Phase diagram for the Gaudin model

Gaudin has determined the partition function at special temperatures for a lattice gas of particles confined to a ring and interacting with a potential that is logarithmic in the interparticle distance. This model is also relevant to the understanding of long-range order in a discrete version of the Sutherland model. Gaudin found that the pressure changes sign at β=2; it will be shown here that no phase transition is associated with this, and that the anomaly is equally present in the continuum version of the problem discussed by Dyson. The partition function and structure factor will be computed for a finite system of up to 32 sites for more general β, and the results interpreted in terms of the theory of a quantum liquid in the presence of an external periodic potential.