Thermodynamics of the Farey Fraction Spin Chain

We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field h. From rigorous and more heuristic arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a “marginal” field is present) despite the unusual nature of the transition for h = 0, and the presence of long-range forces.