The effective bosonic Hamiltonian for excitons reconsidered

The effective bosonic Hamiltonian for excitons, extensively quoted up to now, cannot be correct because it is (surprisingly) non-Hermitian. The oversight physically originates from the intrinsic difficulty of properly defining electron-hole interactions between excitons when dealing with exchange terms. By using our commutation technique, we show that the fermionic character of the excitons cannot be forced into a dressed Coulomb interaction only: The effective bosonic Hamiltonian must contain purely fermionic terms of the same order as the Coulomb terms. They are necessary to ensure hermiticity, and they do not reduce to a two-body interaction, Pauli exclusion being N-body by essence.