Development of New Numerical Renormalization Group Method and Its Application to Low-Dimensional Phase Transitions

1. (1)对于具有非对称转移矩阵的系统，我们引入了“重整单位”，使我们能够构造一个仅涉及对称密度矩阵的数值重整方案。(2)建立了密度矩阵特征值与保留基数之间的普遍渐近关系。(3)对于三维经典统计系统，我们找到了局部张量的显式构造方案，这是传递矩阵的最大特征值状态波函数的重要组成部分，建立了DMRG的高维扩展。我们还证明了二维量子系统可以用三维经典系统的各向异性极限来处理。(1)在nacl型晶体的表面模型中，我们证实了阶跃张力对温度的反常依赖。(2)计算了各向异性台阶张力和晶面形状，验证了台阶在表面上的归一化自由随机行走图像的有效性。(3)对邻近表面的吸附进行了进一步的研究。我们发现，由于吸附质密度的波动，产生了吸引的台阶间相互作用，并且由于台阶间的吸引，发生了一级相变。(1)通过对量子自旋链M-H曲线的精确计算，证明了临界场附近δ函数玻色-气体图的有效性。(2)在S=1/2自旋阶梯和S=1双线性双二次链的M-H曲线中发现了场尖奇异性。(3)对于蜂窝晶格的vbs型AF自旋模型，我们计算了相关长度和亚晶格磁化强度。在S=3/2双线性双二次自旋模型中，我们基于张量积分析对基态波函数进行了pwfrg辅助的变分计算。我们发现了尼尔态和无序态之间的一阶跃迁。(4)计算了二维类伊辛模型的横向磁化率，发现其温度导数与比热具有相同的临界奇异性。(1)建立了一种利用PWFRG/DMRG计算自由能和熵的方法，并将其应用于完全受挫反铁磁体的零点熵计算。(2)研究了具有线性缺陷的二维类ising体系的临界行为，发现即使存在4-自旋相互作用，临界指数的非普适行为仍然存在。同时，确定了缺陷指数之间的标度关系。少

1. (1) For systems with asymmetric transfer matrices, we introduce the "renormalized identity" which allows us to constructed a numerical renormalization scheme where only the symmetric density matrices are involved. (2) We establish a universal asymptotic relation between the density matrix eigenvalue and the number of retained bases. (3) For 3D classical statistical system, we find an explicit construction scheme of local tensor which is an essential building block of the maximum-eigenvalue-state wavefunction of the transfer matrix, establishing a higher-dimensional extension of DMRG. We also show that 2D quantum system can be treated by an anisotropic limit of a 3D classical system.2. (1) We confirm the anomalous temperature dependence of the step tension in the surface model of NaCl-type crystal. (2) We calculate the anisotropic step tension and the crystal facet shape, from which we verified the validity of the renrmalized free random-walk picture for the step on a surface. (3) We … More study the vicinal surface with adsorption. We find that the attractive inter-step interaction emerges due to the adsorbate density fluctuation, and that there occurs a first-order phase transition due to the inter-step attraction.3. (1) From the accurate calculation of the M-H curve of the quantum spin chains, we show the validity of the δ-function bose-gas picture near the critical fields. (2) We discover the middle-field cusp singularity in the M-H curve for the S=1/2 spin ladder and S=1 bilinear-biquadratic chain. (3) For the honeycom latttice VBS-type AF spin model, we calculated correlation length and the sublattice magnetization. The S=3/2 bilinear-biquadratic AF spin model, we perform PWFRG-assisted variational calculation based on the product-of-tensors ansatz for the ground-state wavefunction. We find the first-order transition between the Neel and the disordered states. (4) We calculate transverse susceptibility for the 2D Ising-like models, to show that its temperature derivative has the same critical singularity as the specific heat.4. (1) We establish a method to calculate the free energy and the entropy by the PWFRG/DMRG, and apply it to obtain the zero-point entropy for fully-frustrated antiferromagnets. (2) We study the critical behavior of the 2D Ising-like system with linear defect, to find that the non-universal behavior of the critical exponents persists even if the 4-spin interactions are present. Also, the scaling relations between the defect exponents are confirmed. Less