Effective Hamiltonian for excitons with spin degrees of freedom

Starting from the conventional electron-hole Hamiltonian H-eh, we derive an effective Hamiltonian H-1s 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 H-eh does not give the correct form of (H) over bar(1s), which we obtain by a projection onto the subspace spanned by the Is 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.