Effective Hamiltonian for Excitons with Spin Degrees of Freedom

Abstract

Starting from the conventional electron-hole Hamiltonian ${\mathcal{H}}_{eh}$, we derive an effective Hamiltonian ${\tilde{\mathcal{H}}}_{1s}$ 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 ${\mathcal{H}}_{eh}$ does not give the correct form of ${\tilde{\mathcal{H}}}_{1s}$, 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.

12 pages, no figures, RevTeX

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