Renormalized Bosonic Interaction of Excitons

TL;DR

This paper derives a renormalized bosonic Hamiltonian for 1s excitons with spin in two dimensions, revealing how higher exciton states influence interactions and optical responses in semiconductors.

Contribution

It introduces a projection-based method to derive an effective bosonic Hamiltonian from the electron-hole Hamiltonian, highlighting the renormalization of exciton interactions.

Findings

Interaction between different spin excitons arises from higher state renormalization.

Renormalization significantly alters same-spin exciton interactions.

The theory underpins polarization-dependent optical phenomena in semiconductors.

Abstract

An effective bosonic Hamiltonian of $1s$ excitons with ``spin'' degrees of freedom in two dimension is obtained through a projection procedure, starting from a conventional electron-hole Hamiltonian ${\cal H}_{eh}$. We first demonstrate that a straightforward transformation of ${\cal H}_{eh}$ into a Hamiltonian of bosonic excitons does not give the two-body interaction between an ``up-spin'' exciton and a ``down-spin'' exciton, which are created by the left- and right-circularly polarized light beams, respectively. We then show that this interaction is generated through a projection procedure onto the subspace spanned by $1s$ excitons, as a renormalization effect coming from higher exciton states. The projection also renormalizes the interaction between $1s$ excitons with the same spins by a large amount. These renormalization effects are crucial for the polarization dependence of the optical responses from semiconductors. The present theory gives the microscopic foundation of the phenomenology that was successfully applied to the analysis of four-wave mixing experiments in GaAs quantum wells strongly coupled to the radiation field in a high-Q micro cavity.