Communications in Mathematical Physics

, Volume 203, Issue 3, pp 635–647 | Cite as

A Farey Fraction Spin Chain

  • P. Kleban
  • A.E. Özlük


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.


Phase Transition Free Energy Chain Length Computer Result Number Theory 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • P. Kleban
    • 1
  • A.E. Özlük
    • 2
  1. 1.Department of Physics and Astronomy and Laboratory for Surface Science and Technology, University of Maine, Orono, ME 04469, USA. E-mail: kleban@maine.eduUS
  2. 2.Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USAUS

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