# Symmetry of asymmetric quantum Rabi models

## TL;DR

This paper uses Lie algebra representations to analyze the eigenstates of the asymmetric quantum Rabi model, revealing symmetry properties and level crossings related to the symmetry-breaking parameter.

## Contribution

It introduces a Lie algebraic framework for the asymmetric quantum Rabi model, proving level crossings at specific parameter values and conjecturing their occurrence for a broader set.

## Key findings

- Proves existence of level crossings at  = 1/2
- Conjectures level crossings for all   Z
- Identifies -symmetry in the spectrum for certain parameters

## Abstract

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.