Regular and irregular dynamics of spin-polarized wavepackets in a mesoscopic quantum dot at the edge of topological insulator

The dynamics of Dirac–Weyl spin-polarized wavepackets driven by a periodic electric field is considered for the electrons in a mesoscopic quantum dot formed at the edge of the two-dimensional HgTe/CdTe topological insulator with Dirac–Weyl massless energy spectra, where the motion of carriers is less sensitive to disorder and impurity potentials. It is observed that the interplay of strongly coupled spin and charge degrees of freedom creates the regimes of irregular dynamics in both coordinate and spin channels. The border between the regular and irregular regimes determined by the strength and frequency of the driving field is found analytically within the quasiclassical approach by means of the Ince–Strutt diagram for the Mathieu equation, and is supported by full quantum-mechanical simulations of the driven dynamics. The investigation of quasienergy spectrum by Floquet approach reveals the presence of non-Poissonian level statistics, which indicates the possibility of chaotic quantum dynamics and corresponds to the areas of parameters for irregular regimes within the quasiclassical approach. We find that the influence of weak disorder leads to partial suppression of the dynamical chaos. Our findings are of interest both for progress in the fundamental field of quantum chaotic dynamics and for further experimental and technological applications of spindependent phenomena in nanostructures based on topological insulators.     

Characteristics of level-spacing statistics in chaotic graphene billiards

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.     

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [1] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincar′e symmetry P(1,5)= SO(1,5) P~(1,5) as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.     

Realization of Graphene Physics Through a Fully Optical System

The two-dimensional form of carbon known as graphene awaken the scientific community interest due to its exotic electronic properties, emerging from the behavior of electrons near the Fermi level as massless Dirac fermions in a (1+2)-dimensional “relativistic” space-time, which renders a bridge between condensed matter and relativistic quantum field theory. Optical systems are also prodigal in providing analogues of complex quantum mechanical systems. Here, it is proposed an optical realization capable of capturing the essential physics of the Dirac equation in (1+2)-D dimensions, simulating the properties of graphene through the use of lightwave technology.     

Two-dimensional quantum ring in a graphene layer in the presence of a Aharonov–Bohm flux

In this paper we study the relativistic quantum dynamics of a massless fermion confined in a quantum ring. We use a model of confining potential and introduce the interaction via Dirac oscillator coupling, which provides ring confinement for massless Dirac fermions. The energy levels and corresponding eigenfunctions for this model in graphene layer in the presence of Aharonov–Bohm flux in the centre of the ring and the expression for persistent current in this model are derived. We also investigate the model for quantum ring in graphene layer in the presence of a disclination and a magnetic flux. The energy spectrum and wave function are obtained exactly for this case. We see that the persistent current depends on parameters characterizing the topological defect.     

Landau levels for relativistic particles: Einstein–Brillouin–Keller quantization approach

The effects of a transverse magnetic field on relativistic particles in two dimensions are treated by using the semiclassical quantization rules and the role played by the spin is emphasized. The Landau levels? energies are analyzed by focusing on the square-root dependence on level index obtained for relativistic spinless particles. This result will be compared to the energies calculated for relativistic particles with spin that are governed by the Dirac equation in relativistic quantum mechanics. Then relativistic massless fermions are discussed. The approach provides a conceptual and intuitive introduction to the grounds of quantum Hall effect in carbon based nanostructures.     

Magnetic confinement of massless Dirac fermions in graphene

Because of Klein tunneling, electrostatic potentials are unable to confine Dirac electrons. We show that it is possible to confine massless Dirac fermions in a monolayer graphene sheet by inhomogeneous magnetic fields. This allows one to design mesoscopic structures in graphene by magnetic barriers, e.g., quantum dots or quantum point contacts.     

Maxwell’s Equations as the One-Photon Quantum Equation

Maxwell's equations (the Faraday and Ampère-Maxwell laws) can be presented as a three-component equation in a way similar to the two-component neutrino equation. However, in this case, the electric and magnetic Gauss laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have also shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable and well understood quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum theories.     

