Semiclassical mechanics of a nonintegrable spin cluster

Physical Review B - Condensed Matter and Materials Physics

Volumn 60, Issue 22, 1999, Pages 15179-15186

  Houle, P A ^a   Zhang, N G ^a   Henley, C L ^a  

^a Laboratory of Atomic and Solid State Physics and Sibley School of Mechanical and Aerospace Engineering   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001095233     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.60.15179     Document Type: Article

References (32)

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11. This is an example of the general phenomenon of symmetry-breaking bifurcations of orbits. For a systematic discussion, see M. A. M. de Aguiar and C. P. Malta, Ann. Phys. (N.Y.) 180, 167 (1987); our case corresponds to (b) in their Table I.
12. As Ref. 13 notes, the third-order expansion of the Hamiltonian near a ground state is the same for our model (in the right coordinates) as for the famous Hénon-Heiles model [M. Hénon and C. Heiles, Astron. J. 69, 73 (1964)]. Hence, we see similar behaviors near the limit [e.g., the dependence on excitation energy of the frequency splitting between orbits (b) and (c)].
13. P. A. Houle, Ph.D. thesis, Cornell University, 1998.
14. P. A. Houle and C. L. Henley (unpublished).
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16. (Formula presented) contains a phase factor, (Formula presented) where (Formula presented) is the Maslov index and depends on the topology of the linearized dynamics near the orbit; see J. M. Robbins, Nonlinearity 4, 343 (1991). As we do not consider amplitude or phase, the Maslov index is irrelevant for this paper.
17. Equation (7) is for illustration. The square window aggravates artifacts of the Fourier transform that could be reduced by using a different window function (see Ref. 19).
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22. In fact, the classical dynamics of our system do scale if one rescales the spin length S simultaneously. However, in contrast to the mentioned scaling systems, S in a spin system is not just a numerical parameter, but is a discrete quantum number. In effect, constructing an orbit spectrum by varying S is a mixture of scaled-energy spectroscopy and inverse-(Formula presented) spectroscopy [see J. Main, C. Jung, and H. S. Taylor, J. Chem. Phys. 107, 6577 (1997)]. Such an approach would give poor energy resolution in our system, since we can perform diagonalizations only for discrete values of S in a limited range.
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25. Due to this slowness, any convention to define a Poincaré section gives a topologically equivalent picture. Such pictures for the AFM or FM limits are presented in Ref. 12, Figs. 4.17–4.19, and in Ref. 13.
26. We expect I could be calculated explicitly by some perturbation theory; it is closely related to the approximate invariants provided by the Gustavson normal form construction, or by averaging methods: see R. H. Rand and D. Armbruster, Perturbation Methods, Bifurcation Theory, and Computer Algebra (Springer, New York, 1987).
27. Intriguingly, Fig. 66 is also equivalent to the phase sphere in Fig. 1 of Ref. 5, for an antiferromagnetically coupled cluster of four spins (or four sublattices of spins). The figures look different until one remembers that in Ref. 5, all points related by twofold rotations around the x, y, or z axes are to be identified; really the sphere of Ref. 5 has just two distinct octants, equivalent to the “northern” and “southern” hemispheres of Fig. 66 in this paper. The special points labeled (Formula presented) (Formula presented) and (Formula presented) in Ref. 5 correspond, respectively, to the two “poles,” plus the three stable and three unstable points on the “equator,” in Fig. 66.
28. Observe that the “north pole” symmetry axis in Fig. 66 is the z axis. On the other hand, the axis of the polar coordinates (Formula presented) in Eq. (10) is the x axis and passes through two stationary points of types (b) and (c) in the figure. The equator in Fig. 66 includes all motions with (Formula presented) or (Formula presented) i.e., the oscillations are in phase.
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31. This is plausible in view of the topological equivalence of the reduced dynamics, suggested by Ref. 26. However, this remains a speculation since we have not performed a microscopic calculation of the topological phase for the three-spin problem, analogous to the calculation in Ref. 5. Note that, in comparing to that work, all singlet levels for a given spin length in Ref. 5 correspond to just one polyad cluster of levels in the present problem.
32. In terms of the two oscillators, (Formula presented) implies that polyads are allowed only with a net even number of quanta; we do not yet understand this constraint.