| Quantum spin systerm in low dimensions has been one of the most attractive field in condensed matter physics for many decades . Not only because of many meaningful properties of such systerms, but also in the field the existence of simplified models, which serve as paradigms for understanding and studying the role of strong interactions in many-electron systems. Although simple , these models quite often can not be solved exactly . It is just the difficulty that the strongly correlated electron systems can not be studied by analytical methods, a variety of simulative numerical methods come out sequentially. There are a few typical methods such as the Exact Diagonalization (ED), the Qutum Monte Carlo (QMC), the Variation Monte Carlo (QMC), the Renormalization Group (RG) , and the Density Matrix Renormalization Group (DMRG) and so on . In this thesis, we mainly employed ED and DMRG methods to sumulate and investigate quai-one dimensional quantum spin systerm systerms.In the first chapter , we introduce briefly the idea of the ED method and its advantages and limitations , meanwhile ,by using ED , we investigate the total spin of ground state (GS) of the finite Heisenberg antiferromagnet (HAF)cluster and the effect of side spins on the spin-spin correlation on HAF chain . For clusters of 13 nd 25 sites with the special shape , the numerical results show that their GS are respectively 6-fold and 8-fold degeneracy ( while the GS of macroscopic antiferromagnet is singlet) . So we can conclude that the ground-state total spin (GSTS) is related to its shape for finite bipartite HAF . It is possible that the GS take a much highter value of TS than the lowest possible one(LPST) and higher degenerate. Combining our theoretical analysis , we extend Lieb and Mattis's approaches for finite systems. Moreover, the effect of the ground-state degeneracy on the spin status and physical properties are investigated and discussed. For a class of quasi-one-dimensional cluster, we study the effect of side spins on the spin-spincorrelation on chain, Our calculations show that the side spins added in the same sublattice can effectively strengthen the spin-spin correlation in large distance region and make the change tend to flat. It is exactly proved that periodically adding side spins can set up magnetic long-range order in the ground state. Also, the effect of density of side spins on correlation strength is investigated.The second chapter of the thesis discuss emphatically the development, algorithm, remarks and present problems of DMRG. Besides, we try to add odd or even sites to the rung of a two-leg coupled spin ladder , and to calculate the energy and spin gap of the class of models .In the third chapter , we give a conclusion and discussion. At last we point out the problems of the DMRG procedure itself. |
