Semiclassical quantization of localized lattice solitons

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 68, Issue 5, 2003, Pages 521091-5210915

  Schulman, L S ^a  

^a CLARKSON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CRYSTALS; DOPING (ADDITIVES); ERROR ANALYSIS; FUNCTIONS; INTEGRATION; INTERPOLATION; POTASSIUM COMPOUNDS; TOPOLOGY;

SEMICLASSICAL QUANTIZATION; WAVE FUNCTIONS;

SOLITONS;

EID: 0942300248     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (66)

1. L. S. Schulman, E. Mihóková, A. Scardicchio, P. Facchi, M. Nikl, K. Polák, and B. Gaveau, Phys. Rev. Lett. 88, 224101 (2002).
2. E. Fermi, J. Pasta, and S. Ulam, Technical Report, Los Alamos 1955, reprinted in Enrico Fermi, Collected Papers, Volume II (University of Chicago Press, Chicago, 1965), Article 266, p. 978.
3. E. Fermi, J. Pasta, and S. Ulam, Technical Report, Los Alamos 1955, reprinted in Enrico Fermi, Collected Papers, Volume II (University of Chicago Press, Chicago, 1965), Article 266, p. 978.
4. A. Einstein, Verh. Dtsch. Phys. Ges. 19, 82 (1917).
5. L. Brillouin, J. Phys. Radium 7, 353 (1926).
6. J. B. Keller, Ann. Phys. (N.Y.) 4, 180 (1958).
7. W. Eastes and R. A. Marcus, J. Chem. Phys. 61, 4301 (1974).
8. D. W. Noid and R. A. Marcus, J. Chem. Phys. 62, 2119 (1975).
9. D. W. Noid, M. L. Koszykowski, and R. A. Marcus, J. Chem. Phys. 67, 404 (1977).
10. D. W. Noid and R. A. Marcus, J. Chem. Phys. 67, 559 (1977).
11. K. Sohlberg, R. E. Tuzun, B. G. Sumpter, and D. W. Noid, Phys. Rev. A 57, 906 (1998).
12. I. C. Percival, in Advances in Chemical Physics, edited by I. Prigogine and S. A. Rice (Wiley-Interscience, New York, 1977).
13. C. C. Martens and G. S. Ezra, J. Chem. Phys. 83, 2990 (1985).
14. C. C. Martens and G. S. Ezra, J. Chem. Phys. 86, 279 (1987).
15. C. C. Martens and G. S. Ezra, J. Chem. Phys. 87, 284 (1987).
16. G. S. Ezra, K. Richter, G. Tanner, and D. Wintgen, J. Phys. B 24, L413 (1991).
17. L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981), Wiley Classics edition, 1996.
18. L. S. Schulman, J. Math. Phys. 12, 304 (1971).
19. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983).
20. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978).
21. K. Polák, M. Nikl, and E. Mihóková, J. Lumin. 54, 189 (1992).
22. Actually there are three states, but two are nearly degenerate and the distinction can be ignored.
23. B. Gaveau, E. Mihóková, M. Nikl, K. Polák, and L. S. Schulman, Phys. Rev. B 58, 6938 (1998).
24. B. Gaveau, E. Mihóková, M. Nikl, K. Polák, and L. S. Schulman, J. Lumin. 92, 311 (2001).
25. E. Mihóková, L. S. Schulman, M. Nikl, B. Gaveau, K. Polák, K. Nitsch, and D. Zimmerman, Phys. Rev. B 66, 155102 (2002).
26. In [22-24] we used a continuous relaxation process, A sudden but randomly distributed release or a release depending on a small number of random events were considered, but provided fits of inferior quality.
27. See, for example, .[64], pages 311 and 321 for these results. Note that the ion-separation distance d, has been inserted in our formula, since in [64] it is taken to be unity.
28. Note that "d" is being used here for the nearest-neighbor distance in the fee lattice; this is half of what most authors call the lattice spacing.
29. L. S. Schulman, E. Mihóková, M. Nikl, K. Polák, B. Gaveau, and W. Williams, Clarkson Institute of Statistical Physics technical report.
30. A. J. Sievers and S. Takeno, Phys. Rev. Lett. 61, 970 (1988).
31. S. A. Kiselev, S. R. Bickham, and A. J. Sievers, Phys. Rev. B 50, 9135 (1994).
32. J. B. Page, Phys. Rev. B 41, 7835 (1990).
33. R. Reigada, A. Sarmiento, and K. Lindenberg, Phys. Rev. E 64, 066608 (2001).
34. R. Dusi, R. Viliani, and M. Wagner, Phys. Rev. B 54, 9809 (1996).
35. V. Fleurov, R. Schilling, and S. Flach, Phys. Rev. E 58, 339 (1998).
