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Our review paper is dedicated to studies of two-dimensional (2D) Wannier–Mott excitons in transition metal dichalcogenides and 2D magnetoexcitons in GaAs-type quantum wells subjected to the action of a strong perpendicular magnetic field, which both revealed the Dirac cone dispersion laws. It is shown that necessary conditions for the implementation of this property to be carried out are the taking into account of the electron–hole (e–h) exchange Coulomb interaction and the interdependence between the center-of-mass and relative e–h motions in the frame of the excitons. A short review describing the effect of these two factors on the Dirac cone dispersion law formation is presented.

The theory of two-dimensional (2D) magnetoexcitons was enlarged taking into account the electron–hole (e–h) exchange Coulomb interaction, which appears when the conduction and valence electrons belong partially to both bands. This Coulomb exchange interaction leads to the linear dispersion law of para-magnetoexcitons. Spectral properties and the new luminescence band, which arise owing to the existence of the metastable bound state, are discussed. The thermodynamic properties and the Bose–Einstein condensation conditions for para- and ortho-magnetoexcitons are described. Taking into account the vector potential of the Chern–Simons gauge field the anisotropy of the 2D magnetoexciton magnetic mass was revealed.

The possible existence of the two-dimensional bimagnetoexcitons and metastable bound states formed by two magnetoexcitons with opposite in-plane wave vectors and has been studied. Magnetoexcitons taking part in the formation of molecules look as two electric dipoles with the arms oriented in-plane perpendicular to the respective wave vectors and with the length of the arms where is the magnetic length. Two antiparallel dipoles moving with equal, yet antiparallel, wave vectors have the possibility of moving with equal probability in any direction of the plane, which is determined by the trial wave function of relative motion depending on modulus The magnetoexcitons are composed of electrons and holes situated on the lowest Landau levels with the cyclotron energies greater than the binding energy of the 2D Wannier–Mott exciton. The description has been made in Landau gauge. The spin states of two electrons have been chosen in the form of antisymmetric or symmetric combinations with parameter The effective spins of two heavy holes have been combined in the same resultant spinor states as the spin of the electrons. Because the projections of the both spinor states with are equal to zero, the influence of the Zeeman splitting effect vanishes. In the case of trial wave function , the maximal density of the magnetoexcitons in the momentum space is concentrated on the in-plane ring with radius In the approximation of the lowest Landau levels, when the influence of the excited Landau levels is neglected, stable bound states of bimagnetoexcitons do not exist for both spin orientations. Instead, in the case of and , a deep metastable bound state with the activation barrier comparable with two magnetoexciton ionization potentials has been revealed. In the case of and , only a shallow metastable bound state can appear.

The review is focused on the energy spectrum of two-dimensional electrons, holes, magnetoexcitons, and cavity polaritons under the action of strong magnetic and electric fields perpendicular to the surface of GaAs-type quantum wells (QWs) with a p-type valence band embedded into resonators. As the first step in this direction, the Landau quantization (LQ) of electrons and heavy holes (hhs) has been studied taking into account the Rashba spin-orbit coupling (RSOC) with third-order chirality terms for hhs and nonparabolicity terms in their dispersion law, including the Zeeman splitting (ZS) effects. The nonparabolicity terms proportional to the electric field strength have been introduced to avoid the collapse of the semiconductor energy gap under the action of the third-order chirality terms. Exact solutions for the eigenfunctions and eigenenergies for the LQ task have been obtained on the basis of the Rashba method [1]. In the next steps of our review paper, we have deduced Hamiltonians describing the Coulomb electron–electron and electron–radiation interactions in the second quantization representation. They make it possible to determine the magnetoexciton energy branches and deduce the Hamiltonian of the magnetoexciton–photon interaction. The fifth-order dispersion equation describing the energy spectrum of the cavity magnetoexciton–polaritons has been studied. It takes into account the interaction of the cavity photons with two dipole-active and two quadrupole-active 2D magnetoexciton energy branches. The cavity photons have circular polarizations k s ± oriented along their wave vectors k with quantized longitudinal components kz = ±p Lc , where c L is the resonator length, and with small transverse components || k oriented in the plane of the QW. The 2D magnetoexcitons are characterized by in-plane wave vectors || k and circular polarizations M s arising in the p-type valence band with magnetic momentum projection M = ±1 in the direction of the magnetic field. The selection rules of the exciton– photon interaction have two origins. One of them, of geometrical type, is expressed in terms of the scalar products of the two types of circular polarizations. They depend on in-plane wave vectors || k even in the case of dipole-active transitions, because the cavity photons have an oblique incidence to the surface of the QW. The other origin is related with numbers e n and h n of the LQ levels of electrons and hhs involved in the magnetoexciton formation. Thus, the dipole-active transitions take place under condition , e h n = n whereas in the quadrupole-active transitions the relationship is as follows: 1. e h n = n ± The optical gyrotropy effects appear changing the sign of the photon circular polarization at a given sign of wave vector longitudinal projection z k or equivalently changing the sign of longitudinal projection z k at the same selected circular polarization of light.

