分形上自旋系统相变和临界现象的理论研究
摘要
相变与临界现象是统计物理中极为重要的研究领域，如铁磁体-顺磁体间的转变，导体和超导体的转变，正常流体和超流体的转变等，都属于临界现象。分形是具有自相似对称性的几何图形，可用来模拟自然界中在一定尺度范围内具有自相似对称性的不规则结构，如Koch曲线可用来模拟海岸线，无规行走模型可模拟蛋白质分子链，自回避无规行走模型可模拟线性聚合物等。要描述这些不规则结构系统，尤其是磁性系统的物理性质和规律，研究分形上自旋模型的相变与临界现象问题无疑是很有意义的。
自从20世纪80年代初Gefen等人开创性地解决了分形晶格上lsing模型的相变问题以来，分形晶格上自旋模型的相变研究引起了人们极大的兴趣，20多年来已逐渐发展成为一个重要的研究领域。在研究这类问题时，常用的理论方法有转移矩阵法、组合解法、重整化群方法及图形展开法等。由于分形具有自相似对称性，实空间重整化群方法已证明是一种较为有效的理论方法。本文应用实空间重整化群变换的方法，在外磁场存在的情况下，对三种分形晶格，即Sierpinski镂垫晶格、X分形晶格和钻石型等级晶格上自旋模型的铁磁相变做了一些理论研究，求出了临界点和临界指数，并对同一种晶格上不同模型的研究结果进行了比较，在此基础上，探讨了相变的两个基本问题——标度性和普适性，论文的主要内容包括以下几个方面：
1．采用部分格点消约的实空间重整化群方法，我们考察了外磁场中Sicrpinski镂垫晶格上lsing自旋模型的相变和临界性质，求出了临界点和临界指数，结果发现，外磁场的存在并不影响lsing模型的相变点，即系统的临界温度为零，临界磁场也为零。从零温相变的特征可知，这时系统发生的是一级相变而非二级相变。
2．利用部分格点消约重整化群和自旋重标相结合的方法，在外磁场中考察了Sierpinski镂垫晶格上Gauss自旋模型的相变和临界性质。结果得出，在临界点处，与温度有关的最近邻相互作用参数K~φ=b／4
JN＆
一（b是 Gauss分布常数），夕磁场 h”。0．显然，此乡果与上述 Ising
——
模型的结果有很大的差异．
3．运用与上面相同的方法，研究了外场中X分形晶格上推广后
的0＊uss模型的相变和临界性质．我们发现X分形晶格作为一种非均
匀分形晶格，其临界点也可写为 K”＝b。／q；，h‘＝0的形式，其中 q；
和b。分别是晶格上格点i的配位数和 Gauss分布常数；同时也发现其
临界点不随分形晶格所在空间维数的变化而变化，但其临界指数随空
间维数的增加分别呈现一定的变化趋势．
4．应用部分格点消约实空间重整《匕群方法的同时，我们运用了
累积展开近似的方法，在外磁场中研究了2棱m分支的钻石型等级晶
格上宁模型的相变和临界性质．在研究过程中，对此模型进行了推广，
即假设S‘模型中的CSLIS。分布常数、四自旋相互作用参量及外磁场都
与晶格格点的配位数有关，并且满足关系n；，八；；＝q；q，，
U／U＝U切 和h／h＝O切，其中o是格点i的配位数．h、11
Ill／ qll 17 ｝’gi 1917二’””’－4 —’”—－‘一o－y
和h。分别是格点。上的Gauss分布常数、四自旋相互作用参量和外磁
场．结果表明，这类分形晶格上 S‘模型的临界性质与晶格的分支数 m
或分形维数寸有关：当。＞4即分形维数J．＞3时，晶格上守模型只存
在不动点 K’＝b，／2，u；＝0，h：＝0，RF为 Gauss不动点，其 II＄ 界指数
与 G＞USS模型勺 Ijs界指数相一致，换言之，当工＞4 BF d；＞3时，S‘
模型与 Gauss模型具有相同的临界行为，属于同一普适类，而且当
m＝8（d；＝4）时，S‘模型的结果与平均场理论的结果也是一致的；当
m S 4即 d；5 3，除了存在 Gauss不动点外，还存在 Wi！son一Fisher（W，卜．）
不动点，而且这时只有W．F．不动点对系统的临界性质产生影响．我们
发现这类分形晶格上S‘模型的结果与平移对称晶格上的结果类似．
Phase transitions and critical phenomena in statistical physics are a quite important field of inquiry. As is well known to us, the conversion of iron from paramagnetic to ferromagnetic form, the transitions from conductor to superconductor and from normal fluid to superfluid are examples of critical phenomena. The fractal is a geometrical figure with self-similar symmetry, and it is an important tool for characterizing irregular structures in nature that are self-similar on certain length scales. For example, the Koch curves can be viewed as a mathematical model for coastlines, random walks can be used to mimic protein molecular chains, and self-avoiding walks can serve as a model for linear polymers, and so on. In order to describe explicitly the physical properties and laws of irregular structure systems, especially magnetic systems, it is no doubt necessary for us to study phase transitions of fractal lattices.
