A Farey Fraction Spin Chain
| dc.creator | Kleban, P. | |
| dc.creator | Ozluk, A. E. | |
| dc.date | 1998-08-17 | |
| dc.date.accessioned | 2026-07-07T08:11:37Z | |
| dc.date.available | 2026-07-07T08:11:37Z | |
| dc.description | 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. | |
| dc.description | 15 pages + 5 figures, postscript. Contact: kleban@maine.edu | |
| dc.identifier | https://arxiv.org/abs/cond-mat/9808182 | |
| dc.identifier | http://arxiv.org/abs/cond-mat/9808182 | |
| dc.identifier | Commun. Math. Phys. (1999) 203, pp. 635-647 | |
| dc.identifier | doi:10.1007/s002200050629 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132174 | |
| dc.subject | Statistical Mechanics | |
| dc.subject | Number Theory | |
| dc.title | A Farey Fraction Spin Chain | |
| dc.type | text |