New criteria for bosonic behavior of excitons

Dimensional arguments lead to say that $N$ excitons in a sample of volume $V$ behave as bosons for $\lambda N a_x^3/V \ll 1$, $a_x$ being the exciton radius and $\lambda$ a dimensionless factor. The Mott criterion, which is based on the disappearance of all exciton bound states because of screening, gives $\lambda \simeq 1$. Since excitons feel each other through both Coulomb interaction and Pauli exclusion between their electrons and holes, criteria based on the underlying fermionic character of the exciton should be even more relevant. Two significant quantities are the boson departure $1- [B,B^{\dagger}]$ and boson number $B^{\dagger} B$, where $B^{\dagger}$ is the exciton creation operator. Their expectation values in the $N$-exciton state are close to their boson values for $\lambda \sim 100$ and $\lambda \sim 50$, respectively. By contrast, the expectation value of $B^N (B^{\dagger})^N$ in the vacuum state is close to $N!$ for $25 N^2 a_x^3/V \ll 1$ only. This surprising $N^2$ dependence comes from the intrinsic $N$-body character of Pauli exclusion. Consequences of these new criteria on the Bose condensation of excitons are discussed.