Dirac Equation Research Articles

Numerical Methods for the Nonlinear Dirac Equation in the Massless Nonrelativistic Regime

Numerical Methods for the Nonlinear Dirac Equation in the Massless Nonrelativistic Regime

Twistronics of Kekulé Graphene: Honeycomb and Kagome Flat Bands.

Kekulé-O order in graphene, which has recently been realized experimentally, induces Dirac electron masses on the order of m∼100 meV. We show that twisted bilayer graphene in which one or both layershave Kekulé-O order exhibits nontrivial flat electronic bands on honeycomb and kagome lattices. When only one layer has Kekulé-O order, there is a parameter regime for which the lowest four bands at charge neutrality form an isolated two-orbital honeycomb lattice model with two flat bands. The bandwidths are minimal at a magic twist angle θ≈0.7° and Dirac mass m≈100 meV. When both layers have Kekulé-O order, there is a large parameter regime around θ≈1° and m≳100 meV in which the lowest three valence and conduction bands at charge neutrality each realize isolated kagome lattice models with one flat band, while the next three valence and conduction bands are flat bands on triangular lattices. These flat band systems may provide a new platform for strongly correlated phases ofmatter.

Stationary Schrödinger Equation and Darwin Term from Maximal Entropy Random Walk

We describe particles in a potential by a special diffusion process, the maximal entropy random walk (MERW) on a lattice. Since MERW originates in a variational problem, it shares the linear algebra of Hilbert spaces with quantum mechanics. The Born rule appears from measurements between equilibrium states in the past and the same equilibrium states in the future. Introducing potentials by the observation that time, in a gravitational field running in different heights with a different speed, MERW respects the rule that all trajectories of the same duration are counted with equal probability. In this way, MERW allows us to derive the Schrödinger equation for a particle in a potential and the Darwin term of the nonrelativistic expansion of the Dirac equation. Finally, we discuss why quantum mechanics cannot be simply a result of MERW, but, due to the many analogies, MERW may pave the way for further understanding.

On the Dirac-like equation in 7-component space-time and generalized Clifford-Dirac algebra

The generalized Dirac equation related to 7-component space-time with one time coordinate and six space coordinates has been introduced. Three 8-component Dirac equations have been derived from the same 256-dimensional Clifford-Dirac matrix algebra. Corresponding Clifford-Dirac algebra is considered in the Pauli-Dirac representation of $8 \times 8$ gamma matrices. It is proved that this matrix algebra over the field of real numbers has 256-dimensional basis and it is isomorphic to geometric $\textit{C}\ell^{\texttt{R}}$(1,7) algebra. The corresponding gamma matrix representation of 45-dimensional $\mathrm{SO}(1,9)$ algebra is derived and the way of its generalization to the $\mathrm{SO}(m,n)$ algebra is demonstrated. The Klein-Gordon equation in 7-component space-time is considered as well. The way of corresponding consideration of the Maxwell equations and of equations for an arbitrary spin is indicated.

Completing Dirac’s work. The Dirac electron is a 2D hologram

The QED-physical (QED-P) theory [J. H. Wilson, Phys. Essays 35, 5 (2022)] is actually the theory Dirac sought in 1962 in his attempt to predict the muon as an “extensible model of the electron.” Recently, Lerche attempted to produce a classical solution to Dirac’s equation for the radial motion of an extensible, basically classical, model of the electron. Both Dirac and Lerche proceeded in the wrong, classical direction in this effort. The QED-P center of charge (CoC) position operator is derived directly from the Dirac equation (DE) CoC velocity operator, cα with no ad hoc assumptions. QED-P was integrated with QED into a single theory, and that integration is proved by the highly accurate estimates of QED that are dependent on the DE velocity operator cα. Both QED and QED-P are based directly on the same Dirac Equation (DE) four current c(α,I) that QED couples with an external electromagnetic field. QED uses covariant perturbation theory to produce highly accurate results, except for the electron selfenergy, which is infinite. The DE velocity operator, cα, is the spatial part of the free electron four current, and has highly unusual properties compared to classical velocity vectors. QED could not produce highly accurate answers without the 4 × 4 complex matrix cα as the electron CoC “velocity” operator. QED-P simply integrates the same Dirac equation four current used so successfully in QED, and produces the discrete internal spatial and time coordinate operators (ISaTCOs) to give the electron field’s internal structure a very specific, but highly, nonclassical geometric description, with no ad hoc assumptions. QED and QED-P are complementary theories, and both are proven to be true by the accurate results of QED. The physical interpretation of QED-P is discussed in this paper as a two-dimensional, rapidly vibrating “point” charge, that is always located on a 2D CoC sphere in the electron rest frame, oscillating rapidly through eight eigenvalues with an ISaTCO period of ∼10−22 s. The fact that the CoC’s eight ISaTCO spatial eigenvectors are always are located on a 2D shell encompassing the electron’s 3D “space” inside the CoC’s 2D shell is a direct consequence of the DE, and nothing else. In this paper, it is shown that the “discrete” ISaTCOs produce a one dimensional, discrete quantum harmonic oscillator with its ISaTCOs always located on a 2D CoC shell. The CoC shell is an “2D hologram” emerging from a 3D volume inside the 2D CoC shell with a vibrational electronic clock” producing an internal phase that is propagated throughout space/time. The QED-P point electron charge rotates 720° to complete one internal electron discrete period. The electron’s spin and magnetic moment [J. H. Wilson, Phys. Essays 29, 402 (2016); ibid. 31(1), 59‐67 (2018); ibid. 34, 17 (2021)] are generated by the CoC ISaTCO in QED-P, and there is no need for “intrinsic” properties. The QED-P electron properties described above d are far different than the standard model’s very small point particle with intrinsic properties of spin and magnetic moment.

