Two-dimensional systems in strong magnetic field are studied intensively since the discovery of integer and fractional quantum Hall effects [1–3]. For a long time, such systems were represented by gallium arsenide heterostructures with 2D electron motion within each subband .
New and very interesting realization of 2D electron system appeared when graphene, a monoatomic layer of carbon, was successfully isolated [5, 6]. The most spectacular property of graphene is the fact that its electrons behave as massless chiral particles, obeying Dirac equation. Intensive experimental and theoretical studies of this material over several recent years yielded a plethora of interesting results [7–9]. In particular, graphene demonstrates unusual half-integer quantum Hall effect , which can be observed even at room temperature .
In external perpendicular magnetic field, the motion of electrons along cyclotron orbits acquires zero-dimensional character and, as a result, electrons fill discrete Landau levels . In semiconductor quantum wells, Landau levels are equidistant and separation between them is determined by the cyclotron frequency ωc = eH/mc. In graphene, due to massless nature of electrons, "ultra-relativistic" Landau levels appear, which are non-equidistant and located symmetrically astride the Dirac point [12, 13]. Energies of these levels are , where n = 0, ±1, ±2,..., v
≈106 m/s is the Fermi velocity of electrons and is magnetic length, or radius of the cyclotron orbit (here and below we assume ħ = 1).
In the case of integer filling, when several Landau levels are completely filled by electrons and all higher levels are empty, elementary excitations in the system are caused by electron transitions from one of the filled Landau levels to one of the empty levels . Such transitions can be observed in cyclotron resonance or Raman scattering experiments as absorption peaks at certain energies. With neglect of Coulomb interaction, energy of the excited electron-hole pair is just a distance between Landau levels of electron and hole. Coulomb interaction leads to mixing of transitions between different pairs of Landau levels, changing the resulting energies of elementary excitations.
Characteristic energy of Coulomb interaction in magnetic field is e2/εl
, where ε is a dielectric permittivity of surrounding medium. The relative strength of Coulomb interaction can be estimated as ratio of its characteristic value to a characteristic distance between Landau levels. For massive electrons in semiconductor quantum wells, this ratio is proportional to H-1/2, thus in asymptotically strong magnetic field the Coulomb interaction becomes a weak perturbation [15, 16]. In this case, the lowest Landau level approximation, neglecting Landau level mixing, is often used. It was shown that Bose-condensate of noninteracting magnetoexcitons in the lowest Landau level is an exact ground state in semiconductor quantum well in strong magnetic field .
A different situation arises in graphene. The relative strength of Coulomb interaction in this system can be expressed as rs = e2/εv
and does not depend on magnetic field . The only parameter which can influence it is the dielectric permittivity of surrounding medium ε. At small enough ε, mixing between different Landau levels can significantly change properties of elementary excitations in graphene.
Coulomb interaction leads to appearance of two types of elementary excitations from the filled Landau levels. From summation of "ladder" diagrams we get magnetoexcitons, which can be imagined as bound states of electron and hole in magnetic field [14, 16, 19]. Properties of magnetoexcitons in graphene were considered in several works, mainly in the lowest Landau level approximation [20–24]. At ε ≈ 3, Landau level mixing was shown to be weak in the works [20, 25].
Note that influence of Landau level mixing on properties of an insulating ground state of neutral graphene was considered in  by means of tight-binding Hartree-Fock approximation. It was shown that Landau level mixing favors formation of insulating charge-density wave state instead of ferromagnetic and spin-density wave states in suspended graphene, i.e., at weak enough background dielectric screening.
From the experimental point of view, the most interesting are magnetoexcitons with zero total momentum, which are only able to couple with electromagnetic radiation due to very small photon momentum. For usual non-relativistic electrons, magnetoexciton energy at zero momentum is protected against corrections due to electron interactions by the Kohn theorem . However, for electrons with linear dispersion in graphene the Kohn theorem is not applicable [21, 24, 28–32]. Thus, observable energies of excitonic spectral lines can be seriously renormalized relatively to the bare values, calculated without taking into account Coulomb interaction.
The other type of excitations can be derived using the random phase approximation, corresponding to summation of "bubble" diagrams. These excitations, called magnetoplasmons, are analog of plasmons and have been studied both in 2D electron gas [14, 33] and graphene [18, 20, 21, 24, 34–39] (both with and without taking into account Landau level mixing).
In the present article, we consider magnetoexcitons and magnetoplasmons with taking into account Landau level mixing and show how the properties of these excitations change in comparison with the lowest Landau level approximation. For magnetoexcitons, we take into account the mixing of asymptotically large number of Landau levels and find the limiting values of cyclotron resonance energies.
For simplicity and in order to stress the role of virtual transitions between different pairs of electron and hole Landau levels (i.e., the role of two-particle processes), here we do not take into account renormalization of single-particle energies via exchange with the filled levels. This issue have been considered in several theoretical studies [20, 21, 24, 30, 40]. Correction of Landau level energies can be treated as renormalization of the Fermi velocity, dependent on the ultraviolet cutoff for a number of the filled Landau levels taken into account in exchange processes.
The rest of this article is organized as follows. In Section 2, we present a formalism for description of magnetoexcitons in graphene, which is applied in Section 3 to study influence of Coulomb interaction and Landau level mixing on their properties. In Section 4, we study magnetoplasmons in graphene in the random phase approximation and in Section 5 we formulate the conclusions.