Abstract
We have studied the Frenkel-Kontorova (FK) model with Toda interactions. The phase diagram is found to be asymmetric due to the exponential form of the Toda interaction. The reflection symmetry observed in the standard FK model is broken here. The singularity spectrum and the generalized dimension are calculated and their dependence on the nonlinearity parameter is discussed. The critical exponents of the gap in the phonon spectrum, the correlation length, and the Peierls-Nabarro barrier are found to be the same as those in the standard FK model and are independent of the nonlinearity parameter.
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References
Y. I. Frenkel and T. Kontorova,Zh. Eksp. Tear. Fiz. 8:1340 (1938).
P. Bak,Rep. Prog. Phys. 45:587 (1982).
G. Gruner,Commun. Solid State Phys. 10:183 (1983).
I. E. Dzyaloshinskii,Zh. Eksp. Teor. Fiz. 46:1420 (1964);Sov. Phys. JETP 23:960 (1964).
M. S. Dresselhaus and G. Dresselhaus,Adv. Phys. 30:139 (1981).
D. Schechtman, I. Blech, D. Gratios, and J. W. Cahn,Phys. Rev. Lett. 53:1951 (1984).
F. Nabarro,Theory of Crystal Dislocation (Clarendon, Oxford, 1967).
S. C. Ying,Phys. Rev. B 3:4160 (1971).
M. Toda,J. Phys. Soc. Jpn. 22:431 (1967).
A. Milchev,Phys. Rev. B 33:2062 (1986); A. Milchev and G. M. Mazzucchelli,Phys. Rev. B 38:2808 (1988).
M. Peyrard and S. Aubry,J. Phys. C 16:1593 (1983); S. Aubry, inProceedings of the Les Hauches Lectures, R. Balian and R. Stora, eds. (North-Holland, Amsterdam, 1981).
S. Aubry,Physica 7D:240 (1983).
S. Aubry and P. Y. Le Daeron,Physica 8D:381 (1983).
O. Biham and D. Mukamel,Phys. Rev. A 39:5326 (1989).
H. J. Schellnhuber, H. Urbschat, and A. Block,Phys. Rev. A 33:2856 (1986); H. J. Schellnhuber, H. Urbschat, and J. Wilbrink,Z. Phys. B 80:305 (1990).
I. Niven,Diophantine Approximations (Wiley, New York, 1960).
I. Niven and H. S. Zuckerman,An Introduction to the Theory of Numbers (Wiley, New York, 1980).
S. Aubry, inSolitons and Condensed Matter Physics, A. R. Bishop and T. Schneider, eds. (Springer-Verlag, Berlin, 1979).
L. de Seze and S. Aubry,J. Phys. C 17:389 (1984).
T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman,Phys. Rev. A 33:1141 (1986).
J. M. Greene,J. Math. Phys. 20:1183 (1979).
S. J. Shenker and L. P. Kadanoff,J. Stat. Phys. 27:631 (1982).
J. M. Greene, R. S. Mackay, F. Vivaldi, and M. J. Feigenbaum,Physica 3D:468 (1981).
H. Johannesson, B. Schaub, and H. Suhl,Phys. Rev. B 37:9625 (1988).
R. S. MacKay,Physica D 50:71 (1991).
J. Shi and B. Hu,Phys. Rev. A 45:5455 (1992).
J. M. Greene, H. Johannesson, B. Schaub, and H. Suhl,Phys. Rev. A 36:5858 (1987).
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Lin, B., Hu, B. Frenkel-Kontorova model with toda interactions. J Stat Phys 69, 1047–1068 (1992). https://doi.org/10.1007/BF01058761
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DOI: https://doi.org/10.1007/BF01058761