A Farey Fraction Spin Chain

Дата и время публикации : 1998-08-17T21:49:22Z

Авторы публикации и институты :
P. Kleban (University of Maine)
A. E. Ozluk (University of Maine)

Ссылка на журнал-издание: Commun. Math. Phys. (1999) 203, pp. 635-647
Коментарии к cтатье: 15 pages + 5 figures, postscript. Contact: kleban@maine.edu
Первичная категория: cond-mat.stat-mech

Все категории : cond-mat.stat-mech, math.NT

Краткий обзор статьи: 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.

Category: Physics