Physics – Condensed Matter – Statistical Mechanics
Scientific paper
1998-08-17
Commun. Math. Phys. (1999) 203, pp. 635-647
Physics
Condensed Matter
Statistical Mechanics
15 pages + 5 figures, postscript. Contact: kleban@maine.edu
Scientific paper
10.1007/s002200050629
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 beta = 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 (beta > 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 beta = 2, and suggesting a possible connection with the Riemann zeta 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.
Kleban Peter
Ozluk Ali E.
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