In the early 1980s Gefen et al have presented their series work of critical phenomena on fractals. Since then, an investigation into phase transitions of spin models on fractals aroused people's great interest and developed gradually an important research subject. The usual ways to study the subject are the transfer-matrix method, combination solution, the renormalization-group technique, and graphic expansion, and so forth. But due to the fact that the fractal has the property of self-similarity, the technique of real-space renormalization-group is proved to a comparatively powerful means. In this thesis, some theoretical studies of phase transition of ferromagnetic systems on three fractal lattices, namely, the Sierpinski Gasket, the X fractal lattice and the Diamond-type Hierarchial lattice are performed in the external magnetic field by means of real-space renormalization group. We calculate the critical points and critical exponents, and compare the results obtained from different models on the
same fractal lattices. Furthermore, we discuss two fundamental questions involved in phase transitions ?universality and scaling property. The thesis's main contents are composed of five parts below:
1. We studied the critical properties of the Ising model on Sierpinski Gasket lattices in the presence of external field by using the real-space renormalizaion-group method. The results show that even in external field, the phase transition still occurs at zero-temperature, which can be known to be the first-order rather than second-order phase transition in terms of the feature of zero-temperature phase transition.
2. A technique of spin-rescaling is introduced in the procedure of renormalizaion-group transformation. With such two methods, we investigate the critical properties of the Gauss model on Sierpinski Gasket lattices in the presence of external field. It is found that, at the critical point, the nearest interaction parameter associated with temperature is K'=b/4 and the external field h'=0. The result is much different from that of the Ising model.
3. In the same way as that used above, the critical behavior of the Gauss model on X fractal lattices is investigated in the case of external field. It is shown that the X fractal lattice, an inhomogeneous fractal lattice, has the critical point K' = bifJq, ,h' =0, where qt and 6(/ are the coordination number and Gaussian distribution constant of the site /. In addition, we find that the critical points do not vary with the space dimensionality of the fractal lattices but the critical exponents reveal certain tendency in change as the space dimensionality increases.
