Effective Hamiltonian for Excitons with Spin Degrees of Freedom

Starting from the conventional electron-hole Hamiltonian Heh, we derive an effective Hamiltonian H̃1s for 1s excitons with spin degrees of freedom. The Hamiltonian describes optical processes close to the exciton resonance for the case of weak excitation. We show that straightforward bosonization of Heh does not give the correct form of H̃1s, which we obtain by a projection onto the subspace spanned by the 1s excitons. The resulting relaxation and renormalization terms generate an interaction between excitons with opposite spin. Moreover, exciton-exciton repulsive interaction is greatly reduced by the renormalization. The agreement of the present theory with the experiment supports the validity of the description of a fermionic system by bosonic fields in two dimensions. PACS numbers: 71.35.-y, 71.10.-w, 78.66.-w Typeset using REVTEX 1 Properties of a fermionic system can sometimes be described by bosonic fields, by which theoretical analyses may be greatly simplified. The most successful example is the bosonization of one-dimensional (d = 1) conductors [1], where success is due to the specific feature of the pair spectrum [2], i.e., the low-energy pair spectra consist of discrete branches in the energy versus momentum plane. For d ≥ 2 conductors, discrete branches overlap with continuous spectra, hence the bosonization is nontrivial and still in progress [3]. For insulating solids, on the other hand, discrete branches of excitons are separated from the continuum for any d. From this point of view, it has been suggested that a useful bosonic theory may be constructed if one focuses on exciton states, even for d ≥ 2 [4–7]. However, the validity of the bosonic description of excitons is nontrivial, because the binding energy (in, e.g., GaAs) is comparable to other relevant energies [8]. Under an optical excitation, excitons and free electron-hole (e-h) pairs (continuous spectra) will be created. As the excitation intensity (and thus the e-h density) is increased, the fermionic nature of the system becomes more important, and bosonization requires more bosonic fields. Such a strong-excitation regime has been successfully analyzed without the use of bosonization [9–13]. In a weak-excitation regime, on the other hand, it is expected that the system is well described by a small number of bosonic fields. If this is the case, the bosonic theory will provide a powerful theoretical tool as well as a transparent physical view, as in the case of the d = 1 conductors [1]. To demonstrate the effectiveness of the bosonized theory, optical experiments may be more convenient than the electron transport experiments, because one can easily produce and detect two or more light beams, and obtain rich information from responses to the multi beams. Moreover, one can easily control the polarization of each individual light beam, which gives more detailed information. Recently, by controlling the polarizations of two light beams, Kuwata-Gonokami et al. demonstrated experimentally that the polaritonpolariton scatterings in a quantum well (QW) in an optical micro cavity are well described by a phenomenological bosonic Hamiltonian in the weak-excitation regime [5,14]. Their experiment strongly indicated the validity of a bosonic description for d = 2. However, no 2 theoretical studies were reported which derive their phenomenological Hamiltonian from a microscopic fermionic theory. In this letter, we derive an effective Hamiltonian of excitons from the conventional e-h Hamiltonian. In particular, we will show that the derivation of an effective bosonic theory is nontrivial and not straightforward because a direct bosonization does not give correct results. Our calculation and the agreement with the experimental data [5] directly prove the validity and relevance of a bosonic description for d = 2. Model — We start from a Hamiltonian Heh for an interacting electron-hole system in a QW. Heh = ∑