in the absence of dynamical gauge fields.

Bosonization, mass generation, and the pseudo-Chern-Simons action

We discuss several aspects of a generalization of the Chern-Simons action containing the pseudo-differential operator$\sqrt{-\Box}$, which we shall call pseudo Chern-Simons (PCS). Firstly, we derive the PCS from the bosonization of free massive Dirac particles in (2+1)D in the limit when $m^2\ll p^2$, where $m$ is the fermion mass and $p$ is its momentum. In this regime, the whole bosonized action also has a modified Maxwell term, involving the same pseudo-differential operator. Furthermore, the large-mass $m^2\gg p^2$ regime is also considered. We also investigate the main effects of the PCS term into the Pseudo quantum electrodynamics (PQED), which describes the electromagnetic interactions between charged particles in (2+1)D. We show that the massless gauge field of PQED becomes massive in the presence of a PCS term, without the need of a Higgs mechanism. In the nonrelativistic limit, we show that the static potential has a repulsive term (given by the Coulomb potential) and an attractive part (given by a sum of special functions), whose competition generates bound states of particles with the same charge. Having in mind two-dimensional materials, we also conclude that the presence of a PCS term does not affect the renormalization either of the Fermi velocity and of the band gap in a Dirac-like material.

Chiral oscillations in the non-relativistic regime

Massive Dirac particles are a superposition of left and right chiral components. Since chirality is not a conserved quantity, the free Dirac Hamiltonian evolution induces chiral quantum oscillations, a phenomenon related to the Zitterbewegung, the trembling motion of free propagating particles. While not observable for particles in relativistic dynamical regimes, chiral oscillations become relevant when the particle’s rest energy is comparable to its momentum. In this paper, we quantify the effect of chiral oscillations on the non-relativistic evolution of a particle state described as a Dirac bispinor and specialize our results to describe the interplay between chiral and flavor oscillations of non-relativistic neutrinos: we compute the time-averaged survival probability and observe an energy-dependent depletion of the quantity when compared to the standard oscillation formula. In the non-relativistic regime, this depletion due to chiral oscillations can be as large as 40%. Finally, we discuss the relevance of chiral oscillations in upcoming experiments which will probe the cosmic neutrino background.

Energy spectrum of massive Dirac particles in gapped graphene with Morse potential

In this paper, we study the massive Dirac equation with the presence of the Morse potential in polar coordinate. The Dirac Hamiltonian is written as two second-order differential equations in terms of two spinor wavefunctions. Since the motion of electrons in graphene is propagated like relativistic fermionic quasi-particles, then one is considered only with pseudospin symmetry for aligned spin and unaligned spin by arbitrary k. Next, we use the confluent Heun’s function for calculating the wavefunctions and the eigenvalues. Then, the corresponding energy spectrum obtains in terms of N and k. Afterward, we plot the graphs of the energy spectrum and the wavefunctions in terms of k and r, respectively. Moreover, we investigate the graphene band structure by a linear dispersion relation which creates an energy gap in the Dirac points called gapped graphene. Finally, we plot the graph of the valence and conduction bands in terms of wavevectors.

Sauter-Schwinger Effect in a Bardeen-Cooper-Schrieffer Superconductor.

Since the 1960s a deep and surprising connection has followed the development of superconductivity and quantum field theory. The Anderson-Higgs mechanism and the similarities between the Dirac and Bogoliubov-de Gennes equations are the most intriguing examples. In this last analogy, the massive Dirac particle is identified with a quasiparticle excitation and the fermion mass energy with the superconducting gap energy. Here we follow further this parallelism and show that it predicts an outstanding phenomenon: the superconducting Sauter-Schwinger effect. As in the quantum electrodynamics Schwinger effect, where an electron-positron couple is created from the vacuum by an intense electric field, we show that an electrostatic field can generate two coherent excitations from the superconducting ground-state condensate. Differently from the dissipative thermal excitation, these form a new macroscopically coherent and dissipationless state. We discuss how the superconducting state is weakened by the creation of this kind of excitations. In addition to shedding a different light and suggesting a method for the experimental verification of the Sauter-Schwinger effect, our results pave the way to the understanding and exploitation of the interaction between superconductors and electric fields.

Quantum gravity correction to the thermodynamic quantities of the charged dRGT black hole

In this study, in the framework of the Hamilton approach, the quantum gravity effect on Hawking temperature, on the specific heat capacity, and on the stability conditions of the charged de Rham, Gabadadze, and Tolley (dRGT) black hole are investigated by using the tunneling processes of both charged massive scalar and Dirac particles. It is shown that quantum corrected Hawking temperature depends not only on the black hole properties but also on the properties of the tunnelling particles. It is also observed that quantum corrected Hawking temperature is lower than that of the standard Hawking temperature. Also, the specific heat capacity and the local stability conditions of the black hole are discussed in the context of the tunneling processes of both charged massive scalar and Dirac particles in the presence of quantum gravity effect. It is observed that the black hole might undergo second-type phase transitions to become stable during the tunneling processes of both scalar and Dirac particles. However, in the absence of the quantum gravity effect, the black hole might undergo the both first-type and second-type phase transitions to become stable.

