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 a phase transition occurs for positive inverse temperature β= 2. The free energy is the same as that of 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.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 20 August 1998/ Accepted: 17 December 1998
Rights and permissions
About this article
Cite this article
Kleban, P., Özlük, A. A Farey Fraction Spin Chain. Comm Math Phys 203, 635–647 (1999). https://doi.org/10.1007/s002200050629
Issue Date:
DOI: https://doi.org/10.1007/s002200050629