Absorption and quasinormal modes of classical fields propagating on 3D and 4D de Sitter spacetime

We extensively study the exact solutions of the massless Dirac equation in 3D de Sitter spacetime that we published recently. Using the Newman-Penrose formalism, we find exact solutions of the equations of motion for the massless classical fields of spin s= 12,1,2 and to the massive Dirac equation in 4D de Sitter metric. Employing these solutions, we analyze the absorption by the cosmological horizon and de Sitter quasinormal modes. We also comment on the results given by other authors.     

The Construction of Dirac Wave Packets for a Fermionic Particle Non-Minimally Coupling with an External Magnetic Field

We shall proceed with the construction of normalizable Dirac wave packets for fermionic particles (neutrinos) with dynamics governed by a “modified” Dirac equation with a non-minimal coupling with an external magnetic field. We are not only interested on the analytic solutions of the “modified” Dirac wave equation but also on the construction of Dirac wave packets which can be used for describing the dynamics of some observable physical quantities which are relevant in the context of the quantum oscillation phenomena. To conclude, we discuss qualitatively the applicability of this formal construction in the treatment of chiral (and flavor) oscillations in the theoretical context of neutrino physics. PACS numbers: 02.30.Cj, 03.65.Pm     

Perspectives on relativistic quantum chaos

Quantum Chaos has been investigated for about a half century.It is an old yet vigorous interdisciplinary field with new concepts and interesting topics emerging constantly.Recent years have witnessed a growing interest in quantum chaos in relativistic quantum systems,leading to the still developing field of relativistic quantum chaos.The purpose of this paper is not to provide a thorough review of this area,but rather to outline the basics and introduce the key concepts and methods in a concise way.A few representative topics are discussed,which may help the readers to quickly grasp the essentials of relativistic quantum chaos.A brief overview of the general topics in quantum chaos has also been provided with rich references.     

Bound states of Dirac fermions in monolayer gapped graphene in the presence of local perturbations

In graphene,conductance electrons behave as massless relativistic particles and obey an analogue of the Dirac equation in two dimensions with a chiral nature.For this reason,the bounding of electrons in graphene in the form of geometries of quantum dots is impossible.In gapless graphene,due to its unique electronic band structure,there is a minimal conductivity at Dirac points,that is,in the limit of zero doping.This creates a problem for using such a highly motivated new material in electronic devices.One of the ways to overcome this problem is the creation of a band gap in the graphene band structure,which is made by inversion symmetry breaking(symmetry of sublattices).We investigate the confined states of the massless Dirac fermions in an impured graphene by the short-range perturbations for "local chemical potential" and "local gap".The calculated energy spectrum exhibits quite different features with and without the perturbations.A characteristic equation for bound states(BSs) has been obtained.It is surprisingly found that the relation between the radial functions of sublattices wave functions,i.e.,f_m~+(r),g_m~+(r),and f_m~-(r),g_m~-(r),can be established by SO(2) group.     

Dirac quasinormal modes of <Emphasis Type="Italic">D</Emphasis>-dimensional de Sitter spacetime

We find exact solutions to the Dirac equation in D-dimensional de Sitter spacetime. Using these solutions we analytically calculate the de Sitter quasinormal (QN) frequencies of the Dirac field. For the massive Dirac field this computation is similar to that previously published for massive fields of half-integer spin moving in four dimensions. However to calculate the QN frequencies of the massless Dirac field we must use distinct methods in odd and even dimensions, therefore the computation is different from that already known for other massless fields of integer spin.     

Scattering of massless fermions by Schwarzschild and Reissner-Nordstrm black holes

We study the scattering of massless Dirac fermions by Schwarzschild and Reissner-Nordstrm black holes. This is done by applying partial wave analysis to the scattering modes obtained after solving the massless Dirac equation in the asymptotic regions of the two black hole geometries. We successfully obtain analytic phase shifts, with the help of which the scattering cross section is computed. The glory and spiral scattering phenomena are shown to be present, as in the case of massive fermion scattering by black holes.     

