Massive Dirac Equation Research Articles

Symmetries of modified Dirac operators in supergravity flux backgrounds

Modifications of Dirac operators in supergravity flux backgrounds are considered. Modified spin curvature operators and squares of modified Dirac operators corresponding to Schrödinger-Lichnerowicz-like formulas are obtained for different types of flux modifications. Symmetry operators of modified massless and massive Dirac equations are found in terms of modified Killing-Yano and modified conformal Killing-Yano forms. Extra constraints for symmetry operators in terms of different types of fluxes and modified Killing-Yano forms are determined.

Massive Dirac equation in static Bumblebee black hole space-times, detailed derivations and novel exact solutions

In this work, we construct and investigate the relativistic spin-12 fermionic fields quantum dynamics in static spherically symmetric Bumblebee black hole background. The derivation of the Dirac equation in a general static spherically symmetric black hole space-time is carried out in detail via tetrad formalism. With the help of total angular momentum operator, the angular equation can be separated from the radial part where the solution is given in terms of the spinor harmonics. The radial Dirac equation has a well-known problem due to the presence of the square root terms appearing simultaneously with the rest mass that prevents us to find exact solutions. In this work, we present exact solutions of light mass fermion's wave function and energy levels bound in the static Bumblebee black hole. We discover the exact solutions of the massive Dirac's radial equation in terms of the Confluent Heun functions. Moreover, thanks to the well-known polynomial condition of the Confluent Heun functions, we also derive the energy quantization.

The Dirac equation across the horizons of the 5D Myers–Perry geometry: separation of variables, radial asymptotic behaviour and Hamiltonian formalism

We analytically extend the 5D Myers–Perry metric through the event and Cauchy horizons by defining Eddington–Finkelstein-type coordinates. Then, we use the orthonormal frame formalism to formulate and perform separation of variables on the massive Dirac equation, and analyse the asymptotic behaviour at the horizons and at infinity of the solutions to the radial ordinary differential equation (ODE) thus obtained. Using the essential self-adjointness result of Finster–Röken and Stone’s formula, we obtain an integral spectral representation of the Dirac propagator for spinors with low masses and suitably bounded frequency spectra in terms of resolvents of the Dirac Hamiltonian, which can in turn be expressed in terms of Green’s functions of the radial ODE.

Solutions of Noncommutative Two-Dimensional Position–Dependent Mass Dirac Equation in the Presence of Rashba Spin-Orbit Interaction by Using the Nikiforov–Uvarov Method

Solutions of Noncommutative Two-Dimensional Position–Dependent Mass Dirac Equation in the Presence of Rashba Spin-Orbit Interaction by Using the Nikiforov–Uvarov Method

Traveling edge states in massive Dirac equations along slowly varying edges

Abstract Topologically protected wave motion has attracted considerable research interest due to its chirality and potential applications in many applied fields. We construct quasi-traveling wave solutions to the 2D Dirac equation with a domain wall mass in this work. It is known that the system admits exact and explicit traveling wave solutions, which are termed edge states if the interface is a straight line. By modifying such explicit solutions, we construct quasi-traveling-wave solutions if the interface is nearly straight. The approximate solutions in two scenarios are given. One is the circular edge with a large radius, and the second is a straight line edge with the slowly varying along the perpendicular direction. We show the quasi-traveling wave solutions are valid in a long lifespan by energy estimates. Numerical simulations are provided to support our analysis both qualitatively and quantitatively.

Second order symmetry operators for the massive Dirac equation

Employing the covariant language of two-spinors, we find what conditions a curved four-dimensional Lorentzian spacetime must satisfy for existence of a second order symmetry operator for the massive Dirac equation. The conditions are formulated as existence of a set of Killing spinors satisfying a set of covariant linear differential equations. Using these Killing spinors, we then state the most general form of such an operator. Partial results for the zeroth and first order are presented and interpreted as well. Computer algebra tools from the Mathematica package suite xAct were used for the calculations.

Local Dirac energy decay in the 5D Myers-Perry geometry using an integral spectral representation for the Dirac propagator

We consider the massive Dirac equation in the exterior region of the five-dimensional Myers-Perry black hole. Using the resulting ordinary differential equations (ODEs) obtained from the separation of variables of the Dirac equation, we construct an integral spectral representation for the solution of the Cauchy problem with compactly supported smooth initial data. We then prove that the probability of presence of a Dirac particle to be in any compact region of space decays to zero as , in analogy with the case of the Dirac operator in the Kerr–Newman geometry (Finster et al 2003 Adv. Theor. Math. Phys. 7 25–52).

