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研究生:連峻儀
研究生(外文):Xanatos Lien
論文名稱:海森堡模型的非線性研究
論文名稱(外文):the Nonlinear Research of Heisenberg Model
指導教授:張為民張為民引用關係
指導教授(外文):Wei-Min Zhang
學位類別:碩士
校院名稱:國立成功大學
系所名稱:物理學系碩博士班
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:中文
論文頁數:36
中文關鍵詞:相變繞數週期性魔梯
外文關鍵詞:Frenkel-Kontorova ModelCantor Functiondevil's staircaseCantor SetStandard Mappingwinding numberground stateANNNI Model
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非線性系統中魔梯現象在許多本質上完全不同的系統中都被觀察到,
例如Frenkel-Kontorova模型的Ground States與ANNNI模型中自旋狀態隨不同參數產生的相變。
本文將選擇一個海森堡模型,
能夠與Frenkel-Kontorova模型同樣以最小能量所得到的mapping equation求解,
並且與ANNNI模型中自旋系統的相變能夠對應,
求得絕對零度下的Ground State,並討論魔梯現象出現的可能性;
希望以此作為將Frenkel-Kontorova模型與ANNNI模型的魔梯現象連結的出發點。
We found the ground state of a Heisenberg model which has a phase transition at 0k similar to the ANNNI model by some strategy similar to the Frenkel-Kontorova model.
And we discuss the possibilities of the devil's staircase of this system,in order to connect the devil's staircase phenomenon from different nonlinear systems.
1 引言 4
2 Frenkel-Kontorova Model 與 Standard Mapping 6
3 Extended Frenkel-Kontorova Model 與 Devil's Staircase 11
4 圓映像(Circle Map) 18
5 ANNNI Model(Axial Nearist-Next Nearist Ising Model) 21
6 最近鄰交互作用伴隨外加磁場的古典Heisenberg Model 26
7 與Frenkel-Kontorova Model及ANNNI模型的比較 31
1.馮端 金國鈞, ch6.1 "魔梯", 凝聚態物理學新論, 凡異出版社, 新竹, 339-362, (2000).
2.P. Bak, "Commensurate Phases, Incommensurate Phases and the Devil's Staircase", Rep. Prog. Phys., 45, 587, (1982).
3.R.L. Wheeden, A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Marcel Dekker, N. Y. and Basel, 33-37, (1977); D. R. Chalice, "A Characterization of the Cantor Function", Amer. Math. Monthly 98, 255-258, (1991); R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City, CA, (1987); S. Wagon, "The Cantor Function" and "Complex Cantor Sets", ch4.2 and ch5.1, Mathematica in Action, W. H. Freeman, New York, 102-108 and 143-149, (1991).
4.T. Kontorova and Y. I. Frenkel, Zh. Eksp. & Teor. Fiz. 8, 89, 1340, 1349, (1938); F. C. Frank and J. M. Van der Merwe, Proc. Roy. Soc., London, A198, 2045, (1949). S. C. Ying, Phys. Rev. B3, 4160, (1971).
5.S. Aubry, "The new concept of transition by breaking of analyticity", Solitons and Condensed Matter Physics, Edited by A. R. Bishop and T. Schneider, Solid. State. Sciences 8, Springer Verlag, 264, (1978).
6.V. I. Arnold and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin Inc., (1968).
7.J. Moser, Stable and Random motions in Dynamical Systems, Princeton University Press, N. J., (1973).
8.S. Aubry and P. Y. Le Daeron, "The discrete Frenkel-Kontorova model and its extensions", Physica, 8D, 381-422, (1983).
9.S. Aubry, "The twist map, the extended Frenkel-Kontorova model and the devil's staircase", Physica, 7D, 240-258, (1983).
10.S. Aubry, G. Andre, "Analyticity Breaking and Anderson Localization in Incommensurate Lattices", Ann. of the Israel Phys. Soc. 3, 133-164, (1980); reprinted in: P.J. Steinhardt, S. Ostlund (Eds.), The Physics of Quasicrystals, World Scientific, Singapore, 554-593, (1987).
11.M. H. Jensen, P. Bak, and T. Bohr, "Complete Devil's Staircase, Fractal Dimension, and Universality of Mode-Locking Structure in the Circle Map", Phys. Rev. Lett. 50, 1637,(1983); "Transition to Chaos by Interaction of Resonances in Dissipative Systems. 1. Circle Maps", Phys. Rev. A30, 1960, (1984).
12.J. von Boehm and P. Bak, "Devil's Staircase and the Commensurate-Commensurate Transitions in CeSb", Phys. Rev. Lett. 42, 122, (1987).
13.P. Bak and J. von Boehm, "Ising Model with Solitons, Phasons, and the Devil's Staircase", Phys. Rev. B21, 5297, (1980).
14.M. H. Jensen and P. Bak, "Mean-Field Theory of the Three-Dimensional Anisotropic Ising Model as a Four-Dimensional Mapping", Phys. Rev. B27, 6853, (1983).
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