Comment on "Integrability of the Rabi model"

TL;DR

This paper challenges Braak's claim by demonstrating that the regular spectrum of the Rabi model can be obtained from a different transcendental function without relying on its discrete symmetry.

Contribution

It introduces an alternative transcendental function $F_0(x)$ to determine the Rabi model's spectrum, bypassing the need for the model's $ ext{Z}_2$-symmetry.

Findings

The spectrum can be derived from $F_0(x)$ without using symmetry.

The proposed method offers a different perspective on the Rabi model's spectral analysis.

Numerical results confirm the validity of the new approach.

Abstract

In his recent letter, Braak suggested that a regular spectrum of the Rabi model was given by the zeros of a transcendental function $G_\pm(x)$ (cf Eqs. (3)-(5) of Ref. [1]) and highlighted the role of the discrete $\mathbb{Z}_2$-symmetry, or parity, in determining $G_\pm(x)$. We show here to the contrary that one can define a transcendental function $F_0(x)$ and obtain the regular spectrum of the Rabi model as the zeros of $F_0(x)$ (see Fig. 1) without ever making use of the underlying $\mathbb{Z}_2$-symmetry of the model.