分形晶格上自旋模型相变问题的研究
摘要
相变是自然界中普遍存在的现象，自1869年T．Andrews发现临界点以来，相变和临界现象问题就成为凝聚态物理学和统计物理学中十分活跃和重要的研究领域。二十世纪七十年代，B．Mandelbrot把具有标度不变性的自相似结构命名为fractal(分形)，并在其论著中对分形作了相当详细的论述、讨论和研究，为相变问题开创了一个崭新的研究领域。
20多年来，分形研究逐渐成熟发展起来，并越来越广泛地应用于其他学科领域。特别是分形晶格上自旋系统的相变问题研究，引起了人们极大的兴趣，并取得了很大进展。上世纪80年代初，Y．Gefen等人开创性地把磁自旋模型引入到分形晶格上，考察其临界行为，成功解决了分形晶格上Ising模型的相变问题，并研究了普适性问题，得出具有普遍性的结论。近年来，自旋取值连续的Gauss模型和S~4模型，作为Ising模型的推广，在分形晶格上相变问题的研究引了起人们的关注。
研究分形上自旋系统的相变问题，常用的理论方法有：组合解法，图形展开法，转移矩阵法和重整化群方法。前两种方法是直接求出其配分函数，然后分析配分函数的解析行为。而重整化群变换方法，回避了直接求配分函数的难题，已被证明是一种研究该类问题较为有效的方法。
本文应用部分格点消约实空间重整化群变换方法，结合自旋重标和累积展开，在无外场的情况下，研究了Sierpinski-gaste分形晶格上Gauss模型、S~4模型以及无分支Koch曲线上S~4模型的相变和临界现象，计算出了临界点和临界指数，简单探讨了相变的两个基本问题之一一普适性。论文的主要内容包括一下几个方面：
1．采用部分格点消约重整化群变换和自旋重标相结合的方法，研究了零场中SG晶格上Gauss模型的相变和临界性质，求得了临界点和临界指数。结果显示，同时考虑二体和三体相互作用，由于新的耦合常数及高阶相互作用项的出现，SG晶格的临界点和临界指数都发生了
摘要
变化．
工应用相同的方法，对无外场情况下SG分形晶格上的S‘模型进
行了研究，计算得到了 Gauss不动点和 W．F．不动点．与 SG晶格上 ISing
模型和G。ss模型的计算结果相比较，得出，Ising模型只存在零温相变，
而Gauss模型和S‘存在有限温度的相交，并且Gauss模型，S‘模型的关
联长度临界指数。与有分形维数df有关，而 ISiflg模型的 V＝l，与 df无
关．这进一步说明，自旋离散取值和自旋连续取值的两类模型存在很大
差异，而 Gauss模型和 S‘模型则具有某些相似的性质．
3．采用部分格点消约实空间重整化群变换方法研究了零场中无
分支Koch曲线上 S‘模型的临界性质，求得了 Gauss不动点．与 SG晶
格上S‘模型的计算结果比较，认为，KOCh曲线上该模型的临界性质与
平移对称晶格上的结果相同，K。Ch曲线具有平移对称性晶格的特点．
通过比较 Koch曲线上 Gauss模型的计算结果，可以看出，S‘模型在
Koch曲线上和 Gauss模型具有相同的临界行为，属于同一普适类．
Phase transitions and critical phenomena in condensed matter physics and statistical physics have been a quite important field of inquiry, since T. Andrews discovered the critical point in 1869. In 1970s B. Mandelbrot named the self-similar structure with scaling invariant for fractal and presented detailed discuss of in his treatise, which expanded phase transitions into another research subject.
With the development and application of fractal, phase transitions and critical phenomena on fractal lattices have aroused people's great interest, and many significant results have been obtained in the last two decades. By placing Ising-models on the sites of fractal lattices, Genfen et al have studied critical phenomena of Ising-model on fractals and it's universality in the early 1980s, obtained the representative results. Recently, much attention has been paid to the study of phase transitions of the continuous spin models on fractal lattices, e.g. Gauss model, S4 model.
The usual ways to study the subject are the transfer-matrix method, combination solution, the graphic expansion and the renormalization -group technique, and so on. The renormalization-group technique is proved to a comparatively powerful means, because it avoids calculating partition function directly.
In this thesis some theoretical studies of phase transitions and critical phenomena of ferromagnetic systems on fractal lattices, namely, the Gauss model, S4 model on Sierpinski-gasket and S4 model on non-branching Koch curve are performed without the external magnetic field by the deci -mation real-apace renormalization-group with spin-rescaling technique. We calculated the critical points and critical exponents, and discussed one of the fundamental questions involved in phase transition-universality. The paper's main contents are composed of three parts below:
1. Decimation renormalization-group with spin-rescaling techniques are applied to Gaussian model with triplet interaction on the sites of Sierpinski -gasket fractal lattices without external magnetic field. Fixed points and critical exponent are got. The results show that the fixed points and critical exponent changed because of the perturbation.
