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Analysis of Higher Dimensional Systems by use of the Tensor Product Variational Approach

使用张量积变分方法分析高维系统

基本信息

批准号：
13640383
负责人：
NISHINO Tomotoshi
金额：
$ 2.24万
依托单位：
kobe University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

We have developed a new numerical renormalization group (RG) method for two-dimensional (2D) quantum systems and 3D classical systems, using a variational state represented as a product of local weights. As an example, we employed 2D IRF model, that contains 3 parameters, as a variational state for the square lattice S=1/2 XXZ model, which is one of the representative 2D quantum systems. We obtained a good energy estimate, even though we have only 3 parameters. In this case the variational formulation works better in the anisotroplc limit, the XY mdoel.For the application for 3D Classical systems, we choose 3D lsing model as a reference system, and prepare a variational state that contains 162 variational parameters. In this case the local factor has auxiliary spin variable, that can be interpreted as the renormalized spin. Since there are so many parameters, one has to survey them automaticaliy. For this purpouse we developed a self-consistent equation for the local weight, and improv … More e them vie iterative numerical procedure. As a result, we showed that the phase transition temperature is obtained accurately within the error of 1%.In the above cases, the variational state is uniform. This Is because CTMRG, the numerical RG method we have used, can treat uniform 2D models only. In order to improve this restriction, we considered a usage of the density matrix renormalization group (DMRG) for the system that have non-uniform ground state. As an example, we have started calculations of thermal state hi the ANNNI model, which has been considered to have complex structure in the ordered phase that appears in the Intermediate temperature. At present, we get a partial phase diagram, that suggests the suppression of the area of comensulate phase.As a bi-product of these researches, we unexpectedly get a new usage of CTMRG for the stochastic systems. The system has a kind of speed of light and information can be transferred within the light cone. We find a new targeting scheme for this case, and proposed a new numerical RG method, "the Ught Cone CTMRG method". Depending on the parameter condition the method becomes instable, and to improve this drawback is one of the task in the future studies Less

我们发展了一种新的数值重整化群（RG）方法，用于二维（2D）量子系统和3D经典系统，使用变分状态表示为局部权重的乘积。作为一个例子，我们采用了包含三个参数的二维IRF模型作为正方形晶格S=1/2XXZ模型的变分态，该模型是二维量子系统的代表之一。我们得到了一个很好的能量估计，即使我们只有3个参数。对于3D经典系统的应用，我们选择3D lsing模型作为参考系，准备了一个包含162个变分参数的变分状态。在这种情况下，局部因子有辅助自旋变量，可以解释为重整化自旋。由于有这么多的参数，一个人必须调查他们自动。为此，我们建立了一个自洽的局部权方程，并改进了局部权方程。 ...更多信息用迭代数值方法求解。结果表明，该方法可以准确地得到相变温度，误差在1%以内，且在上述情况下，相变温度的变化状态是一致的.这是因为我们使用的数值RG方法CTMRG只能处理均匀的2D模型。为了改善这一限制，我们考虑了密度矩阵重整化群（DMRG）的使用具有非均匀基态的系统。作为一个例子，我们已经开始在ANNNI模型中的热状态的计算，这被认为是在有序相，出现在中间温度的复杂结构。目前，我们得到了一个部分相图，它表明了共囊相的面积被抑制，作为这些研究的一个副产品，我们意外地得到了CTMRG在随机系统中的一个新的用法。该系统具有光速，信息可以在光锥内传输。针对这种情况，我们提出了一种新的目标选择方案，并提出了一种新的数值RG方法--直锥CTMRG方法。该方法受参数条件的影响而变得不稳定，改善这一缺点是今后研究的任务之一