In our review, the dispersion laws of two-dimensional (2D) cavity magnetoexcitonpolaritons and the interaction of 2D magnetoexcitons with wave vectors k 0 are discussed. Magnetoexcitons do appear in a system of coplanar electron–hole pairs subjected to the action of strong magnetic and electric fields both being oriented along the axis of the microcavity and perpendicular to the surface of the quantum well embedded in it. Cavity polaritons are formed as a result of the interaction between 2D magnetoexcitons and cavity photons that arbitrary propagate inside the microcavity relative to the plain of the quantum well.

The properties of two-dimensional (2D) cavity polaritons subjected to the action of strong perpendicular magnetic and electric fields giving rise to the Landau quantization (LQ) of the 2D electrons and holes accompanied by the Rashba spin-orbit coupling (RSOC) and by the Zeeman splitting (ZS) have been investigated. A strong magnetic field, where the electron and the hole cyclotron energy frequencies are greater than the binding energy of the 2D Wannier-Mott excitons, completely reconstructs it transforming into a magnetoexciton, the structure of which is determined by the Lorentz force rather than by the Coulomb electronhole (eh) interaction. We predict drastic changes in the optical properties of the cavity polaritons including those in the state of Bose-Einstein condensation. The main of them is the existence of a multitude of the polariton energy levels closely adjacent on the energy scale, their origin being related with the LQ of the electrons and holes. Most of these levels exhibit nonmonotonous dependences on magnetic field strength B with overlapping and intersections. More so, the selection rules for the band-to-band optical quantum transitions, as well as the quantum transitions from the ground state of the crystal to the magnetoexciton states, essentially depend on numbers ne and nh of the LQ levels of the eh pair forming the magnetoexciton. By slowly changing the external magnetic and electric fields, it is possible to change the lowest polariton energy level, its oscillator strength, the probability of the quantum transition, and the Rabi frequency of the polariton dispersion law. They depend on the relation between numbers ne and nh and can lead to dipole-active, quadrupole-active, or forbidden optical transitions. Our results are based on the exact solutions for the eigenfunctions and the eigenvalues of the Pauli-type Hamilonian with third order chirality terms and a nonparabolic dispersion law for heavy-holes and with first order chirality terms for electrons. They were obtained using the method proposed by Rashba [1]. We expect that these results will also determine the collective behavior of the cavity polaritons, for example, in the GaAs-type quantum wells embedded into a microcavity, which have recently revealed the phenomenon of the Bose-Einstein condensation in the state of the thermodynamic quasi-equilibrium but in the absence of a strong perpendicular magnetic field.

