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

我们发展了一种新的二维(2D)量子系统和3D经典系统的数值重整化群(RG)方法，它使用一个表示为局部权重乘积的变分状态。作为例子，我们采用含有3个参数的二维IRF模型作为二维量子系统的典型代表之一--正方形晶格S=1/2XXZ模型的变化态。我们得到了一个很好的能量估计，尽管我们只有3个参数。在这种情况下，变分公式在各向异性极限XY模型下工作得更好。对于3D经典系统的应用，我们选择3D lsing模型作为参考系，并准备了包含162个变分参数的变分状态。在这种情况下，局域因子有辅助自旋变量，可以解释为重整化自旋。由于有如此多的参数，人们必须自动测量它们。为此，我们建立了局部重量的自洽方程，并改进了…它们更多地依赖于迭代数值过程。结果表明，相变温度的测量精度在1%以内，且变化态是一致的。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)用于具有非均匀基态的系统。作为一个例子，我们已经开始在ANNNI模型中计算热态，该模型被认为在中温出现的有序相中具有复杂的结构。目前，我们得到了一个部分相图，它表明了补偿相位的面积被抑制。作为这些研究的一个双积，我们出人意料地得到了CTMRG在随机系统中的新用途。该系统具有一种光速，可以在光锥内传递信息。针对这种情况，我们找到了一种新的目标定位方法，并提出了一种新的数值RG方法--随机锥CTMRG方法。依赖于参数条件，该方法变得不稳定，而改善这一缺陷是今后研究较少的任务之一

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