Validity of Wang–Hioe operator-series convergence assumptions underlying partition-function decoupling

Establish whether the specific convergence properties of operator series (assumptions i and ii on page 832 of Wang and Hioe, Phys. Rev. A 7, 831–836 (1973)) hold in the thermodynamic limit for a system of N atoms interacting with a single-mode electromagnetic field in the dipole approximation, which are used to argue that the partition function is independent of the field–matter coupling and thus precludes field condensation.

Background

A widely cited argument against superradiant phase transitions in the dipole approximation (often referred to as a no-go theorem) asserts that, in the thermodynamic limit, the partition function becomes independent of the field–matter coupling, thereby preventing coherent field condensation. That argument (as presented by Bialynicki-Birula and Rzazewski, 1979) relies on a result by Wang and Hioe (1973) that invokes specific convergence properties of operator series (their assumptions i and ii on page 832).

In this paper, the authors construct explicit coherent states with energy lower than any zero-field configuration, countering the no-go conclusion. Nonetheless, they note that the operator-series convergence assumptions used in the decoupling argument have not been proven valid, leaving their status an unresolved technical question in the rigorous treatment of the thermodynamic limit for the atom–field system in the dipole approximation.