Cluster Approximation for the Farey Fraction Spin Chain

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.

17 pages, 3 figures

References in corpus (5)

• On the spectrum of Farey and Gauss maps
• A Farey Fraction Spin Chain
• The Phase Transition in Statistical Models Defined on Farey Fractions
• Generalized Number Theoretic Spin Chain-Connections to Dynamical Systems and Expectation Values
• Thermodynamics of the Farey Fraction Spin Chain

Cited by in corpus (5)

• A multidimensional continued fraction generalization of Stern's diatomic sequence
• Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point
• A thermodynamic classification of pairs of real numbers via the Triangle Multi-dimensional continued fraction
• Wild attractors and thermodynamic formalism
• A Thermodynamic Classification of Real Numbers