Integrability vs exact solvability in the quantum Rabi and Dicke models
  The Rabi model describes the simplest interaction between light and matter via a two-level quantum system interacting with a bosonic field. We demonstrate that the fully quantised version of the Rabi model is integrable in the Yang-Baxter sense at two parameter values. The model is argued to be not Yang-Baxter integrable in general. This is in contrast to the claim that the quantum Rabi model is integrable based on a phenomenological criterion of quantum integrability not presupposing the existence of a set of commuting operators. Similar Yang-Baxter integrable points are identified for the generalised Rabi model and the fully quantised Dicke model. The integrable points have particular implications for the level statistics of the Dicke model. 
Murray T. Batchelor and Huan-Qiang Zhou
quant-ph