A Farey Fraction Spin Chain |
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Authors: | P Kleban AE Özlük |
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Institution: | (1) Department of Physics and Astronomy and Laboratory for Surface Science and Technology, University of Maine, Orono, ME 04469, USA. E-mail: kleban@maine.edu, US;(2) Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA, US |
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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.
Received: 20 August 1998/ Accepted: 17 December 1998 |
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