Symmetry of asymmetric quantum Rabi models

The aim of this paper is a better understanding for the eigenstates of the asymmetric quantum Rabi model by Lie algebra representations of $\mathfrak{sl}_2$. We define a second order element of the universal enveloping algebra $\mathcal{U}(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2(\mathbb{R})$, which, through the action of a certain infinite dimensional representation of $\mathfrak{sl}_2(\mathbb{R})$, provides a picture of the asymmetric quantum Rabi model equivalent to the one drawn by confluent Heun ordinary differential equations. Using this description, we prove the existence of level crossings in the spectral graph of the asymmetric quantum Rabi model when the symmetry-breaking parameter $\epsilon$ is equal to $\frac12$, and conjecture a formula that ensures likewise the presence of level crossings for general $\epsilon \in \frac12\mathbb{Z}$. This result on level crossings was demonstrated numerically by Li and Batchelor in 2015, investigating an earlier empirical observation by Braak (2011). The first analysis of the degenerate spectrum was given for the symmetric quantum Rabi model by Ku\'s in 1985. In our picture, we find a certain reciprocity (or $\mathbb{Z}_2$-symmetry) for $\epsilon \in \frac12\mathbb{Z}$ if the spectrum is described by representations of $\frak{sl}_2$. We further discuss briefly the non-degenerate part of the exceptional spectrum from the viewpoint of infinite dimensional representations of $\mathfrak{sl}_2(\mathbb{R})$ having lowest weight vectors.