On the Existence of Monodromies for the Rabi model

TL;DR

This paper investigates the monodromy properties of the Rabi model's eigenvalue problem, linking monodromy data to the model's parameters through isomonodromy and Painlevé V, to systematically determine its spectrum.

Contribution

It introduces a novel approach using isomonodromy and Painlevé V tau-functions to analyze the Rabi model's spectral quantization via monodromy data.

Findings

Monodromy data can be expressed in terms of model parameters.

The approach provides a systematic way to determine the spectrum.

Monodromy analysis relates to the Stokes phenomenon and irregular singularities.

Abstract

We discuss the existence of monodromies associated with the singular points of the eigenvalue problem for the Rabi model. The complete control of the full monodromy data requires the taming of the Stokes phenomenon associated with the unique irregular singular point. The monodromy data, in particular the composite monodromy, are written in terms of the parameters of the model via the isomonodromy method and the tau-function of the Painlev\'e V. These data provide a systematic way to obtain the quantized spectrum of the Rabi model.