2. In the same way, the critical behavior of the S4 model on SG fractal lattices is investigated without external field. We obtained Gaussian and W.F. fixed points. Comparing the result with that of Gaussian model and Ising model obtained recently, we propose that the phase transition of Ising model on Sierpinski-gaskit occur at zero-temperature, but the phase of Gauss model occur at finitude-temperature, and that the critical exponent of S4 depend on the fractal dimensionality, while the critical exponent of Ising model has nothing to do with the fractal dimensionality, v = 1. These show S model has some similar property with Gaussian model.
3. We studied the critical behavior of the S4 model on non-branching Koch curve and only obtained the Gaussian fixed point. The result reflects the S4 model and the Gaussian model belongs to the same universality class. In comparison with the results on Sierpinski-gasket, it shows that this model on Koch has the same properties as on translation symmetric lattices.
20多年来，分形研究逐渐成熟发展起来，并越来越广泛地应用于其他学科领域。特别是分形晶格上自旋系统的相变问题研究，引起了人们极大的兴趣，并取得了很大进展。上世纪80年代初，Y．Gefen等人开创性地把磁自旋模型引入到分形晶格上，考察其临界行为，成功解决了分形晶格上Ising模型的相变问题，并研究了普适性问题，得出具有普遍性的结论。近年来，自旋取值连续的Gauss模型和S~4模型，作为Ising模型的推广，在分形晶格上相变问题的研究引了起人们的关注。
研究分形上自旋系统的相变问题，常用的理论方法有：组合解法，图形展开法，转移矩阵法和重整化群方法。前两种方法是直接求出其配分函数，然后分析配分函数的解析行为。而重整化群变换方法，回避了直接求配分函数的难题，已被证明是一种研究该类问题较为有效的方法。
本文应用部分格点消约实空间重整化群变换方法，结合自旋重标和累积展开，在无外场的情况下，研究了Sierpinski-gaste分形晶格上Gauss模型、S~4模型以及无分支Koch曲线上S~4模型的相变和临界现象，计算出了临界点和临界指数，简单探讨了相变的两个基本问题之一一普适性。论文的主要内容包括一下几个方面：
1．采用部分格点消约重整化群变换和自旋重标相结合的方法，研究了零场中SG晶格上Gauss模型的相变和临界性质，求得了临界点和临界指数。结果显示，同时考虑二体和三体相互作用，由于新的耦合常数及高阶相互作用项的出现，SG晶格的临界点和临界指数都发生了
摘要
变化．
工应用相同的方法，对无外场情况下SG分形晶格上的S‘模型进
行了研究，计算得到了 Gauss不动点和 W．F．不动点．与 SG晶格上 ISing
模型和G。ss模型的计算结果相比较，得出，Ising模型只存在零温相变，
而Gauss模型和S‘存在有限温度的相交，并且Gauss模型，S‘模型的关
联长度临界指数。与有分形维数df有关，而 ISiflg模型的 V＝l，与 df无
关．这进一步说明，自旋离散取值和自旋连续取值的两类模型存在很大
差异，而 Gauss模型和 S‘模型则具有某些相似的性质．
3．采用部分格点消约实空间重整化群变换方法研究了零场中无
分支Koch曲线上 S‘模型的临界性质，求得了 Gauss不动点．与 SG晶
格上S‘模型的计算结果比较，认为，KOCh曲线上该模型的临界性质与
平移对称晶格上的结果相同，K。Ch曲线具有平移对称性晶格的特点．
通过比较 Koch曲线上 Gauss模型的计算结果，可以看出，S‘模型在
Koch曲线上和 Gauss模型具有相同的临界行为，属于同一普适类．
Phase transitions and critical phenomena in condensed matter physics and statistical physics have been a quite important field of inquiry, since T. Andrews discovered the critical point in 1869. In 1970s B. Mandelbrot named the self-similar structure with scaling invariant for fractal and presented detailed discuss of in his treatise, which expanded phase transitions into another research subject.
With the development and application of fractal, phase transitions and critical phenomena on fractal lattices have aroused people's great interest, and many significant results have been obtained in the last two decades. By placing Ising-models on the sites of fractal lattices, Genfen et al have studied critical phenomena of Ising-model on fractals and it's universality in the early 1980s, obtained the representative results. Recently, much attention has been paid to the study of phase transitions of the continuous spin models on fractal lattices, e.g. Gauss model, S4 model.
The usual ways to study the subject are the transfer-matrix method, combination solution, the graphic expansion and the renormalization -group technique, and so on. The renormalization-group technique is proved to a comparatively powerful means, because it avoids calculating partition function directly.
