A MONTE CARLO RENORMALIZATION GROUP STUDY OF ISING MODEL ON 2-DIMENSIONAL RANDOM TRIANGLE LATTICE

Ⅰ. INTRODUCTIONMonte Carlo renormalization group method (MCRG) has become a powerful technical tool for the study of critical behaviour and continuum limit of discrete systems since 1976. The method is therefore widely applied to the study of fixed point and critical exponents in statistical models, and to the study of phase transitions, β function and scaling behaviour for some gauge fields in lattice gauge theories.

Entanglement and quantum phase transition in the Heisenberg-Ising model

We use the quantum renormalization-group（QRG） method to study the entanglement and quantum phase transition（QPT） in the one-dimensional spin-1/2 Heisenberg-Ising model [Lieb E,Schultz T and Mattis D 1961 Ann.Phys.（N.Y.） 16 407].We find the quantum phase boundary of this model by investigating the evolution of concurrence in terms of QRG iterations.We also investigate the scaling behavior of the system close to the quantum critical point,which shows that the minimum value of the first derivative of concurrence and the position of the minimum scale with an exponent of the system size.Also,the first derivative of concurrence between two blocks diverges at the quantum critical point,which is directly associated with the divergence of the correlation length.

Two-Dimensional Saddle Point Equation of Ginzburg-Landau Hamiltonian for the Diluted Ising Model

The saddle point equation of Ginzburg-Landau Hamiltonian for the diluted Ising model is developed. The ground state is solved numerically in two dimensions. The result is partly explained by the coarse-grained approximation.

Exact Calculation of Antiferromagnetic Ising Model on an Inhomogeneous Surface Recursive Lattice to Investigate Thermodynamics and Glass Transition on Surface/Thin Film

An inhomogeneous 2-dimensional recursive lattice formed by planar elements has been designed to investigate the thermodynamics of Ising spin system on the surface/thin film. The lattice is constructed as a hybrid of partial Husimi square lattice representing the bulk and 1D single bonds representing the surface. Exact calculations can be achieved with the recursive property of the lattice. The model has an anti-ferromagnetic interaction to give rise to an ordered phase identified as crystal, and a solution with higher energy to represent the amorphous/metastable phase.Free energy and entropy of the ideal crystal and supercooled liquid state of the model on the surface are calculated by the partial partition function. By analyzing the free energies and entropies of the crystal and supercooled liquid state,we are able to identify the melting and ideal glass transition on the surface. The results show that due to the variation of coordination number, the transition temperatures on the surface decrease significantly compared to the bulk system.Our calculation qualitatively agrees with both experimental and simulation works on the thermodynamics of surfaces and thin films conducted by others. Interactions between particles farther than the nearest neighbor distance are taken into consideration, and their effects are investigated.

Critical Behavior of Gaussian Model on X Fractal Lattices in External Magnetic Fields

Using the renormalization group method, the critical behavior of Gaussian model is studied in external magnetic fields on X fractal lattices embedded in two-dimensional and -dimensional Euclidean spaces, respectively. Critical points and exponents are calculated. It is found that there is long-range order at finite temperature for this model, and that the critical points do not change with the space dimensionality (or the fractal dimensionality ). It is also found that the critical exponents are very different from results of Ising model on the same lattices, and that the exponents on X lattices are different from the exact results on translationally symmetric lattices.

Critical Behavior of the Gaussian Model with Periodic Interactions on Diamond—Type Hierarchical Lattices in External Magnetic Fields

The Gaussian spin model with periodic interactions on the diamond-type hierarchical lattices is constructed by generalizing that with uniform interactions on translationally invariant lattices according to a class of substitution sequences. The Gaussian distribution constants and imposed external magnetic fields are also periodic depending on the periodic characteristic of the interaction bonds. The critical behaviors of this generalized Gaussian model in external magnetic fields are studied by the exact renormalization-group approach and spin rescaling method. The critical points and all the critical exponents are obtained. The critical behaviors are found to be determined by the Gaussian distribution constants and the fractal dimensions of the lattices. When all the Gaussian distribution constants are the same, the dependence of the critical exponents on the dimensions of the lattices is the same as that of the Gaussian model with uniform interactions on translationally invariant lattices.

Equilibrium dynamics of the sub-ohmic spin-boson model at finite temperature

We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σ_(z) at finite temperature for the sub-ohmic spin-boson model.A peak is observed at the frequency ω_(T)~T in the curve of C(ω).The curve merges with the zero-temperature C(ω) in ω>>ω_(T) and deviates significantly from the zero-temperature curve in ω<<ω_(T).

