Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point

We consider the Farey fraction spin chain in an external field \(h\). Using ideas from dynamical systems and functional analysis, we show that the free energy \(f\) in the vicinity of the second-order phase transition is given, exactly, by $$ f \sim \frac t{\log t}-\frac1{2} \frac{h^2}t \quad \text{for} \quad h^2\ll t \ll 1 . $$ Here \(t=\lambda_{G}\log(2)(1-\frac{\beta}{\beta_c})\) is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, \(\lambda_G\). It follows that \(\lambda_G\) determines the amplitude of both the specific heat and susceptibility singularities. To our knowledge, there is only one other microscopically defined interacting model for which the free energy near a phase transition is known as a function of two variables. Our results confirm what was found previously with a cluster approximation, and show that a clustering mechanism is in fact responsible for the transition. However, the results disagree in part with a renormalisation group treatment.