Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice

The spatial discretization of the single-cone Dirac Hamiltonian on the surface of a topological insulator or superconductor needs a special ``staggered’’ grid, to avoid the appearance of a spurious second cone in the Brillouin zone. We adapt the Stacey discretization from lattice gauge theory to produce a generalized eigenvalue problem, of the form \bm{\mathcal H}\bm{\psi}=\bm{E}\bm{\mathcal P}\bm{\psi}ℋ𝛙=𝐄𝒫𝛙, with Hermitian tight-binding operators \bm{\mathcal H}ℋ, \bm{\mathcal P}𝒫, a locally conserved particle current, and preserved chiral and symplectic symmetries. This permits the study of the spectral statistics of Dirac fermions in each of the four symmetry classes A, AII, AIII, and D.

Geometric phase for Dirac Hamiltonian under gravitational fields in the nonrelativistic regime

We show the appearance of geometric phase in a Dirac particle traversing in nonrelativistic limit in a time-independent gravitational field. This turns out to be similar to the one originally described as a geometric phase in magnetic fields. We explore the geometric phase in the Kerr and Schwarzschild geometries, which have significant astrophysical implications. Nevertheless, the work can be extended to any spacetime background including that of time-dependent. In the Kerr background, i.e. around a rotating black hole, geometric phase reveals both the Aharonov–Bohm effect and Pancharatnam–Berry phase. However, in a Schwarzschild geometry, i.e. around a nonrotating black hole, only the latter emerges. We expect that our assertions can be validated in both the strong gravity scenarios, like the spacetime around black holes, and weak gravity environment around Earth.

Dirac 4x1 Wavefunction Recast into a 4x4 Type Wavefunction

As currently understood, the Dirac theory employs a 4 x1 type wavefunction. This 4x1 Dirac wavefunction is acted upon by a 4x4 Dirac Hamiltonian operator, in which process, four independent particle solutions result. Insofar as the real physical meaning and distinction of these four solutions, it is not clear what these solutions really mean. We demonstrate herein that these four independent particle solutions can be brought together under a single roof wherein the Dirac wavefunction takes a new form as a 4x4 wavefunction. In this new formation of the Dirac wavefunction, these four particle solutions precipitate into three distinct and mutuality dependent particles that are eternally bound in the same region of space. Given that Quarks are readily found in a mysterious threesome cohabitation-state eternally bound inside the Proton and Neutron, we make the suggestion that these Dirac particles might be Quarks. For the avoidance of speculation, we do not herein explore this idea further but merely present it as a very interesting idea worthy of further investigation. We however must say that, in the meantime, we are looking further into this very interesting idea, with the hope of making inroads in the immediate future.

Fermion number 1/2 of sphalerons and spectral mirror symmetry

We present a rederivation of the baryon and lepton numbers $$\frac{1}{2}$$ 1 2 of the $$SU(2)_L$$ S U ( 2 ) L S sphaleron of the standard electroweak theory based on spectral mirror symmetry. We explore the properties of a fermionic Hamiltonian under discrete transformations along a noncontractible loop of field configurations that passes through the sphaleron and whose endpoints are the vacuum. As is well known, CP transformation is not a symmetry of the system anywhere on the loop, except at the endpoints. By augmenting CP with a chirality transformation, we observe that the Dirac Hamiltonian is odd under the new transformation precisely at the sphaleron, and this ensures the mirror symmetry of the spectrum, including the continua. As a consistency check, we show that the fermionic zero mode presented by Ringwald in the sphaleron background is invariant under the new transformation. The spectral mirror symmetry which we establish here, together with the presence of the zero mode, are the two necessary conditions whence the fermion number $$\frac{1}{2}$$ 1 2 of the sphaleron can be inferred using the reasoning presented by Jackiw and Rebbi or, equivalently, using the spectral deficiency $$\frac{1}{2}$$ 1 2 of the Dirac sea. The relevance of this analysis to other solutions is also discussed.

Electrodynamics in Fock–Lorentz linear fractional relativity

In this paper, we have explored the effect of Fock–Lorentz linear fractional relativity on the electrodynamics laws, where the radius of universe [Formula: see text] emerges as a consequence in the new formulation of Fock’s transformation. By employing the Dirac Hamiltonian analysis scheme, we have studied the case of the free particle, as well as the charged particle in presence of an external electromagnetic field in the new deformed phase space “[Formula: see text]-Minkowski.” The Lorentz force is obtained up to the first-order [Formula: see text]. Furthermore, we have discussed the modified form of Leinard–Wiechart potentials.

ANALYTICAL SOLUTION OF (2+1) DIMENSIONAL DIRAC EQUATION IN TIME-DEPENDENT NONCOMMUTATIVE PHASE-SPACE

In this article, we studied the system of a (2+1) dimensional Dirac equation in a time-dependent noncommutative phase-space. More specifically, we investigated the analytical solution of the corresponding system by the Lewis-Riesenfeld invariant method based on the construction of the Lewis-Riesenfeld invariant. Knowing that we obtained the time-dependent Dirac Hamiltonian of the problem in question from a time-dependent Bopp-Shift translation, then used it to set the Lewis- Riesenfeld invariant operators. Thereafter, the obtained results were used to express the eigenfunctions that lead to determining the general solution of the system.

Defect Dynamics in Graphene

The experimental and theoretical study of graphene, two-dimensional (2D) graphite, is an extremely rapidly growing field of today's condensed matter research. Different types of disorder in graphene modify the Dirac equation leading to unusual spectroscopic and transport properties. The authors studied one of the disorders (i.e., grain boundaries) and formulated a theoretical model of graphene grain boundary by generalizing the two-dimensional graphene Dirac Hamiltonian model. In this model only, the authors considered the long-wavelength limit of the particle transport, which provides the main contribution to the graphene conductance. In this work, they derived the Hamiltonian in a rotated side dependent reference frame describing crystallographic axes mismatching at a grain boundary junction and showed that properties like energy spectrum are an independent reference frame. Also, they showed one of the topological property of graphene.