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

我们开发了一种新的二维（2D）量子系统和三维经典系统的数值重整化群（RG）方法，使用变分状态表示为局部权重的乘积。作为例子，我们采用包含3个参数的二维IRF模型作为具有代表性的二维量子系统之一的方形晶格S=1/2 XXZ模型的变分状态。我们得到了一个很好的能量估计，即使我们只有3个参数。在这种情况下，变分公式在各向异性极限下效果更好，即XY模型。为了应用于三维经典系统，我们选择三维lsing模型作为参照系，并准备了包含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，可以解释为重整自旋。由于有如此多的参数，人们必须自动地对它们进行调查。为此，我们建立了局部权值的自洽方程，并对其进行了改进。结果表明，得到的相变温度误差在1%以内。在上述情况下，变分态是均匀的。这是因为我们使用的数值RG方法CTMRG只能处理均匀的二维模型。为了改善这一限制，我们考虑对具有非均匀基态的系统使用密度矩阵重整化群（DMRG）。作为一个例子，我们已经开始计算ANNNI模型的热态，它被认为在中间温度出现的有序相中具有复杂的结构。目前，我们得到了一个局部相图，表明了对相接面积的抑制。作为这些研究的副产品，我们意外地在随机系统中获得了CTMRG的新用途。该系统具有一种光速，可以在光锥内传输信息。针对这种情况，我们找到了一种新的定位方案，并提出了一种新的数值RG方法——“光锥CTMRG方法”。随着参数条件的变化，该方法变得不稳定，改进这一缺点是今后研究的任务之一

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