Foundation of the Numerical Renormalization Group and Its Higher-Dimensional Generalization

2000: DMRG的高维推广及其在简单系统中的应用推广DMRG（一维量子，二维经典）中定点波函数的矩阵积结构，引入高阶张量来表示三维经典统计系统的“目标状态”（传递矩阵的最大特征值状态）。波函数的张量积和结构允许我们使用低维算法计算密度矩阵和其他物理量；我们的高维算法是一个“嵌套”结构。2001：高维DMRG在特定模型中的应用及密度-矩阵算法（特别是特征值分布）的基础。以一类以三维顶点模型表示的聚合物模型为例，应用三维DMRG，研究了其相变等热行为。对于另一类顶点模型，我们计算了零点熵，并与通过Pauling近似得到的结果进行了比较，结果吻合得很好。对于密度矩阵特征值的普遍渐近性，我们分析了精确可解模型，并提取了两个指数（主指数和次指数）来表征渐近性。在物理上，发现主指数与半无限一维自旋系统的熵有关，其中位置的“权重”随着与原点的距离线性增长；因此，指数的值预计是相当稳健的。至于子指数，我们对可解模型的研究表明，它们之间的值具有显著的一致性，尽管还没有得到简单的物理解释。2002：物理量的精度与密度矩阵的普遍渐近的关系。期望普遍渐近会导致关于保留基的“普遍截断误差”，我们进行了分析和数值计算，但没有决定性的答案，迄今为止

2000 : Higher-dimensional generalization of DMRG and its application to simple systemsGeneralizing the matrix-product structure of the fixed-point wave function in the DMRG (1D quantum, 2D classical), we introduced higher-rank tensors toexpress the "target state" (largest-eigenvalue state of the transfer matrix) for 3D classical statistical systems. The sum-of-the-products-of-tensors structure ofthe wave function allows us to calculate the density matrix and other physical quantities by using the lower-dimensional algorithm ; our higher-dimensionalalgorithm is the one with "nested" structure.2001 : Application of the higher-dimensional DMRG to specific models and foundation of the density-matrix algorithm (esp. eigenvalue distribution)Taking a class of polymer models, expressed as 3D vertex models, we applied the 3D DMRG and studied their thermal behaviorincluding the phase transitions. For another class of vertex models, we calculated the zero-point entropy and compared the resultto t … More he one derived via the Pauling approximation, seeing a good agreement. As for the universal asymptotics of the density-matrixeigenvalues, we analyzed the exactly solvable models and extracted two exponents (main and sub) to characterize the asymptotics.Physically, the main exponent is found to be related to the entropy of a semi-infinite 1D spin system where "weight" of sites growslinearly with the distance from the origin ; thus the value of the exponent is expected to be fairly robust. As for the sub exponent, ourstudies made on the solvable models shows a remarkable coincidence of the value among them, though a simple physical interpretationhas not been attained yet.2002 : Relation between the accuracy of the physical quantities and the universal asymptotics of the density matrix.Expecting that the universal asymptotics leads to the "universal truncation error" with respect to the retained bases, we performedboth analytical and numerical calculations, but with no decisive answer, hitherto Less