Landau level spectroscopy of Bi2Te3

Here we report on Landau level spectroscopy in magnetic fields up to 34 T performed on a thin film of topological insulator Bi$_2$Te$_3$ epitaxially grown on a BaF$_2$ substrate. The observed response is consistent with the picture of a direct-gap semiconductor in which charge carriers closely resemble massive Dirac particles. The fundamental band gap reaches $E_g=(175\pm 5)$~meV at low temperatures and it is not located on the trigonal axis, thus displaying either six or twelvefold valley degeneracy. Notably, our magneto-optical data do not indicate any band inversion. This suggests that the fundamental band gap is relatively distant from the $\Gamma$ point where profound inversion exists andgives rise to relativistic-like surface states of Bi$_2$Te$_3$.

GUP effect on thermodynamical properties of the noncommutative rotating BTZ black hole

In this paper, we investigated the quantum gravity effects on the thermal properties of the [Formula: see text]-dimensional noncommutative rotating Banados–Teitelboim–Zanelli (NCR-BTZ) black hole in the context of quantum tunneling of relativistic particles. These include Hawking temperature, the thermally local and global stability conditions, and the phase transitions. For this purpose, in the framework of the generalized uncertainty principle (GUP), we used the Hamilton–Jacobi approach to calculate the tunneling probability for a massive scalar, Dirac, and vector boson particles from the [Formula: see text]-dimensional NCR-BTZ black hole. We found that the modified Hawking temperature of the black hole depends on the black hole properties, on the tunneling particle properties, on the noncommutative parameter, and on the GUP parameter. Using the modified Hawking temperature, we calculated the modified heat capacity, and then we discussed the local thermodynamic stability conditions for the black hole. The black hole may undergo a first-type phase transition to become stable under the scalar particle tunneling whereas, it might undergoes both the first and the second-type phase transitions under the both Dirac and vector boson particles tunneling process. Furthermore, we calculated the Gibbs free energy of the black hole, and we investigated the global stability conditions. We observed that Hawking–Page phase transition may occur in the presence of the quantum gravity effect under the tunneling process of scalar, Dirac, and vector boson particles. In the context of quantum gravity effect, we also derived the modified equation of state to investigate the critical behavior of the commutative rotating BTZ black hole. Finally, we shown that Van der Waals-like phase transition may occur in the context of tunneling process of both Dirac and vector boson particle, whereas it may not occur for the tunneling of scalar particle.

Simulating Dirac Hamiltonian in curved space-time by split-step quantum walk

Dirac particle represents a fundamental constituent of our nature. Simulation of Dirac particle dynamics by a controllable quantum system using quantum walks will allow us to investigate the non-classical nature of dynamics in its discrete form. In this work, starting from a modified version of one-spatial dimensional general inhomogeneous split-step discrete quantum walk we derive an effective Hamiltonian which mimics a single massive Dirac particle dynamics in curved (1 + 1) space-time dimension coupled to U(1) gauge potential—which is a forward step towards the simulation of the unification of electromagnetic and gravitational forces in lower dimension and at the single particle level. Implementation of this simulation scheme in simple qubit-system has been demonstrated. We show that the same Hamiltonian can represent (2 + 1) space-time dimensional Dirac particle dynamics when one of the spatial momenta remains fixed. We also discuss how we can include U(N) gauge potential in our scheme, in order to capture other fundamental force effects on the Dirac particle. The emergence of curvature in the two-particle split-step quantum walk has also been investigated while the particles are interacting through their entangled coin operation.

Momentum-induced Zitterbewegung of massless relativistic quasiparticles emreged in an optical lattice

We have investigated dynamic characteristics of massless Dirac and hybrid quasiparticles in an optical lattice. Zitterbewegung (ZB) of the quasiparticles has been realized and well manipulated. The results reveal that massless Dirac quasiparticles can also generate ZB oscillation, only with an initial velocity being introduced to the quasiparticles as the price to pay. Besides, we noticed an essential distinction between this momentum-induced ZB and an ordinary ZB generated by massive Dirac particles, i.e., instead of all directions, ZB induced by momentum in a 2D system will emerge exclusively in the direction orthogonal to where the momentum is introduced. On the other hand, momentum-induced ZB will be attenuated more rapidly since the initial momentum causes greater expansion rate of wave packets. Therefore, an optimal range of parameters is necessary in detecting such ZB. We have confidence that momentum-induced ZB for massless Dirac quasiparticles will be observed in the near future since the frequency, amplitude and lifespan of ZB can now be controlled in a detectable range in the cold atom system.

