Effective bosonic Hamiltonian for excitons: A too naive concept

Excitons, being made of two fermions, may appear from far as bosons. Their close-to-boson character is however quite tricky to handle properly. Using our commutation technique especially designed to deal with interacting close-to-boson particles, we here calculate the exact expansion in Coulomb interaction of the exciton-exciton correlations, and show that a naive effective bosonic Hamiltonian for excitons cannot produce these X-X correlations correctly.