36. Y. Zolotaryuk, S. Flach, and V. Fleurov, Phys. Rev. B 63, 214422 (2001).
37. W. Z. Wang, J. T. Gammel, A. R. Bishop, and M. I. Salkola, Phys. Rev. Lett. 76, 3598 (1996).
38. V. V. Konotop and S. Takeno, Phys. Rev. E 63, 066606 (2001).
39. B. Horovitz, A. R. Bishop, and S. R. Phillpot, Phys. Rev. Lett. 60, 2210 (1988).
40. B. Horovitz, A. R. Bishop, and S. R. Phillpot, Phys. Rev. A 40, 1240 (1989).
41. S. R. Phillpot, A. R. Bishop, and B. Horovitz, J. Phys. C 20, L485 (1987).
42. U. T. Schwarz, L. Q. English, and A. J. Sievers, Phys. Rev. Lett. 83, 223 (1999).
43. B. I. Swanson, J. A. Brozik, S. P. Love, G. F. Strouse, A. P. Shreve, A. R. Bishop, W.-Z. Wang, and M. I. Salkola, Phys. Rev. Lett. 82, 3288 (1999).
44. By saying that the orbit occupies a two-dimensional torus, I mean that only two of the J's for the corresponding orbit are nonzero.
45. M. Berry, in Chaotic Behavior of Deterministic Systems, edited by G. Iooss, R. Helleman, and R. Stora (North-Holland, Amsterdam, 1983).
46. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
47. K. Hotta and K. Takatsuka, J. Phys. A 36, 4785 (2003).
48. A similar derivation is given in [65], The extension of the EBK method given in the present paper based on multiple connectedness (of the underlying coordinate space) is not considered in [65].
49. Note that terms in the polynomials are taken to be Hermitian with real coefficients. Otherwise i(pq-qp) provides an immediate counterexample.
50. n values. As to KAM's dense singular structure in phase space due to frequency commensurability, this casts a shadow on the entire enterprise. Perhaps one should aim for a nonuniform approximation in which one stops KAM iterations at a finite stage, avoiding the singular structure, but nevertheless achieving sufficient accuracy in terms of hℏ.
51. 2) assertion was established for polynomials, but for a power series errors might not converge. A second defect of our approach is that it does not go beyond 0(hℏ).
52. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1980).
53. L. S. Schulman, Phys. Rev. 176, 1558 (1968).
54. Higher harmonics can also appear, due to nonlinearity. Their low amplitude in our system indicates that linear combinations of the p 's and q's should show simple behavior. We did not pursue this (analytically, for example) because we sought general numerical methods, valid without this property.
55. Other cases: There may be no intersection - no surprise, since the torus is of limited extent and need not intersect the plane at all. Or the line may enter on one side and leave on an adjacent one (or may touch a single corner - a degenerate version of this). This represents a closed curve (or a point on the surface of the torus), but a curve that (on the torus) is topologically equivalent to a point. Hence the action evaluated along this curve will be zero. A nongeneric possibility is for the line to pass through two corners, but this is an unwanted and unlikely complication.
56. L. S. Schulman, Phys. Status Solidi B 237, 124 (2003).
57. M. C. Gutzwiller, J. Math. Phys. 12, 343 (1971).
58. M. C. Gutzwiller, J. Math. Phys. 11, 1791 (1970).
59. K. Takatsuka, J. Comput. Phys. 102, 374 (1992).
60. 1). Energies reported here are energies above the equilibrium energy just described.
61. This correlates with the form of the excitation, just as optical and acoustic phonons are distinguished.
62. The calculation is entirely classical; the hℏ enters only because the action (j) values we use are spaced by hℏ.
63. A. M. Karo and J. R. Hardy, Phys. Rev. 129, 2024 (1963).
64. S. R. Bickham and A. J. Sievers, Phys. Rev. B 43, 2339 (1991).
65. M. Marder, Condensed Matter Physics (Wiley, New York, 2000).
66. H. Hasegawa, M. Robnik, and G. Wunner, Suppl. Prog. Theor. Phys. 98, 198 (1989).