{"paper":{"title":"Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.DS","math.MP","math.NT"],"primary_cat":"cond-mat.stat-mech","authors_text":"J. Fiala, O.F. Bandtlow, P. Kleban, T. Prellberg","submitted_at":"2009-09-15T20:36:41Z","abstract_excerpt":"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 $$ f \\sim \\frac t{\\log t}-\\frac1{2} \\frac{h^2}t \\quad \\text{for} \\quad h^2\\ll t \\ll 1 . $$\n  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 a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.2878","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}