Quantum gravity effect on the Hawking radiation of charged rotating BTZ black hole

In this study, the quantum gravity effect on the tunnelling radiation of charged massive spin-0 scalar particle from 2+1 dimensional charged rotating Banados-Teitelboim-Zanelli (BTZ) black hole is looked into by using the Hamilton-Jacobi approach. For this, we calculate the modified Hawking temperature of the black hole by using the modified Klein-Gordon equation based on the Generalized Uncertainty Principle (GUP), and we noticed that the modified Hawking temperature of the black hole depends not only on the black hole properties, but also on the angular momentum, energy, charge and mass of the tunnelling scalar particle. Using the modified Hawking temperature, we discussed the stability of the black hole in the context of the modified heat capacity, and observed that it might undergo both first and second-type phase transitions in the presence of the quantum gravity effect, but just a first-type transition in the absence of the quantum gravity effect. Furthermore, we investigated the modified Hawking temperature of the black hole by using the tunnelling processes of the charged massive Dirac and vector boson particles. We observed that scalar, Dirac and vector particles are tunnelled from the black hole completely differently from each other in the presence of the quantum gravity effect.

Electron scattering in two-dimensional semiconductors: Contrasting massive Dirac and Schrödinger behavior

Electronic transport through a material depends on the response to local perturbations induced by defects or impurities in the material. The scattering processes can be described in terms of phase shifts and corresponding cross sections. The multiorbital nature of the spinor states in transition metal dichalcogenides would naturally suggest the consideration of a massive Dirac equation to describe the problem, while the parabolic dispersion of its conduction and valence bands would invite a simpler Schr\"odinger equation description. Here, we contrast the scattering of massive Dirac particles and Schr\"odinger electrons, in order to assess different asymptotic regimes (low and high Fermi energy) for each one of the electronic models and describe their regime of validity or transition. At low energies, where the dispersion is approximately parabolic, the scattering processes are dominated by low angular momentum channels, which results in nearly isotropic scattering amplitudes. On the other hand, the differential cross section at high Fermi energies exhibits clear signatures of the linear band dispersion, as the partial phase shifts approach a non-zero value. We analyze the electronic dynamics by presenting differential cross sections for both attractive and repulsive scattering centers. The dissimilar behavior between Dirac and Schr\"odinger carriers points to the limits and conditions over which different descriptions are required for the reliable treatment of scattering processes in these materials.

Nonexistence of time-periodic solutions of the Dirac equation in non-extreme Kerr-Newman-AdS spacetime

In non-extreme Kerr-Newman-AdS spacetime, we prove that there is no nontrivial Dirac particle which is $L^p$ for $0 |Q|+q~\kappa~$, outside and away from the event horizon. By taking $q=\frac{1}{2}$, we show that there is no normalizable massive Dirac particle with mass greater than $|Q|+\frac{\kappa}{2}$ outside and away from the event horizon in non-extreme Kerr-Newman-AdS spacetime, and they must either disappear into the black hole or escape to infinity, and this recovers the same resultof Belgiorno and Cacciatori in the case of $Q=0$ obtained by using spectral methods. Furthermore, we prove that any Dirac particle with eigenvalue $|\lambda|<\frac{\kappa}{2}$ must be $L^2$ outside and away fromthe event horizon.

The massive Dirac equation in Kerr geometry: separability in Eddington–Finkelstein-type coordinates and asymptotics

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr geometry is described in the Newman-Penrose formalism by a regular Carter tetrad and the Dirac spinors and matrices are defined in a chiral Newman-Penrose dyad representation. Applying Chandrasekhar's mode ansatz, the Dirac equation is separated into radial and angular systems of ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are derived, and the decay of the associated errors is analyzed. Moreover, specific aspects of the angular eigenfunctions and eigenvalues are discussed. Finally, as an application, the scattering of massive Dirac particles by the gravitational field of a rotating Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating coordinates and for the construction of an integral spectral representation of the Dirac propagator that yields the dynamics of Dirac particles outside, across, and inside the event horizon, up to the Cauchy horizon.

Dynamical polarizability, screening and plasmons in one, two and three dimensional massive Dirac systems

We study the density–density response function of a collection of charged massive Dirac particles and present analytical expressions for the dynamical polarization function in one, two and three dimensions. The polarization function is then used to find the dispersion of the plasmon modes, and electrostatic screening of Coulomb interactions within the random phase approximation. We find that for massive Dirac systems, the oscillating screened potential (or density) decays as r−2 and r−3 in two and three dimensions respectively, and as r−1 for one dimensional non-interacting systems. However for massless Dirac systems there is no electrostatic screening or Friedel oscillation in one dimension, and the oscillating screened potential decays as r−3 and r−4, in two and three dimensions respectively. Our analytical results for the polarization function will be useful for exploring the physics of massive and massless Dirac electrons in different experimental systems with varying dimensionality.