Thermodynamic Transitions of Antiferromagnetic Ising Model on the Fractional Multi-branched Husimi Recursive Lattice

The multi-branched Husimi recursive lattice is extended to a virtual structure with fractional numbers of branches joined on one site. Although the lattice is undrawable in real space, the concept is consistent with regular Husimi lattice. The Ising spins of antiferromagnetic interaction on such a set of lattices are calculated to check the critical temperatures(Tc) and ideal glass transition temperatures(Tk) variation with fractional branch numbers. Besides the similar results of two solutions representing the stable state(crystal) and metastable state(supercooled liquid)and indicating the phase transition temperatures, the phase transitions show a well-defined shift with branch number variation. Therefore the fractional branch number as a parameter can be used as an adjusting tool in constructing a recursive lattice model to describe real systems.

用一维PCA描述二维Ising模型的探讨

将一维ＣＡ的动态演化时空图与二维Ｉｓｉｎｇ自旋位形等同起来，用重整化群方法计算得到各向异性三角晶格Ｉｓｉｎｇ模型的临界点发生在无序线ｅ－２Ｊ＝ｃｏｓｈ（２Ｄ）上，这与平衡统计力学计算结果一致；同时求得了相关指数的初步结果。

三维Ising模型的蒙特卡罗模拟

采用蒙特卡罗(Monte Carlo)重点抽样法对三维Ising模型进行计算机模拟,测量无外磁场时三维Ising模型中自旋键链的能量、磁化强度、比热及磁化率的统计平均值与标准误差(不确定度).结果表明,三维Ising模型在无外磁场时存在自发磁化现象,铁磁→非铁磁相变临界点在J/(kBTC)=0.222 0,或居里温度TC=4.500 0处.并研究存在外磁场时上述物理量随温度与外磁场的变化规律,给出物理解释.

二维Ising模型重正化群方法集团不同大小划分结果的分析

基于二维三角晶格模型,分别选取包含7、9个格点自旋的集团为Kadanoff集团,通过实空间重正化群变换分析求解临界点及临界指数,对其结果对比分析表明:选取更大的自旋集团可以提高精度.

特殊BCC Ising模型相变现象的重整化群处理

用二维正方格子概率性细胞自动机的演化动力学描述一类特殊的体心立方晶体Ｉｓｉｎｇ模型，通过重整化群处理，得到该模型的相变点发生在曲线ｅｘｐ（－８Ｊ）＝ｃｏｓｈ（４Ｄ）上，数值结果为Ｄｃ＝０．２１６８０９，Ｊｃ＝－０．０４２０７８，该点实际上是一个Ｌｉｆｓｈｉｔｚ三相点．

镶嵌正方格子上铁磁Ising模型的相变

应用等效变换的方法,把镶嵌正方格子铁磁Ising模型转化为可求解的正方格子。采用重整化群变换,得到了正方格子上Ising模型的临界点。通过得到的两个变换关系,求得镶嵌正方晶格上Ising模型的临界点为K*=0.405。

采用正方元胞的Ising模型实空间重正化群解

在二维正方形晶格上,将元胞取为4格点正方形,采用3种不同的规则定义块自旋状态,进行了重正化群计算,得出了更为精确的结果;解决了元胞内格点数为偶数的重正化群计算问题.

用变分累积展开法计算二维Ising模型八阶临界温度

本文用变分累积展开对Ｉｓｉｎｇ模型的临界温度的计算提出一种直观的图示解法，使计算大为简化。应用此法求得各阶临界温度与其他文献推算结果相一致。本文利用此法计算二维Ｉｓｉｎｇ模型八阶临界温度Ｔｃ（８）＝２．４７６７１，此法也可以推广到三维的情形．

Ising模型的临界性质

利用重整化群变换和自旋重标相结合的方法,研究m分支Koch曲线的Ising模型的相变和临界现象.求出了系统的临界指数,发现临界指数只与Koch曲线的分形维数有关.

二维Kagomé系统带有次近邻相互作用Ising模型的实空间重正化群研究

利用实空间重正化群方法，研究了二维 Ｋａｇｏｍｅ晶格上带有次近邻相互作用Ｉｓｉｎｇ模型，得到了其临界温度和各热力学量的临界指数。

Ising模型实空间重正化的Monte Carlo模拟

以二维正方晶格Ising模型为例,采用Monte Carlo方法模拟实空间重正化群操作,分别采用简单抽样和重要性抽样。对于简单抽样,可以得到解析解,但模拟的元胞不大;而对于重要性抽样,虽然只能获得数值解,但是可以模拟很大的元胞,结果也更接近精确解;文中的方法使实空间重正化群操作的精度大大提高。

二维正方晶格Ising模型的重整化群计算

用实空间重整化群方法计算二维正方晶格Ising模型 ,给出临界温度及临界指数 .

分形上Ising模型的临界性质

应用实空间重整化群的方法,研究了一种分形上Ising模型的相变和临界性质,求出了系统的临界点0.264,根据RG理论,得到了系统的临界指数。与该分形上Gauss模型比较,系统的临界点和临界指数都发生了变化。