Cluster Transfer Matrix Method and Quantum Statistical Mechanics of Spin Systems

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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.