In this thesis some theoretical studies of phase transitions and critical phenomena of ferromagnetic systems on fractal lattices, namely, the Gauss model, S4 model on Sierpinski-gasket and S4 model on non-branching Koch curve are performed without the external magnetic field by the deci -mation real-apace renormalization-group with spin-rescaling technique. We calculated the critical points and critical exponents, and discussed one of the fundamental questions involved in phase transition-universality. The paper's main contents are composed of three parts below:
1. Decimation renormalization-group with spin-rescaling techniques are applied to Gaussian model with triplet interaction on the sites of Sierpinski -gasket fractal lattices without external magnetic field. Fixed points and critical exponent are got. The results show that the fixed points and critical exponent changed because of the perturbation.
2. In the same way, the critical behavior of the S4 model on SG fractal lattices is investigated without external field. We obtained Gaussian and W.F. fixed points. Comparing the result with that of Gaussian model and Ising model obtained recently, we propose that the phase transition of Ising model on Sierpinski-gaskit occur at zero-temperature, but the phase of Gauss model occur at finitude-temperature, and that the critical exponent of S4 depend on the fractal dimensionality, while the critical exponent of Ising model has nothing to do with the fractal dimensionality, v = 1. These show S model has some similar property with Gaussian model.
3. We studied the critical behavior of the S4 model on non-branching Koch curve and only obtained the Gaussian fixed point. The result reflects the S4 model and the Gaussian model belongs to the same universality class. In comparison with the results on Sierpinski-gasket, it shows that this model on Koch has the same properties as on translation symmetric lattices.
引文
[1] Gefen Y, Aharony A, and Mandelbrot B B, Phase transition on fractal: Ⅰ. Quasi-Linear lattices, J Phys A: Math Gen 16, 127 (1983).
[2] Gefen Y, Aharony A, and Shapir Y et al, Phase transition on fractals: Ⅱ. Sierpinski gaskets, J Phys A: Math Gen 17, 435 (1984).
[3] Gefen Y, Aharony A, and Mandelbrot B B, Phase transition on fractal: Ⅲ. Infinitely ramified lattices, J Phys A: Math Gen 17, 1277 (1984).
[4] Wang Z D, Gong C D, and Arno H, Critical behavior on some fractals, Phys Rev E 34, 1531 (1986).
[5] Li S and Yang Z R, Real-space renormalization-group study of the phase transition in a Gaussian model of fractals, Phys Rev E 55, 6656 (1997).
[6] Kong X M, Li Z Q and Zhu J Y, Critical behavior of the Gaussian model on fractal lattices in external magnetic field, Science in China A 43,768 (2000).
[7] Kong X M and Li S, The Gaussian model on the inhomogeneous fractal lattices, Commun Theor Phys 33, 63 (2000).
[8] Lin Z Q and Kong X M, Critical behavior of the Gaussian model on a diamond-type hierarchical lattice with periodic and a periodic interactions, Physica A, 271 (1999).
[9] 孔祥木，李崧，钻石型等级晶格上高斯模型的临界性质，中国科学(A辑)28，1129(1998)．
[10] Zhu J Y, Yang Z R, Glauber critical dynamics: exact solution of the kinetic Gaussian model, Phys Rev E, 55, 656 (1997).
[11] Stanley H E，重整化群与渗流理论，物理学进展 5，1985．
[12] 李英，孔祥木，黄家寅，X分形晶格上Gauss模型的临界性质，物理学报 51，1346 (2002)．
[13] Stanley H E, Introduction to Phase transition and critical phenomena,
Clarenden Press, Oxford, (1971).
[14] Widom B, Chcm J, Phys, 43, 3892 (1965).
[15] 汪子丹，龚昌德，凝聚态物理中的分形，物理学进展10，1～55(1990)．
[16] 伍法岳，杨展如，相变与临界现象(Ⅰ)，物理学进展1，1～123(1981)．
[17] 伍法岳，杨展如，相变与临界现象(Ⅱ)，物理学进展1，26～28(1981)．
[18] Kadanoff L P, Physics, 2, 263 (1966).
[19] Wilson K G, Phys Rev B 4, 3174 (1971).
[20] Curie P, Ann Chim Phys 5,289 (1895); J de Phys 4, 263 (1895).
[21] Weiss P, J de Phys 6, 661 (1907).
[22] Landau L D, phys Z Soviet Union, 11,545 (1937).
[23] Gorsky W, Phys Z, 50, 84 (1928).
[24] Ising E, Phys Z, 31,253 (1925).
[25] Onsager L, Phys Rev, 65, 117 (1944).
[26] Wilson K G, Rev. Mod. Phys, 55, 583 (1983).
[27] Berlin T H and Kac M, The spherical model of a ferromagnet, Phys Rev 86, 821 (1952).
