Abstract
The nonlinear dynamics is investigated for a system ofN classical spins. This represents a Hamiltonian system withN degrees of freedom. According to the Liouville theorem, the complete integrability of such a system requires the existence ofN independent integrals of the motion which are mutually in involution. As a basis for the investigation of regular and chaotic spin motions, we have examined in detail the problem of integrability of a two-spin system. It represents the simplest autonomous spin system for which the integrability problem is nontrivial. We have shown that a pair of spins coupled by an anisotropic exchange interaction represents a completely integrable system for any values of the coupling constants. The second integral of the motion (in addition to the Hamiltonian), which ensures the complete integrability, turns out to be quadratic in the spin variables. If, in addition to the exchange anisotropy also singlesite anisotropy terms are included in the two-spin Hamiltonian, a second integral of the motion quadratic in the spin variables exists and thus guarantees integrability, only if the model constants satisfy a certain condition. Our numerical calculations strongly suggest that the violation of this condition implies not only the nonexistence of a quadratic integral, but the nonexistence of a second independent integral of motion in general. Finally, as an example of a completely integrableN-spin system we present the Kittel-Shore model of uniformly interacting spins, for which we have constructed theN independent integrals in involution as well as the action-angle variables explicitly.
This is a preview of subscription content, access via your institution.
References
Lichtenberg, A.J., Lieberman, M.A.: Regular and stochastic motion. Berlin, Heidelberg, New York: Springer 1983
Eilenberger, G.: Solitons: mathematical methods for physicists. Berlin, Heidelberg, New York: Springer 1981
Chaotic Behavior in Quantum Systems: theory and applications. Casati, G. (ed.). New York: Plenum Press 1985
Feshbach, H.: Phys. Today39, (No. 4), 7 (1986)
Müller, G.: Phys. Rev. A (in press)
Feingold, M., Peres, A.: Physica9D, 433 (1983)
Feingold, M. Moiseyev, N., Peres, A.: Phys. Rev. A30, 509 (1984)
Nakamura, K., Nakahara, Y., Bishop, A.R.: Phys. Rev. Lett.54, 861 (1985)
Nakamura, K., Bishop, A.R.: Phys. Rev. B33, 1963 (1986)
Frahm, H., Mikeska, H.J.: Z. Phys. B-Condensed Matter60, 117 (1985)
Månson, M.: Phys. Rev. B12, 400 (1975)
Arnold, V.I.: Mathematical Methods of Classical Mechanics. p. 271. Berlin, Heidelberg, New York: Springer 1978
Holt, C.R.: J. Math. Phys.23, 1037 (1982)
Hietarinta, J.: Phys. Lett.96A, 273 (1983); Phys. Rev. A28, 3670 (1983); Phys. Rev. Lett.52, 1057 (1984)
Fordy, A.P.: Phys. Lett.97 A, 21 (1983)
This deviates from the standard definition of a Poincaré surface of section in that the latter discriminates between points of intersection in which the trajectory crosses this hyperplane from one side or the other
Kittel, C., Shore, H.: Phys. Rev.138, A 1165 (1965) See also Cooke, J.F.: Phys. Rev.141, 390 (1966)
Dekeyser, R., Lee, M.H.: Phys. Rev. B19, 265 (1979)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Magyari, E., Thomas, H., Weber, R. et al. Integrable and nonintegrable classical spin clusters. Z. Physik B - Condensed Matter 65, 363–374 (1987). https://doi.org/10.1007/BF01303725
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01303725
Keywords
- Anisotropy
- Hamiltonian System
- Spin System
- Model Constant
- Spin Motion