A Farey Fraction Spin Chain

TL;DR

This paper introduces a novel number-theoretic spin chain based on Farey fractions, analyzes its thermodynamics, phase transition, and connections to number theory, including conjectures related to the Riemann zeta function.

Contribution

It presents a new Farey fraction-based spin chain model, proves the existence of a phase transition, and explores its deep number-theoretic connections and properties.

Findings

Existence of a phase transition at inverse temperature beta=2.

Low-temperature state is completely magnetized for long chains.

Interaction coefficients are mostly ferromagnetic, except the constant term.

Abstract

We introduce a new number-theoretic spin chain and explore its thermodynamics and connections with number theory. The energy of each spin configuration is defined in a translation-invariant manner in terms of the Farey fractions, and is also expressed using Pauli matrices. We prove that the free energy exists and exhibits a unique phase transition at inverse temperature beta = 2. The free energy is the same as that of a related, non translation-invariant number-theoretic spin chain. Using a number-theoretic argument, the low-temperature (beta > 3) state is shown to be completely magnetized for long chains. The number of states of energy E = log(n) summed over chain length is expressed in terms of a restricted divisor problem. We conjecture that its asymptotic form is (n log n), consistent with the phase transition at beta = 2, and suggesting a possible connection with the Riemann zeta function. The spin interaction coefficients include all even many-body terms and are translation invariant. Computer results indicate that all the interaction coefficients, except the constant term, are ferromagnetic.