4. With the renormalization group method and the technique of cumulate expansion, we studied the critical behavior of the S4 model on Diamond-type Hierarchial lattices. In the course of studies, we make a modification on the form of the S4 model, in which it is assumed that the Gaussian distribution constant b, four spins interaction parameter u and
the external magnetic field h depend on the coordination number q1, of the
自从20世纪80年代初Gefen等人开创性地解决了分形晶格上lsing模型的相变问题以来，分形晶格上自旋模型的相变研究引起了人们极大的兴趣，20多年来已逐渐发展成为一个重要的研究领域。在研究这类问题时，常用的理论方法有转移矩阵法、组合解法、重整化群方法及图形展开法等。由于分形具有自相似对称性，实空间重整化群方法已证明是一种较为有效的理论方法。本文应用实空间重整化群变换的方法，在外磁场存在的情况下，对三种分形晶格，即Sierpinski镂垫晶格、X分形晶格和钻石型等级晶格上自旋模型的铁磁相变做了一些理论研究，求出了临界点和临界指数，并对同一种晶格上不同模型的研究结果进行了比较，在此基础上，探讨了相变的两个基本问题——标度性和普适性，论文的主要内容包括以下几个方面：
1．采用部分格点消约的实空间重整化群方法，我们考察了外磁场中Sicrpinski镂垫晶格上lsing自旋模型的相变和临界性质，求出了临界点和临界指数，结果发现，外磁场的存在并不影响lsing模型的相变点，即系统的临界温度为零，临界磁场也为零。从零温相变的特征可知，这时系统发生的是一级相变而非二级相变。
2．利用部分格点消约重整化群和自旋重标相结合的方法，在外磁场中考察了Sierpinski镂垫晶格上Gauss自旋模型的相变和临界性质。结果得出，在临界点处，与温度有关的最近邻相互作用参数K~φ=b／4
JN＆
一（b是 Gauss分布常数），夕磁场 h”。0．显然，此乡果与上述 Ising
——
模型的结果有很大的差异．
3．运用与上面相同的方法，研究了外场中X分形晶格上推广后
的0＊uss模型的相变和临界性质．我们发现X分形晶格作为一种非均
匀分形晶格，其临界点也可写为 K”＝b。／q；，h‘＝0的形式，其中 q；
和b。分别是晶格上格点i的配位数和 Gauss分布常数；同时也发现其
临界点不随分形晶格所在空间维数的变化而变化，但其临界指数随空
间维数的增加分别呈现一定的变化趋势．
4．应用部分格点消约实空间重整《匕群方法的同时，我们运用了
累积展开近似的方法，在外磁场中研究了2棱m分支的钻石型等级晶
格上宁模型的相变和临界性质．在研究过程中，对此模型进行了推广，
即假设S‘模型中的CSLIS。分布常数、四自旋相互作用参量及外磁场都
与晶格格点的配位数有关，并且满足关系n；，八；；＝q；q，，
U／U＝U切 和h／h＝O切，其中o是格点i的配位数．h、11
Ill／ qll 17 ｝’gi 1917二’””’－4 —’”—－‘一o－y
和h。分别是格点。上的Gauss分布常数、四自旋相互作用参量和外磁
场．结果表明，这类分形晶格上 S‘模型的临界性质与晶格的分支数 m
或分形维数寸有关：当。＞4即分形维数J．＞3时，晶格上守模型只存
在不动点 K’＝b，／2，u；＝0，h：＝0，RF为 Gauss不动点，其 II＄ 界指数
与 G＞USS模型勺 Ijs界指数相一致，换言之，当工＞4 BF d；＞3时，S‘
模型与 Gauss模型具有相同的临界行为，属于同一普适类，而且当
m＝8（d；＝4）时，S‘模型的结果与平均场理论的结果也是一致的；当
m S 4即 d；5 3，除了存在 Gauss不动点外，还存在 Wi！son一Fisher（W，卜．）
不动点，而且这时只有W．F．不动点对系统的临界性质产生影响．我们
发现这类分形晶格上S‘模型的结果与平移对称晶格上的结果类似．
Phase transitions and critical phenomena in statistical physics are a quite important field of inquiry. As is well known to us, the conversion of iron from paramagnetic to ferromagnetic form, the transitions from conductor to superconductor and from normal fluid to superfluid are examples of critical phenomena. The fractal is a geometrical figure with self-similar symmetry, and it is an important tool for characterizing irregular structures in nature that are self-similar on certain length scales. For example, the Koch curves can be viewed as a mathematical model for coastlines, random walks can be used to mimic protein molecular chains, and self-avoiding walks can serve as a model for linear polymers, and so on. In order to describe explicitly the physical properties and laws of irregular structure systems, especially magnetic systems, it is no doubt necessary for us to study phase transitions of fractal lattices.
