The Farey Fraction Spin Chain

Peter Kleban, University of Maine

Abstract:

The Farey Fraction Spin Chain is a set of one-dimensional statistical mechanical models built on the Farey fractions (modified Farey sequence). These models lie between statistical mechanics and number theory, and are of interest in both areas. A direct connection to dynamical systems is used to prove that the models rigorously exhibit a (barely) second-order phase transition. Additionally, one may calculate certain correlation functions. The phase diagram, including an external magnetic field, is determined by means of renormalization group and also via a (dynamical system inspired) cluster approximation. The results, interestingly, are almost consistent. Examination of the partition function at the critical point suggests a subtle, apparently new, property of the Farey fractions. There are also rigorous results by number theorists for the "density of states" and a close connection to the Lewis three- term equation, extensively studied in the theory of the Selberg zeta- function.