On the Integrability of the Rabi Model

TL;DR

This paper analytically determines the spectrum of the Rabi model for all parameters, introduces a criterion linking symmetry groups to integrability, and presents a minimal non-integrable quantum system.

Contribution

It provides an exact spectrum for the Rabi model, proposes a new criterion for quantum integrability based on symmetry groups, and introduces a minimal non-integrable quantum system.

Findings

Exact spectrum of the Rabi model for arbitrary parameters

Finite symmetry groups can imply integrability without conserved charges

A minimal non-integrable quantum system without symmetries

Abstract

The exact spectrum of the Rabi hamiltonian is analytically found for arbitrary coupling strength and detuning. I present a criterion for integrability of quantum systems containing discrete degrees of freedom which shows that in this case a finite symmetry group may entail integrability, even without the presence of conserved charges beyond the hamiltonian itself. Moreover, I introduce and solve a natural generalization of the Rabi model which has no symmetries and is therefore probably the smallest non-integrable physical system.