Four-Particle Anyon Exciton: Boson Approximation

A theory of anyon excitons consisting of a valence hole and three quasielectrons with electric charges $(-e/3)$ is presented. A full symmetry classification of the $k=0$ states is given, where $k$ is the exciton momentum. The energy levels of these states are expressed by quadratures of confluent hypergeometric functions. It is shown that the angular momentum $L$ of the exciton ground state depends on the distance between electron and hole confinement planes and takes the values $L=3n$, where $n$ is an integer. With increasing $k$ the electron density shows a spectacular splitting on bundles. At first a single anyon splits off of the two-anyon core, and finally all anyons become separated.Comment: Revtex 13 pages + 6 uuencoded postscript figure

Localization of massless Dirac particles via spatial modulations of the Fermi velocity

The electrons found in Dirac materials are notorious for being difficult to manipulate due to the Klein phenomenon and absence of backscattering. Here we investigate how spatial modulations of the Fermi velocity in two-dimensional Dirac materials can give rise to localization effects, with either full (zero-dimensional) confinement or partial (one-dimensional) confinement possible depending on the geometry of the velocity modulation. We present several exactly solvable models illustrating the nature of the bound states which arise, revealing how the gradient of the Fermi velocity is crucial for determining fundamental properties of the bound states such as the zero-point energy. We discuss the implications for guiding electronic waves in few-mode waveguides formed by Fermi velocity modulation.Comment: 9 pages, 6 figure

Excitonic Mott transition in double quantum wells

We consider an electron-hole system in double quantum wells theoretically. We demonstrate that there is a temperature interval over which an abrupt jump in the value of the ionization degree occurs with an increase of the carrier density or temperature. The opposite effect - the collapse of the ionized electron-hole plasma into an insulating exciton system - should occur at lower densities. In addition, we predict that under certain conditions there will be a sharp decrease of the ionization degree with increasing temperature - the anomalous Mott transition. We discuss how these effects could be observed experimentally.Comment: 6 pages, 4 figure

One-dimensional Coulomb problem in Dirac materials

We investigate the one-dimensional Coulomb potential with application to a class of quasirelativistic systems, so-called Dirac-Weyl materials, described by matrix Hamiltonians. We obtain the exact solution of the shifted and truncated Coulomb problems, with the wavefunctions expressed in terms of special functions (namely Whittaker functions), whilst the energy spectrum must be determined via solutions to transcendental equations. Most notably, there are critical bandgaps below which certain low-lying quantum states are missing in a manifestation of atomic collapse.Comment: 7 pages, 5 figure

Massless Dirac fermions in two dimensions: Confinement in nonuniform magnetic fields

We show how it is possible to trap two-dimensional massless Dirac fermions in spatially inhomogeneous magnetic fields, as long as the formed magnetic quantum dot (or ring) is of a slowly decaying nature. It is found that a modulation of the depth of the magnetic quantum dot leads to successive confinement-deconfinement transitions of vortexlike states with a certain angular momentum, until a regime is reached where only states with one sign of angular momentum are supported. We illustrate these characteristics with both exact solutions and a hitherto unknown quasi-exactly solvable model utilizing confluent Heun functions.Comment: 7 pages, 3 figure

Bielectron vortices in two-dimensional Dirac semimetals

Searching for new states of matter and unusual quasiparticles in emerging materials and especially low-dimensional systems is one of the major trends in contemporary condensed matter physics. Dirac materials, which host quasiparticles which are described by ultrarelativistic Dirac-like equations, are of a significant current interest from both a fundamental and applied physics perspective. Here we show that a pair of two-dimensional massless Dirac-Weyl fermions can form a bound state independently of the sign of the inter-particle interaction potential, as long as this potential decays at large distances faster than Kepler's inverse distance law. This leads to the emergence of a new type of energetically-favourable quasiparticle: bielectron vortices, which are double-charged and reside at zero-energy. Their bosonic nature allows for condensation and may give rise to Majorana physics without invoking a superconductor. These novel quasiparticles arguably explain a range of poorly understood experiments in gated graphene structures at low doping.Comment: 9 pages, 2 figure

Exactly-solvable problems for two-dimensional excitons

Several problems in mathematical physics relating to excitons in two dimensions are considered. First, a fascinating numerical result from a theoretical treatment of screened excitons stimulates a re-evaluation of the familiar two-dimensional hydrogen atom. Formulating the latter problem in momentum space leads to a new integral relation in terms of special functions, and fresh insights into the dynamical symmetry of the system are also obtained. A discussion of an alternative potential to model screened excitons is given, and the variable phase method is used to compare bound-state energies and scattering phase shifts for this potential with those obtained using the two-dimensional analogue of the Yukawa potential. The second problem relates to excitons in a quantising magnetic field in the fractional quantum Hall regime. An exciton against the background of an incompressible quantum liquid is modelled as a few-particle neutral composite consisting of a positively-charged hole and several quasielectrons with fractional negative charge. A complete set of exciton basis functions is derived, and these functions are classified using a result from the theory of partitions. Some exact results are obtained for this complex few-particle problem.Comment: 66 pages, 9 figure

Momentum alignment and the optical valley Hall effect in low-dimensional Dirac materials

We study the momentum alignment phenomenon and the optical control of valley population in gapless and gapped graphene-like materials. We show that the trigonal warping effect allows for the spatial separation of carriers belonging to different valleys via the application of linearly polarized light. Valley separation in gapped materials can be detected by measuring the degree of circular polarization of band-edge photoluminescence at different sides of the sample or light spot (optical valley Hall effect). We also show that the momentum alignment phenomenon leads to the giant enhancement of near-band-edge interband optical transitions in narrow-gap carbon nanotubes and graphene nanoribbons independent of the mechanism of the gap formation. A detection scheme to observe these giant interband transitions is proposed which opens a route for creating novel terahertz radiation emitters.Comment: 28 pages, 9 figure

Photon emission induced by elastic exciton--carrier scattering in semiconductor quantum wells

We present a study of the elastic exciton--electron ($X-e^-$) and exciton--hole ($X-h$) scattering processes in semiconductor quantum wells, including fermion exchange effects. The balance between the exciton and the free carrier populations within the electron-hole plasma is discussed in terms of ionization degree in the nondegenerate regime. Assuming a two-dimensional Coulomb potential statically screened by the free carrier gas, we apply the variable phase method to obtain the excitonic wavefunctions, which we use to calculate the 1$s$ exciton--free carrier matrix elements that describe the scattering of excitons into the light cone where they can radiatively recombine. The photon emission rates due to the carrier-assisted exciton recombination in semiconductor quantum-wells (QWs) at room temperature and in a low density regime are obtained from Fermi's golden rule, and studied for mid-gap and wide-gap materials. The quantitative comparison of the direct and exchange terms of the scattering matrix elements shows that fermion exchange is the dominant mechanism of the exciton--carrier scattering process. This is confirmed by our analysis of the rates of photon emission induced by electron-assisted and hole-assisted exciton recombinations.Comment: Thoroughly revised version of previous work. Weak and incorrect assumptions have been removed from the paper, and its scope has evolved: see abstract. This is the final version, i.e. as accepted for publication in the European Physical Journal