New Approximate Solutions to a Spatially-Dependent Mass Dirac Equation for Modified Hylleraas Plus Eckart Potential with Improved Yukawa Potential as a Tensor in the DQM Framework

New Approximate Solutions to a Spatially-Dependent Mass Dirac Equation for Modified Hylleraas Plus Eckart Potential with Improved Yukawa Potential as a Tensor in the DQM Framework

So(2, 1) algebra, local Fermi velocity, and position-dependent mass Dirac equation

We investigate the (1 + 1)-dimensional position-dependent mass Dirac equation within the confines of so(2, 1) potential algebra by utilizing the character of a spatial varying Fermi velocity. We examine the combined effects of the two when the Dirac equation is equipped with an external pseudoscalar potential. Solutions of the three cases induced by so(2, 1) are explored by profitably making use of a point canonical transformation.

Hadamard property of the in and out states for Dirac fields on asymptotically static spacetimes

We consider massive Dirac equations on asymptotically static spacetimes with a Cauchy surface of bounded geometry. We prove that the associated quantized Dirac field admits in and out states, which are asymptotic vacuum states when some time coordinate tends to $\mp\infty$. We also show that the in /out states are Hadamard states.(new version with some typos corrected).

Thermodynamics of Graphene beyond the Linear Approximation

Abstract In this paper, we study the thermodynamic correction due to the deformed Dirac equation in a homogeneous magnetic field. We study the deformation of both the massive and massless Dirac equation. The deformation will occur from generalized uncertainty principle, which occurs due to a minimal length. As the massless Dirac equation in (2+1) gives the behaviour of electrons in systems like sheet of graphene, this deformed Dirac equation will be used to analyze the thermal properties of graphene in a magnetic eld. Due to the existence of minimal length the degeneracy of systems gets modied, as there exist states which vanish when minimal length considerations are ignored. Thus, we will also analyze the implication of this additional degeneracy due to these states to the thermodynamics of the systems. In the case of graphene the minimal length or zero point length is the inter-atomic distance, which occurs by going beyond the linear regime of graphene.

Scattering theory for Dirac fields inside a Reissner–Nordström-type black hole

We show asymptotic completeness for the massive charged Dirac equation between the black hole and Cauchy horizons of a sub-extremal [(anti-)de Sitter]–Reissner–Nordström black hole.

Small data scattering of Dirac equations with Yukawa type potentials in $L_x^2(\mathbb R^2)$

We revisit the Cauchy problem of nonlinear massive Dirac equation with Yukawa type potentials $\mathcal F^{-1}\left[(b^2 + |\xi|^2)^{-1}\right]$ in 2 dimensions. The authors of [10, 4] obtained small data scattering and large data global well-posedness in $H^s$ for $s > 0$, respectively. In this paper, we show that the small data scattering occurs in $L_x^2(\mathbb R^2)$. This can be done by combining bilinear estimates and modulation estimates of [12,10].

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.

Renormalization of the band gap in 2D materials through the competition between electromagnetic and four-fermion interactions in largeNexpansion

Recently the renormalization of the band gap $m$, in both WSe$_2$ and MoS$_2$, has been experimentally measured as a function of the carrier concentration $n$. The main result establishes a decreasing of hundreds of meV, in comparison with the bare band gap, as the carrier concentration increases. These materials are known as transition metal dichalcogenides and their low-energy excitations are, approximately, described by the massive Dirac equation. Using Pseudo Quantum Electrodynamics (PQED) to describe the electromagnetic interaction between these quasiparticles and from renormalization group analysis, we obtain that the renormalized mass describes the band gap renormalization with a function given by $m(n)/m_0=(n/n_0)^{C_\lambda/2}$, where $m_0=m(n_0)$ and $C_\lambda$ is a function of the coupling constant $\lambda$. We compare our theoretical results with the experimental findings for WSe$_2$ and MoS$_2$, and we conclude that our approach is in agreement with these experimental results for reasonable values of $\lambda$. In addition we introduced a Gross-Neveu (GN) interaction which could simulate an disorder/impurity-like microscopic interaction. In this case, we show that there exists a critical coupling constant, namely, $\lambda_c \approx 0,66$ in which the beta function of the mass vanishes, providing a stable fixed point in the ultraviolet limit. For $\lambda>\lambda_c$, the renormalized mass decreases while for $\lambda<\lambda_c$ it increases with the carrier concentration.

