Cluster Approximation for the Farey Fraction Spin Chain

Details Cluster Approximation for the Farey Fraction Spin Chain Cluster Approximation for the Farey Fraction Spin Chain

2005-07-28

arxiv.org/abs/cond-mat/0507662v1

J. Stat. Phys. (2006) 123, pp. 455-471

Physics

Condensed Matter

Statistical Mechanics

17 pages, 3 figures

Scientific paper

10.1007/s10955-006-9034-x

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.

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