Relative Morse index and multiple solutions for a non-periodic Dirac equation with external fields

This paper is concerned with the stationary solutions of the Dirac equation −i∑k=13αk∂ku+aβu+ωu+V(x)u=Gu(x,u),where G is asymptotically quadratic and is not assumed to be C2. We present a new approach to construct an index theory for the associated linear Dirac equation and define the relative Morse index to measure the difference between the nonlinearity at the origin and at infinity. By building upon the idea of combining the index theory and a generalized linking theorem, we obtain existence and multiplicity of stationary solutions.

(Invited) Modeling of the Electric Double Layer (EDL) at Li/SEI/Electrolyte Interfaces

Liquid electrolytes, consisting of salts, solvents, and additives, must form a stable solid electrolyte interphase (SEI) to ensure the performance and durability of lithium(Li)-ion batteries. However, the electric double layer (EDL) structure near charged surfaces is still unsolved, despite its importance in dictating the species being reduced for SEI formation near a negative electrode. Recently, we have developed an interactive Molecular Dynamics -Density Functional Theory -data statistics (MD-DFT-data) model to investigate the reduction reactions of multicomponent electrolytes within the EDL. We will illustrate the effect of EDL on SEI formation in two essential electrolytes, the carbonate-based electrolyte for Li-ion batteries and the ether-based electrolyte for batteries with Li-metal anodes. Our results reveal that the role of fluoroethylene carbonate (FEC) additive differs drastically in the two electrolytes as an SEI modifier to form the beneficial F-containing SEI component (e.g., LiF). The competition among the cations, anions, and various species in the solvents with a charged surface at different temperatures can all jointly determine the EDL structure and therefore the SEI compositions. [1]While the classical Poisson–Boltzmann EDL model developed for fully solvated ions face new challenges in high-concentration liquid electrolytes (HCE), localized high-concentration liquid electrolytes (LHCE) as well as solid electrolytes (SE), new theoretical developments are required. We will introduce a new DEL model in SE and SEI by solving the DFT-informed Poisson–Fermi–Dirac equation and demonstrate how it can be used for interlayer thickness design. [2][1] Wu, Q.S, McDowell, M.T., & Qi, Y., Journal of the American Chemical Society 145 (4), 2473-2484 (2023)[2] Swift, M.W., Swift, J.W. & Qi, Y. Modeling the electrical double layer at solid-state electrochemical interfaces. Nat Comput Sci 1, 212–220 (2021)

On Clifford multivector-valued actions, generalized Dirac equation and quantization of branes

We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford multivector-valued actions, which was inspired by the work of Kanatchikov [Kanatchikov IV. De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory. Rept Math Phys. 1999;43:157–170; Kanatchikov IV. De Donder-Weyl Hamiltonian formulation and precanonical quantization of Vielbein gravity. J Phys Conf Ser. 2013;442:012041] on the De Donder-Weyl formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in D = 2 , 3 space–time dimensions are found. This formalism can be extended to curved space–time backgrounds. We conclude by proposing a wave-functional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder–Weyl Hamiltonian formulation of field theory.