In the early 1980s Gefen et al have presented their series work of critical phenomena on fractals. Since then, an investigation into phase transitions of spin models on fractals aroused people's great interest and developed gradually an important research subject. The usual ways to study the subject are the transfer-matrix method, combination solution, the renormalization-group technique, and graphic expansion, and so forth. But due to the fact that the fractal has the property of self-similarity, the technique of real-space renormalization-group is proved to a comparatively powerful means. In this thesis, some theoretical studies of phase transition of ferromagnetic systems on three fractal lattices, namely, the Sierpinski Gasket, the X fractal lattice and the Diamond-type Hierarchial lattice are performed in the external magnetic field by means of real-space renormalization group. We calculate the critical points and critical exponents, and compare the results obtained from different models on the
same fractal lattices. Furthermore, we discuss two fundamental questions involved in phase transitions ?universality and scaling property. The thesis's main contents are composed of five parts below:
1. We studied the critical properties of the Ising model on Sierpinski Gasket lattices in the presence of external field by using the real-space renormalizaion-group method. The results show that even in external field, the phase transition still occurs at zero-temperature, which can be known to be the first-order rather than second-order phase transition in terms of the feature of zero-temperature phase transition.
2. A technique of spin-rescaling is introduced in the procedure of renormalizaion-group transformation. With such two methods, we investigate the critical properties of the Gauss model on Sierpinski Gasket lattices in the presence of external field. It is found that, at the critical point, the nearest interaction parameter associated with temperature is K'=b/4 and the external field h'=0. The result is much different from that of the Ising model.
3. In the same way as that used above, the critical behavior of the Gauss model on X fractal lattices is investigated in the case of external field. It is shown that the X fractal lattice, an inhomogeneous fractal lattice, has the critical point K' = bifJq, ,h' =0, where qt and 6(/ are the coordination number and Gaussian distribution constant of the site /. In addition, we find that the critical points do not vary with the space dimensionality of the fractal lattices but the critical exponents reveal certain tendency in change as the space dimensionality increases.
4. With the renormalization group method and the technique of cumulate expansion, we studied the critical behavior of the S4 model on Diamond-type Hierarchial lattices. In the course of studies, we make a modification on the form of the S4 model, in which it is assumed that the Gaussian distribution constant b, four spins interaction parameter u and
the external magnetic field h depend on the coordination number q1, of the
引文
[1] Gefen Y, Aharony A, and Mandelbrot B B, Phase transition on fractals:Ⅰ. Quasi-Linear lattices, J Phys A: Math Gen 16, 1267 (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 fractals: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);汪子丹，龚昌德，凝聚态物理中的分形，物理学进展10, 1～55(1990)．
[5] Yang Z R, Family of diamond-type hierarchical lattices, Phys Rev B 38, 728(1988).
[6] 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).
[7] 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)．
[8] Lin Z Q and Kong X M, Critical behavior of the Gaussian model on a diamond-type hierarchial lattice with periodic and aperiodic interactions, Physica A, 271 (1999).
[9] Kong X M and Li S, The Gaussian model on inhomogenous fractal lattices, Commun Theor Phys 33, 63(2000).
[10] Kong X M, Lin Z Q et al, Critical behavior of the Gaussian model on fractal lattices in external magnetic field, Science in China A 43,768(2000).
[11] Curie P, Ann Chim Phys 5,289(1895); J de Phys 4,263(1895).
[12] Weiss P, J de Phys 6, 661 (1907).
[13] Landau L D, Phys Z Soviet Union 11,545(1937).
[14] Widom B, J Chem Phy 43, 3898(1965).
[15] Kadanoff L P, Physics 2, 263(1966).
[16] Wilson K G, Phys Rex, B 4, 3174(1971).
[17] Ising E, Z Phys 31,253(1925).
[18] Berlin T H and Kac M, The spherical model of a ferromagnet, Phys Rev 86, 821 (1952).