Coupling of a biquaternionic Dirac field to a bosonic field

We extend the biquaternionic Dirac equation to include interactions with a background bosonic Geld. The obtained biquaternionic Dirac equation yields Maxwell-like equations that hold for both a matter Geld and an electromagnetic Geld. We establish that the electric Geld is perpendicular to the matter magnetic Geld and the magnetic Geld is perpendicular to the matter inertial Geld. We show that the inertial and magnetic masses are conserved separately. The magnetic mass density arises as a result of the coupling between the vector potential and the matter inertial Geld. The presence of the vector and scalar potentials and also the matter inertial and magnetic Gelds modify the Standard form of the derived Maxwell equations. The resulting interacting electrodynamics equations are generalizations of the equations of Wilczek or Chern-Simons axion-like Gelds. The coupled Geld in the biquaternioic Dirac Geld reconstructs the Wilczek axion Geld. We show that the electromagnetic Geld vector \(\overrightarrow{F}=\overrightarrow{E}+ic\overrightarrow{B}\), where \(\overrightarrow{E}\) and \(\overrightarrow{B}\) are the respective electric and magnetic Gelds, satisGes the massive Dirac equation and, moreover, \(\overrightarrow{\triangledown}\cdot\overrightarrow{F}=0\).

Higher-Order Topological States in Surface-Wave Photonic Crystals.

Photonic topological states have revolutionized the understanding of the propagation and scattering of light. The recent discovery of higher‐order photonic topological insulators opens an emergent horizon for 0D topological corner states. However, the previous realizations of higher‐order topological insulators in electromagnetic‐wave systems suffer from either a limited operational frequency range due to the lumped components involved or a bulky structure with a large footprint, which are unfavorable for achieving compact photonic devices. To overcome these limitations, a planar surface‐wave photonic crystal realization of 2D higher‐order topological insulators is hereby demonstrated experimentally. The surface‐wave photonic crystals exhibit a very large bulk bandgap (a bandwidth of 28%) due to multiple Bragg scatterings and host 1D gapped edge states described by massive Dirac equations. The topology of those higher‐dimensional photonic bands leads to the emergence of in‐gap 0D corner states, which provide a route toward robust cavity modes for scalable compact photonic devices.

Quasibound states of the Dirac field in Schwarzschild and Reissner–Nordström black hole backgrounds

In this paper, we study the existence of (quasi)bound states in two spacetime geometries describing Schwarzschild and Reissner–Nordström black holes. For obtaining these types of states, we search for discrete quantum modes of the massive Dirac equation in the two geometries. After imposing the quantization condition, an analytical expression for the energy of the ground states is derived. The energy of higher states is then obtained numerically. For very small values of the black hole mass M, we compare the energy of the Reissner–Nordström black hole quasibound state with the Dirac–Coulomb energy and we have found the two to be in good agreement.

Anomalous equilibrium currents for massive Dirac electrons

Existing investigations of the anomalous Hall effect i.e. a current flowing transverse to the electric field in the absence of an external magnetic field) are concerned with the transport current. However, for many applications one needs to know the total current, including its pure magnetization part. In this paper, we employ the two-dimensional massive Dirac equation to find the exact universal total current flowing along a potential step of arbitrary shape. For a spatially slowly varying potential we find the current density $\mathbf{j}(\vec r)$ and the energy distribution of the current density $\mathbf{j}^\varepsilon(\vec r)$. The latter turns out to be unexpectedly nonuniform, behaving like a $\delta$-function at the border of the classically accessible area at energy~$\varepsilon$. To demonstrate explicitly the difference between the magnetization and transport currents we consider the transverse shift of an electron ray in an electric field.

Quasinormal modes of Dirac field in 2 + 1 dimensional gravitational wave background

In this study, we investigate the solutions of the massive and massless Dirac equation, at first, in the 2+1 dimensional gravitational wave background and then in the background with the presence of a constant magnetic field. In both situations, using the solutions obtained in terms of the confluent Heun functions, we calculate the quasinormal modes (QNMs) and discuss the stability conditions of the background. Comparing the quality factors derived from the QNMs in the existence of Dirac field mass and constant magnetic field with together, we see that the magnetic field plays a very important role in reducing the number of the gravitational wave oscillations because of its damping or screening effect, but there is no magnetic field effect on the proper oscillations of the gravitational wave. In addition, we acquire quasi-resonance modes (QRMs) for the massive test field in the absence of magnetic field and massless Dirac field in the presence of magnetic field from these QNMs. In both situations, the gravitational wave (GW) background becomes a good oscillator because the quality factor is very high. Moreover, we discuss the gravitational redshift for the wave background. Finally, we see that the obtained the GW frequencies and redshift values in the context of the study are compatible with the observational data.