Effect of magnetic flux on scattering in a graphene magnetic quantum dot

Our theoretical investigation focuses on the impact of a magnetic flux on two aspects related to the trapping time of electrons within a graphene magnetic quantum dot (GMQD), which results from an applied magnetic field. Specifically, we examine the electron diffusion and trapping time of the quasi-states within the GMQD. Employing the Dirac equation, we derive solutions for the energy spectrum in each region comprising the GMQD, serving as mathematical tools to assess diffusion efficiency and estimate trapping time. Our findings reveal that an increase in magnetic flux enhances scattering efficiency, even in the absence of magnetic field. Through a scattering analysis in real space, we observe a significant improvement in the probability density within the GMQD. We demonstrate the feasibility of extending the trapping time of an electron within a GMQD by adjusting the magnetic flux.

Topological light guiding and trapping via shifted photonic crystal interfaces

The exploration of topological states in photonic crystals has inspired a number of intriguing discoveries, which in turn provide new mechanisms for the manipulation of light in unprecedented ways. Here, we show that light can be effectively guided and trapped at the shifted photonic crystal interfaces (SPCIs). The projected bandgap of SPCIs, which depends on the shift parameter, is characterized by a Dirac mass. Interestingly, the SPCI with zero Dirac mass is a glide-symmetric waveguide featured with gapless interface states that exhibit excellent transmission performance even in the presence of disorders and sharp corners. Moreover, placing two SPCIs with opposite Dirac mass together results in a photonic bound state due to the Jackiw–Rebbi theory. Our work provides an alternative way toward the design of ultracompact photonic devices such as robust waveguides and cavities as well as the cavity-waveguide coupled systems that can serve as high-performance building blocks of miniature integrated topological photonic circuits.

High-order finite element method for atomic structure calculations

We introduce featom, an open source code that implements a high-order finite element solver for the radial Schrödinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the κ=±1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10−8 Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schrödinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (Z) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines. Program summaryProgram Title: featomCPC Library link to program files:https://doi.org/10.17632/962fzmm7f7.1Licensing provisions: MITProgramming language: Fortran with command-line interfacesNature of problem: Solution of the Schrödinger, Dirac, and Kohn–Sham equations of density functional theory for isolated atoms.Solution method: A high-order finite element method is used to assemble a generalized eigenproblem that is then solved for eigenvalues and orbitals. The Poisson equation is solved using the finite element method also. Self-consistent field equations are solved using Pulay mixing.Additional comments including restrictions and unusual features: Our solution methodology for the Dirac equation does not suffer from spurious states. We maintain high accuracy even for κ=±1 states. Unlike other methods, our approach is not limited to Coulombic or self-consistent potentials and can handle non-uniform meshes, including exponential meshes.Restrictions: Spherical symmetry.

Real-Time Observation of Instantaneous ac Stark Shift of a Vacuum Using a Zeptosecond Laser Pulse

Based on the numerical solution of the time-dependent Dirac equation, we propose a method to observe in real time the ac Stark shift of a vacuum driven by an ultra-intense laser field. By overlapping the ultra-intense pump pulse with another zeptosecond probe pulse whose photon energy is smaller than 2mc 2, electron–positron pair creation can be controlled by tuning the time delay between the pump and probe pulses. Since the pair creation rate depends sensitively on the instantaneous vacuum potential, one can reconstruct the ac Stark shift of the vacuum potential according to the time-delay-dependent pair creation rate.

Inevitable monokineticity of strongly singular alignment

We prove that certain types of measure-valued mappings are monokinetic i.e. the distribution of velocity is concentrated in a Dirac mass. These include weak measure-valued solutions to the strongly singular Cucker–Smale model with singularity of order greater or equal to the dimension of the ambient space. Consequently, we are able to answer a couple of open questions related to the singular Cucker–Smale model. First, we prove that weak measure-valued solutions to the strongly singular Cucker–Smale kinetic equation are monokinetic, under very mild assumptions that they are uniformly compactly supported and weakly continuous in time. This can be interpreted as a rigorous derivation of the macroscopic fractional Euler-alignment system from the kinetic Cucker–Smale equation without the need to perform any hydrodynamical limit. This suggests the superior suitability of the macroscopic framework to describe large-crowd limits of strongly singular Cucker–Smale dynamics. Second, we perform a direct micro- to macroscopic mean-field limit from the Cucker–Smale particle system to the fractional Euler-alignment model. This leads to the final result—the existence of weak solutions to the fractional Euler-alignment system with almost arbitrary initial data in 1D, including the possibility of a vacuum. Existence can be extended to 2D under the a priori assumption that the density of the mean-field limit has no atoms.

Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime

Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime

Exact Foldy–Wouthuysen transformation for a Dirac equation describing the interaction of spin-1/2 relativistic particles with an external electromagnetic field

The Thomas–Bargmann–Michel–Telegdi (TBMT) equation is derived using the Exact Foldy–Wouthuysen transformation. Extra new terms were found, and we discuss their possible physical applications. The main point of this work is to detail the procedure to get the general result. We explicitly present the choice of parametrization we used on the initial Hamiltonian and the motivations to take it. We emphasize that the final equations can depend on this choice, and it is possible to prevent the manipulations of the quadratic Hamiltonian that become extremely cumbersome. More importantly, it is done in such a way that the transformed equations allow the direct separation into mass, kinetic, and interaction correction terms to the original TBMT equation.

The interaction of electron with the infinitesimally short electromagnetic pulse

After the classical approach to acceleration of a charged particle by the δ-form impulsive force, we consider the corresponding quantum theory based on the Volkov solution of the Dirac equation. We determine the modified Compton formula for frequency of photons generated by the scattering of the δ-form laser pulse on the electron in a rest. The article follows the physical ideas involved in the author text with the title Electron in an Ultrashort Laser Pulse (Pardy, 2003).

Compton scattering in dielectric medium

We determine the Compton effect from the Volkov solution of the Dirac equation for a process in medium with the index of refraction n. Volkov solution involves the mass shift, or, the mass renormalization of an electron. We determine the modified Compton formula for the considered physical situation. The index of refraction causes that the wave lengths of the scattered photons are shorter for some angles than the wave lengths of the original photons. This is anomalous Compton effect.

New spin and p-spin symmetries of the deformed Dirac equation for the IMRQHPs including the improved YLT interaction in three-dimensional extended relativistic quantum mechanics symmetries

In this paper, the modified Dirac equation (MDE) has been solved approximately for improved Manning–Rosen plus quasi-Hellman potentials (IMRQHPs), including the improved Yukawa tensor interaction under new spin symmetry and pseudospin symmetry for arbitrary spin–orbit number [Formula: see text], an appropriate approximation employed on the centrifugal terms in the symmetries of three-dimensional extended relativistic quantum mechanics (3D-ERQM). This potential is a superposition of Manning–Rosen plus quasi-Hellman potentials and some other exponential terms. Through the application of both new approximations to deal with the centrifugal term, Bopp’s shift method and the independent-time perturbation theory method, the recent modified approximate energy corrections under IMRQHPs are obtained for nuclei [Formula: see text]O and [Formula: see text]F and (O2, I2, N2, H2, LiH, CO and NO) diatomic molecules in 3D-ERQM and three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries. The recent values that we get appear sensitive to [Formula: see text], which are known as the discrete atomic quantum numbers, the mixed potential depths ([Formula: see text], [Formula: see text]) and [Formula: see text], the range of the potential [Formula: see text] and non-commutativity parameters [Formula: see text]. In both 3D-ERQM and 3D-ENRQM symmetries, we show that the recent corrections obtained on the new bound states under IMRQHPs are infinitesimal to the principal part of energy in the ordinary cases of 3D-RQM and 3D-NRQM symmetries. In the new symmetries of 3D-ERQM symmetries, it is not possible to get the exact analytical solutions [Formula: see text] and [Formula: see text], only the approximate solutions are possible. In addition, we discussed in detail the non-relativistic study of the studied mixed potentials. Four special cases, i.e. we investigated some special cases in the context of modified Schrödinger theories. In the framework of known relativistic quantum mechanics, we have clearly demonstrated that the modified Schrödinger equation can be represented as the Dirac equation under the influence of IMRQHPs.

Light antiproton one-electron quasi-molecular ions within the relativistic A-DKB method.

In the present work, two quasi-molecular compounds each involving one antiproton and one electron (p̄), He+-p̄ and H-p̄, are investigated. Using completely relativistic calculations within the finite-basis method adapted to systems with axial symmetry, the adiabatic potential curves are constructed by numerically solving the two-center Dirac equation. The binding energies of electron are obtained as a function of the inter-nuclear distance and compared with the corresponding nonrelativistic values and relativistic leading-order corrections calculated in the framework of other approaches. A semantic analysis of antiproton quasi-molecular ions with compounds containing a proton (p) instead of an antiproton is given. The advantages of the A-DKB method are demonstrated.

A CHARACTERIZATION OF WEAKLY REGULAR LINEAR FUNCTIONALS

A linear functional is said to be weakly-regular if it is not a finite sum of Dirac masses and their derivatives. In this paper, we consider the first-order linear differential equations (Eu)'+Fu=0 where u is a non-zero linear functional and (E,F) is a pair of polynomials, with E monic. The aim of this work is to give weak-regularity conditions on u. Under certain admissibility conditions of the pair (E, F), the weak-regularity of u leads to its regularity. Some examples are analyzed.