Plasma oscillations of edge Dirac fermions

The dispersion law of one-dimensional plasmons in a quasi-one-dimensional system of massless Dirac fermions has been calculated. Two model two-dimensional systems where bands of edge states filled with such Dirac fermions appear at the edge have been considered. Edge states in the first system, topological insulator, are due to topological reasons. Edge states in the second system, system of massive Dirac fermions, have Tamm origin. It has been shown that the dispersion laws of plasmons in both systems in the long-wavelength limit differ only in the definition of the parameters (velocity and localization depth of Dirac fermions). The frequency of plasmons is formally quantum (ω ∝ ? ^?1/2) and, in the case of the Coulomb interaction between electrons, depends slightly on the Fermi level E _F. The dependence on E _F is stronger in the case of short-range interaction. The quantum features of oscillations of massless one-dimensional Dirac fermions are removed by introducing the mass of Dirac fermions at the Fermi level and their density. Correspondence to the dispersion law of classical one-dimensional plasma oscillations in a narrow stripe of “Schrödinger” electrons has been revealed.     

Separating the variables in a massless Dirac equation in Minkowski space

Exact integration of the Dirac equation is a classical topic in mathematical physics, which has been researched for several decades. A basic method is complete segregation of the variables. Such separation can be attained in a Dirac equation containing an external electromagnetic field in Minkowski space by means of complete sets of first-order symmetry matrix operators. The purpose of this paper is to solve an analogous case for a free massless Dirac equation. That task has a special feature because external fields are absent and the massless equation is reduced to a D'Alambert equation by squaring. Nevertheless, interest attaches to states defined by the first-order symmetry-operator matrices that cannot be obtained by setting the mass to zero in systems containing a mass Dirac equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 105–110, January, 1995.     

Indefinite charge for the classical spinor field

We define a conserved Lorentz vector for a two-component spinor field that obeys the Klein-Gordon equation and interpret it as a charge-current density. The corresponding total charge can take negative as well as positive values, which is not the case for the usual charge of the Dirac field. We consequently can define probability amplitudes for a relativistic quantum mechanics, and we solve the inhomogeneous equation by means of the causal Green function. This vector is not invariant under gauge transformations of the spinor field, and we cannot generalize the equation by the gauge invariant substitution to obtain the interaction with an electromagnetic field. In the limit of a massless field that obeys the Weyl equation, the charge vanishes.     

Dirac Equation in the Spatially Flat Friedmann Model

The unitary transformation which diagonalizesthe field-free Dirac Hamiltonian in the spatially flatFriedmann-Robertson-Walker metric is analyzed, and apair of simultaneous first-order nonlinear differential equations is derived for the two parameters(two angles) that characterize the transformation. Theequations are solved approximately for a test particlewhose kinetic energy is small compared to its mass energy, and minimum-uncertainty wave packetsare constructcd from the solutions. It is found thatgeneral relativity limits the quantum mechanical spreadof the wave packets, but forces then to expand with the expanding space, as if they were embeddedin it. The massless Dirac equation is solved exactly forthe two-component neutrino spinor, and yieldsgeneralized nonspreading wave packets which display no quantum mechanical spread at all, but areconstrained to expand with the expanding space as theyfollow null geodesics.     

Statistical properties of eigenvalues for an operating quantum computer with static imperfections

We investigate the transition to quantum chaos, induced by static imperfections, for an operating quantum computer that simulates efficiently a dynamical quantum system, the sawtooth map. For the different dynamical regimes of the map, we discuss the quantum chaos border induced by static imperfections by analyzing the statistical properties of the quantum computer eigenvalues. For small imperfection strengths the level spacing statistics is close to the case of quasi-integrable systems while above the border it is described by the random matrix theory. We have found that the border drops exponentially with the number of qubits, both in the ergodic and quasi-integrable dynamical regimes of the map characterized by a complex phase space structure. On the contrary, the regime with integrable map dynamics remains more stable against static imperfections since in this case the border drops only algebraically with the number of qubits. Received 19 June 2002 / Received in final form 30 September 2002 Published online 17 Decembre 2002 RID="a" ID="a"e-mail: dima@irsamc.ups-tlse.fr RID="b" ID="b"UMR 5626 du CNRS