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Communications in Mathematical Physics

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

A Farey Fraction Spin Chain

  • P. Kleban
  • A.E. Özlük

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.

Keywords

Phase Transition Free Energy Chain Length Computer Result Number Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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