Cluster Approximation for the Farey Fraction Spin Chain

TL;DR

This paper introduces a cluster approximation method to analyze the Farey fraction spin chain in an external field, deriving phase boundaries and free energy scaling, and confirming known asymptotic behaviors at zero field.

Contribution

It develops an effective cluster energy approximation for the Farey spin chain, valid for large clusters, to study phase transitions and free energy scaling.

Findings

Reproduces known asymptotic free energy at zero field

Derives phase boundaries for non-zero external field

Results align with mean field and renormalization group predictions

Abstract

We consider the Farey fraction spin chain in an external field $h$. Utilising ideas from dynamical systems, the free energy of the model is derived by means of an effective cluster energy approximation. This approximation is valid for divergent cluster sizes, and hence appropriate for the discussion of the magnetizing transition. We calculate the phase boundaries and the scaling of the free energy. At $h=0$ we reproduce the rigorously known asymptotic temperature dependence of the free energy. For $h \ne 0$, our results are largely consistent with those found previously using mean field theory and renormalization group arguments.