A Farey Fraction Spin Chain

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 β = 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 (β > 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 β = 2, and suggesting a possible connection with the Riemann ζ–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.