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

数值重正化群的基础及其高维推广

基本信息

批准号：
12640393
负责人：
AKUTSU Yasuhiro
金额：
$ 2.5万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

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

二○年：DMRG的高维推广及其在简单系统中的应用推广了DMRG（一维量子，二维经典）中不动点波函数的矩阵乘积结构，引入高阶张量来表示三维经典统计系统的“目标态”（转移矩阵的最大本征值态）。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;我们的高维算法是具有“嵌套”结构的算法。2001：高维DMRG在具体模型中的应用及密度矩阵算法的建立（特别是本征值分布）以一类聚合物模型为例，采用三维顶点模型，应用三维DMRG方法研究了它们的热行为，包括相变。对于另一类顶点模型，我们计算了零点熵，并将计算结果与其它顶点模型的零点熵进行了比较。 ...更多信息他一个通过鲍林近似推导，看到一个很好的协议。对于密度矩阵本征值的普适渐近性，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）来表征其渐近性，物理上，主指数与半无限一维自旋系统的熵有关，在这个系统中，位置的“权重”与离原点的距离成线性关系，因此该指数的值具有相当的鲁棒性.关于次指数，我们对可解模型的研究表明，尽管还没有得到简单的物理解释，但它们之间的值具有显著的一致性。物理量的精度与密度矩阵的普适渐近性之间的关系。期望普适渐近性导致关于保留基的“普适截断误差”，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，