Cluster Transfer Matrix Method and Quantum Statistical Mechanics of Spin Systems

1. 本项目的主要部分是开发一种基于Suzuki-Trotter公式的方法，通过该方法我们可以计算一维自旋系统在极低温t K_BT/2JS^2-0.1下的热力学量子效应。我们已经观察到，在这种情况下，由最近邻居交换耦合-2J产生的自旋之间的相关性随着t的减小而在长距离上增长，因此应用Suzuki-Trotter公式的分解大小应设置为与要计算热力学的温度下的相关长度一样大。这种思想导致了聚类转移矩阵法的产生。对S = 1/2 XY模型的解析应用表明，即使在3自旋簇加上Trotter数M = 6的情况下，与精确计算结果相比，数值计算也有很好的改进。对于S = 1/2 Heisenberg模型，在铁磁情况下，取4个自旋团+ M = 6，得到了令人满意的磁化率和比热降至t - 0.1的结果，但在反铁磁情况下，结果不太令人满意。在后一种情况下，我们需要扩大集群的大小和M的大小。我们给出了在不达到半经典极限的情况下计算某些量子自旋系统隧穿分裂的解析公式。能级的分裂是量子相干性的特征频率，可以在考虑的自旋系统中看到。对孤子承载系统的统计力学进行了广泛的回顾。这篇综述的目的是作为当前物理学领域的入门课程。

1. The main part of this project is to develope a method on the basis of the Suzuki-Trotter formula, by which we can compute the quantum effect on thermodynamics of one-dimensional spin systems down to very low temperature t K_BT/2JS^2-0.1. We have observed that the correlation among spins, which is produced by the nearest neighbour exchange coupling -2J in the present case, grows up long distance with decreasing t, so that the size of decomposition for applying the Suzuki-Trotter formula should be set to be as large as the correlation length at temperature down to which one want to compute the thermodynamics. This idea has lead us the cluster transfer matrix method. An analytical application to S = 1/2 XY model has shown a good improvement of numerical computation even in the case of 3 spin cluster plus Trotter number M = 6, on comparing the results with exact ones. For S = 1/2 Heisenberg model we have obtained satisfactory results of susceptibility and specific heat down to t - 0.1 in the ferromagnetic case by taking 4 spin cluster plus M = 6, but not so satisfactory results in teh antiferromagnetic case. We have to enlarge both the size of cluster and M in the latter case.2. We have presented an analytic formula of computing the tunnelling splitting in some kinds of quantum spin system without going to the semiclassical limit. The splitting of energy levels is the characteristic frequency of quantum coherence to be seen in spin systems under consideration.3. An extensive review of statistical mechanics of soliton bearing systems has been presented. This review has been intended to be available as an introductory course to the present field of physics.