Thermodynamics of the Farey Fraction Spin Chain

TL;DR

This paper investigates the thermodynamics and phase transitions of the Farey fraction spin chain model, extended with an external field, using rigorous and renormalization group methods to analyze its phase diagram and critical behavior.

Contribution

It introduces an external field to the Farey fraction spin chain and rigorously analyzes the resulting phase diagram and transition behavior using advanced theoretical methods.

Findings

Determined the phase diagram of the extended model.

Identified the nature of phase transitions with the external field.

Results align with scaling theory despite unusual zero-field transition.

Abstract

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 renormalization group 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.