Generalized Number Theoretic Spin Chain-Connections to Dynamical Systems and Expectation Values

Abstract

We generalize the number theoretic spin chain, a one-dimensional statistical model based on the Farey fractions, by introducing a parameter \(x \geqslant 0\). This allows us to write recursion relations in the length of the chain. These relations are closely related to the Lewis three-term equation, which is useful in the study of the Selberg ζ-function. We then make use of these relations and spin orientation transformations. We find a simple connection with the transfer operator of a model of intermittency in dynamical systems. In addition, we are able to calculate certain spin expectation values explicitly in terms of the free energy or correlation length. Some of these expectation values appear to be directly connected with the mechanism of the phase transition