Abstract

The Gaudin model consists of a collection of spins interacting with a single oscillator. It is of physical interest in many fields such as quantum optics or cold atomic gases. Besides, it is known to be integrable in both its classical and quantum-mechanical versions. We will focuss on the behavior of this model system in the vicinity of stable and unstable equilibrium points. We show that the latter present a topological obstruction (called monodromy) to the existence of action-angle coordinates in any phase-space neighbourhood containing them. At the quantum level, this phenomenon is reflected by the presence of a dislocation in the lattice of joint eigenvalues of the mutually commuting Hamiltonians. It also induces a non-trivial braiding of the roots of the Bethe-Ansatz equations in the complex plane, if these joint eigenvalues are varied along a closed path encircling the critical value. This shows that, even in an integrable system, the notion of a global “quantum number” can be problematic.