[19] Stanley H E, J Phys Soc Japan S 26, 102(1969).
[20] Ma S, Modern Theory of Critical Phenomena, New York, 1976.
[21] Stanley H E，重正化群与渗流理论，物理学进展5，1985．
[22] Stanley H E, Introduction to Phase Transition and Critical Phenomena Oxford, 1983.
[23] Berezinskii V L, Sov Phys JETP 32,493(1970).
[24] Takano H and Suzuki M, J stat Phys 26,635(1981).
[25] Mandelbrot B B, The Fractal Geometry of Nature, Freeman, New York, 1983.
[26] Feder J, Fractals, Plenum Press, New York, 1988.
[27] Bunde A and Havlin S, Fractals in Science, Springer, Heidelberg, 1994.
[28] Stauffer D, Scaling Theory of Percolation Cluster, Phys Rep 54,1(1979).
[29] Flory P J, Principles of Polymer Chemistry, cornell University Press,New York, 1971.
[30] 杨展如，分形物理学，上海科技教育出版社，1996．
[31] Riechl L E, A Modern Course in Statistical Physics, Austin: University of Texas Press, 1980.
[32] Bambi H, Introduction to Real-Space Renormalisation-Group Methods in Critical and Chaotic Phenomena, Phys Rep 91,233(1982).
[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 fractals: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);汪子丹，龚昌德，凝聚态物理中的分形，物理学进展10, 1～55(1990)．
[5] Yang Z R, Family of diamond-type hierarchical lattices, Phys Rev B 38, 728(1988).
[6] 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).
[7] 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)．
[8] Lin Z Q and Kong X M, Critical behavior of the Gaussian model on a diamond-type hierarchial lattice with periodic and aperiodic interactions, Physica A, 271 (1999).
[9] Kong X M and Li S, The Gaussian model on inhomogenous fractal lattices, Commun Theor Phys 33, 63(2000).
[10] Kong X M, Lin Z Q et al, Critical behavior of the Gaussian model on fractal lattices in external magnetic field, Science in China A 43,768(2000).
[11] Curie P, Ann Chim Phys 5,289(1895); J de Phys 4,263(1895).
[12] Weiss P, J de Phys 6, 661 (1907).
[13] Landau L D, Phys Z Soviet Union 11,545(1937).
[14] Widom B, J Chem Phy 43, 3898(1965).
[15] Kadanoff L P, Physics 2, 263(1966).
[16] Wilson K G, Phys Rex, B 4, 3174(1971).
[17] Ising E, Z Phys 31,253(1925).
[18] Berlin T H and Kac M, The spherical model of a ferromagnet, Phys Rev 86, 821 (1952).
[19] Stanley H E, J Phys Soc Japan S 26, 102(1969).
[20] Ma S, Modern Theory of Critical Phenomena, New York, 1976.
[21] Stanley H E，重正化群与渗流理论，物理学进展5，1985．
[22] Stanley H E, Introduction to Phase Transition and Critical Phenomena Oxford, 1983.
[23] Berezinskii V L, Sov Phys JETP 32,493(1970).
[24] Takano H and Suzuki M, J stat Phys 26,635(1981).
[25] Mandelbrot B B, The Fractal Geometry of Nature, Freeman, New York, 1983.
[26] Feder J, Fractals, Plenum Press, New York, 1988.
[27] Bunde A and Havlin S, Fractals in Science, Springer, Heidelberg, 1994.
[28] Stauffer D, Scaling Theory of Percolation Cluster, Phys Rep 54,1(1979).
[29] Flory P J, Principles of Polymer Chemistry, cornell University Press,New York, 1971.
[30] 杨展如，分形物理学，上海科技教育出版社，1996．
[31] Riechl L E, A Modern Course in Statistical Physics, Austin: University of Texas Press, 1980.
[32] Bambi H, Introduction to Real-Space Renormalisation-Group Methods in Critical and Chaotic Phenomena, Phys Rep 91,233(1982).