[28] Stauffer D, Scaling theory of percolation cluster, Phys Rep, 54, 1 (1929).
[29] Ma S, Modem theory of critical phenomena, New York, 1976.
[30] Bambi H, Introduction to Real-space renormalization-group methods in critical and chaotic phenomena, Phys Rep, 91,233 (1982).
[31] Mandelbrot B B, The fractal geometry of nature, Freeman, New York, 1983.
[32] Glauber, Time-dependent statistics of the Ising model, J Math Phys, 4, 294(1963).
[33] Qin Y, Yang Z R, Critical dynamics of the kinetic Ising model with
triplet interaction on Sierpinski-gasket-type fractals, Phys Rev B, 46, 284 (1992).
[34] 杨展如，分形物理学，上海科技教育出版社，1996．
[35] 北京大学物理系《量子统计物理学》编写组，量子统计物理学，北京大学出版社，1996．
[36] John Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996.
[2] Gefen Y, Aharony A, and Shapir Y et al, Phase transition on fractals: Ⅱ. Sierpinski gaskets, J Phys A: Math Gen 17, 435 (1984).
[3] Gefen Y, Aharony A, and Mandelbrot B B, Phase transition on fractal: Ⅲ. Infinitely ramified lattices, J Phys A: Math Gen 17, 1277 (1984).
[4] Wang Z D, Gong C D, and Arno H, Critical behavior on some fractals, Phys Rev E 34, 1531 (1986).
[5] Li S and Yang Z R, Real-space renormalization-group study of the phase transition in a Gaussian model of fractals, Phys Rev E 55, 6656 (1997).
[6] Kong X M, Li Z Q and Zhu J Y, Critical behavior of the Gaussian model on fractal lattices in external magnetic field, Science in China A 43,768 (2000).
[7] Kong X M and Li S, The Gaussian model on the inhomogeneous fractal lattices, Commun Theor Phys 33, 63 (2000).
[8] Lin Z Q and Kong X M, Critical behavior of the Gaussian model on a diamond-type hierarchical lattice with periodic and a periodic interactions, Physica A, 271 (1999).
[9] 孔祥木，李崧，钻石型等级晶格上高斯模型的临界性质，中国科学(A辑)28，1129(1998)．
[10] Zhu J Y, Yang Z R, Glauber critical dynamics: exact solution of the kinetic Gaussian model, Phys Rev E, 55, 656 (1997).
[11] Stanley H E，重整化群与渗流理论，物理学进展 5，1985．
[12] 李英，孔祥木，黄家寅，X分形晶格上Gauss模型的临界性质，物理学报 51，1346 (2002)．
[13] Stanley H E, Introduction to Phase transition and critical phenomena,
Clarenden Press, Oxford, (1971).
[14] Widom B, Chcm J, Phys, 43, 3892 (1965).
[15] 汪子丹，龚昌德，凝聚态物理中的分形，物理学进展10，1～55(1990)．
[16] 伍法岳，杨展如，相变与临界现象(Ⅰ)，物理学进展1，1～123(1981)．
[17] 伍法岳，杨展如，相变与临界现象(Ⅱ)，物理学进展1，26～28(1981)．
[18] Kadanoff L P, Physics, 2, 263 (1966).
[19] Wilson K G, Phys Rev B 4, 3174 (1971).
[20] Curie P, Ann Chim Phys 5,289 (1895); J de Phys 4, 263 (1895).
[21] Weiss P, J de Phys 6, 661 (1907).
[22] Landau L D, phys Z Soviet Union, 11,545 (1937).
[23] Gorsky W, Phys Z, 50, 84 (1928).
[24] Ising E, Phys Z, 31,253 (1925).
[25] Onsager L, Phys Rev, 65, 117 (1944).
[26] Wilson K G, Rev. Mod. Phys, 55, 583 (1983).
[27] Berlin T H and Kac M, The spherical model of a ferromagnet, Phys Rev 86, 821 (1952).
[28] Stauffer D, Scaling theory of percolation cluster, Phys Rep, 54, 1 (1929).
[29] Ma S, Modem theory of critical phenomena, New York, 1976.
[30] Bambi H, Introduction to Real-space renormalization-group methods in critical and chaotic phenomena, Phys Rep, 91,233 (1982).
[31] Mandelbrot B B, The fractal geometry of nature, Freeman, New York, 1983.
[32] Glauber, Time-dependent statistics of the Ising model, J Math Phys, 4, 294(1963).
[33] Qin Y, Yang Z R, Critical dynamics of the kinetic Ising model with
triplet interaction on Sierpinski-gasket-type fractals, Phys Rev B, 46, 284 (1992).
[34] 杨展如，分形物理学，上海科技教育出版社，1996．
[35] 北京大学物理系《量子统计物理学》编写组，量子统计物理学，北京大学出版社，1996．
[36] John Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996.