The Bose-Einstein condensation (BEC) of the two-dimensional (2D) magnetoexcitonpolaritons in microcavity, where the Landau quantization of the electron and hole states accompanied by the Rashba spinorbit coupling plays the main role, was investigated. The Landau quantization levels of the 2D heavy holes with nonparabolic dispersion law and third-order chirality terms both induced by the external electric field perpendicular to the semiconductor quantum well and strong magnetic field B give rise to a nonmonotonous dependence of the magnetoexciton energy levels and the polariton energy branches on B.

The Landau quantization of the two-dimensional (2D) heavy holes, its influence on the energy spectrum of 2D magnetoexcitons, and their optical orientation are studied. The Hamiltonian of the heavy holes is written in a two-band model taking into account the Rashba spin-orbit coupling (RSOC) with two spin projections, but with nonparabolic dispersion law and third order chirality terms. The most Landau levels, except three with m = 0, 1, 2, are characterized by two quantum numbers m - 3 and m for m > 3 for two spin projections, respectively. The difference between them is determined by the third order chirality. Four lowest Landau levels (LLLs) for heavy holes were combined with two LLLs for conduction electron, which were taken the same as they were deduced by Rashba in his theory of spin-orbit coupling (SOC) based on the initial parabolic dispersion law and first order chirality terms. As a result of these combinations, eight 2D magnetoexciton states were formed. Their energy spectrum and the selection rules for the quantum transitions from the ground state of the crystal to exciton states were determined. On this base, optical orientation effects, such as spin polarization and magnetoexciton alignment, are discussed.

The Rashba spin-orbit coupling (RSOC) in the case of two-dimensional (2D) electrons and holes in a strong perpendicular magnetic field is studied. The spinor-type wave functions are characterized by different numbers of Landau levels in different spin projections. For electrons, they differ by 1 as was established earlier by Rashba [1], whereas for holes they differ by 3. Two lowest electron states and four lowest hole states of Landau quantization give rise to eight 2D magnetoexciton states. The exchange electron-hole interaction in the frame of these states is investigated.

The Bose-Einstein Condensation (BEC) of the two-dimensional magnetoexcitons on the superposition state, as well as on the excited Landau levels, is studied. The superposition of two excitonic states formed by electron and hole on the lowest Landau levels (0, 0) and on the first excited Landau levels (1, 1) was considered. The generalized Bogoliubov u-v transformation was deduced. The criterion for the exciton concentration was established. These questions will be studied below.

The collective elementary excitations of a system of Bose-Einstein condensed two- dimensional magnetoexcitons interacting with electron-hole(e-h) plasma in a strong perpendicular magnetic field are studied. The breaking of the gauge symmetry is introduced into the Hamiltonian following the Bogoliubov`s theory of quasiaverages. The motion equations for the summary operators describing the creation and annihilation of magnetoexcitons as well as the density fluctuations of the electronhole(e-h) plasma were derived. They suggest the existence of magneto-exciton-plasmon complexes, the energies of which differ by the energies of one or two plasmon quanta. Starting with these motion equations one can study the Bose-Einstein Condensation (BEC) of different magneto-exciton-plasmon complexes introducing different constants of the broken symmetry correlated with their energies. The Green`s functions constructed from these summary operators are two-particle Green`s functions. They obey the chains of equations expressing the two-particle Green`s functions through the four-particle and six-particle Green`s functions. These chains were truncated in such a way that the six-particle Green`s functions, were expressed through the two-particle ones. At the same time the elementary excitations with different wave vectors were decoupled. As a result of these simplifications the Dyson-type equation in a matrix form for the two-particle Green`s functions was obtained. The determinant constructed from the self-energy part 44 × ( , ) ij P ω ∑ G gives rise to dispersion equation. The dispersion relations were obtained in analytical form, when in the self-energy parts ( , ) ij P ω ∑ G only the terms linear in Coulomb interaction were kept. Taking into account also the terms quadratic in Coulomb interaction the dispersion equation becomes cumbersome and it can be solved only numerically.