Two-dimensional Massless Dirac Equation Research Articles

A new class of solvable two-dimensional scalar potentials for graphene

In the present paper, a systematic approach is presented for solution of two-dimensional massless Dirac equation with external electrostatic potential applied. This approach is based on the new - asymmetric - form of SUSY-like intertwining relations. It allows to build a wide variety of pairs of SUSY-partner external scalar potentials. If one of them is simple enough to be solvable, its partner is also solvable although it may have a non-trivial dependency on both coordinates. Physically, this kind of problems is related to the description of graphene and some other materials with external potential. Solvability obtained by means of asymmetric form of SUSY intertwining relations allows to extend the class of analytically solvable two-dimensional models.

On the Effect of Intersection of Characteristics in a Two-Dimensional Massless Dirac Equation with Linear Potential and Localized Initial Condition

On the Effect of Intersection of Characteristics in a Two-Dimensional Massless Dirac Equation with Linear Potential and Localized Initial Condition

Closed-form representations of iterated Darboux transformations for the massless Dirac equation

It is shown that first-order Darboux transformations for the two-dimensional massless Dirac equation with scalar potential and for the Schrödinger equation are the same up to a change of coordinates. As a consequence, we obtain a closed-form representation of iterated, higher-order Darboux transformations for our Dirac equation. We use the formalism to generate several new exactly-solvable Dirac systems through higher-order Darboux transformations.

Fundamental solution of the stationary Dirac equation with a linear potential

We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier transforms of parabolic cylinder functions. This solution describes the flux of quasiparticles in graphene emitted by a pointlike source of electrons that are partially converted into holes (antiparticles). Using our explicit formula, we calculate its semiclassical asymptotic behavior in the hole region.

Darboux partners of Heun-class potentials for the two-dimensional massless Dirac equation

We apply the Darboux transformation to construct new exactly-solvable cases of the two-dimensional massless Dirac equation for potential classes of Lambert-W and inverse exponential type. Both of these classes originate from the Heun equation. Conditions are devised for transformed potentials to be real-valued, and to be in terms of elementary functions.

Higher-order Darboux transformations for the massless Dirac equation at zero energy

We construct higher-order Darboux transformations for the two-dimensional massless Dirac equation at zero energy. Both standard and confluent transformations are applicable. Our construction is based on a higher-order extension of the Darboux transformation for Schrödinger equations with quadratically energy-dependent potentials [J. Lin et al., Phys. Lett. A 362, 212 (2007)].

Solution of the Two-Dimensional Dirac Equation with a Linear Potential and a Localized Initial Condition

In this paper, we consider the Cauchy problem with localized initial data for a two-dimensional massless Dirac equation with a linear potential.

Zero energy states for a class of two-dimensional potentials in graphene

The excitations in graphene and some other materials are described by two-dimensional massless Dirac equation with applied external potential of some kind. Solutions of this zero energy equation are built analytically for a wide class of scalar potentials. In contrast to most publications on analytical solutions of massless two-dimensional Dirac equation, our potentials really depend on both spatial coordinates in some bounded domain. Several examples of such construction are given explicitly.

The Symmetrized Square-Root Potential: Exact Solutions and Application to the Two-Dimensional Massless Dirac Equation

We determine bound-state solutions and associated energy eigenvalues of a boundary-value problem that is governed by the Schrodinger equation for a symmetrized, inverse square-root potential. We use these results to construct a scalar potential for which the two-dimensional massless Dirac equation admits zero-energy solutions.

Simple Exact Solutions and Asymptotic Localized Solutions of the Two-Dimensional Massless Dirac Equation for Graphene

We construct exact solutions and asymptotic localized solutions of the two-dimensional massless Dirac equation for graphene with a small potential.

Rationally extended potentials and Xm-Jacobi–type solutions of the two-dimensional massless Dirac equation

We present closed-form solutions of the two-dimensional massless Dirac equation for hyperbolic potentials that are rationally extended in terms of $ X_{m}$ -Jacobi polynomials. Our method of construction is based on rational extensions of nonrelativistic Scarf and Poschl-Teller models that include $ {PT}$ -symmetric and energy-dependent cases.

Bound states of the two-dimensional Dirac equation for an energy-dependent hyperbolic Scarf potential

We study the two-dimensional massless Dirac equation for a potential that is allowed to depend on the energy and on one of the spatial variables. After determining a modified orthogonality relation and norm for such systems, we present an application involving an energy-dependent version of the hyperbolic Scarf potential. We construct closed-form bound-state solutions of the associated Dirac equation.

Simulating electron wave dynamics in graphene superlattices exploiting parallel processing advantages

This work introduces a parallel computing framework to characterize the propagation of electron waves in graphene-based nanostructures. The electron wave dynamics is modeled using both “microscopic” and effective medium formalisms and the numerical solution of the two-dimensional massless Dirac equation is determined using a Finite-Difference Time-Domain scheme. The propagation of electron waves in graphene superlattices with localized scattering centers is studied, and the role of the symmetry of the microscopic potential in the electron velocity is discussed. The computational methodologies target the parallel capabilities of heterogeneous multi-core CPU and multi-GPU environments and are built with the OpenCL parallel programming framework which provides a portable, vendor agnostic and high throughput-performance solution. The proposed heterogeneous multi-GPU implementation achieves speedup ratios up to 75x when compared to multi-thread and multi-core CPU execution, reducing simulation times from several hours to a couple of minutes. Program summaryProgram title: GslSim.Program Files doi:http://dx.doi.org/10.17632/prmfv63nj6.1Licensing provisions: GPLv3.Programming language: C, OpenCL and Matlab for results analysis.Nature of problem: Computing the time evolution of electron waves in graphene superlattices is a time consuming process due to the high number of necessary nodes to discretize the spatial and time domains.Solution method: We develop a simulator based on the C/OpenCL standards to study the time evolution of electron waves in graphene superlattices by exploiting hardware architectures such as graphics processing units (GPUs) to speedup the computation of the pseudospinor.

Construction of zero-energy states in graphene through the supersymmetry formalism

We devise a supersymmetry-based method for the construction of zero-energy states in graphene. Our method is applied to a two-dimensional massless Dirac equation with a hyperbolic scalar potential. We determine supersymmetric partners of our initial system and derive a reality condition for the transformed potential. The Dirac potentials generated by our method can be used to approximate interactions that are experimentally realizable.

Finite difference method for transport properties of massless Dirac fermions

We adapt a finite difference method of solution of the two-dimensional massless Dirac equation, developed in the context of lattice gauge theory, to the calculation of electrical conduction in a graphene sheet or on the surface of a topological insulator. The discretized Dirac equation retains a single Dirac point (no ``fermion doubling''), avoids intervalley scattering as well as trigonal warping, and preserves the single-valley time-reversal symmetry $(=\text{symplectic}\text{ }\text{symmetry})$ at all length scales and energies\char22{}at the expense of a nonlocal finite difference approximation of the differential operator. We demonstrate the symplectic symmetry by calculating the scaling of the conductivity with sample size, obtaining the logarithmic increase due to antilocalization. We also calculate the sample-to-sample conductance fluctuations as well as the shot-noise power and compare with analytical predictions.

Three-dimensional generalization of anyon superconductivity.

A three-dimensional generalization of the wave function of Girvin et al. is constructed by taking the N\ensuremath{\rightarrow}\ensuremath{\infty} limit of a solution of the two-dimensional massless Dirac equation with an SU(N) gauge field. The resulting wave function is closely related to instanton solutions of the self-dual Einstein quations, and can be used to construct a multiparticle wave function with remarkable holonomy properties and a ground-state wave function with off-diagonal long-range order.