Abstract
In this thesis, we perform numerical studies on two special one-dimensional physical systems. We first investigate the insulating phases of spin-2 bosons loaded in optical lattice by non-Abelian DMRG calculation. Three possible phases are identified in our calculation, namely, ferromagnetic, dimerized and trimerized. By finite-size extrapolations, the exotic trimerized phase is believed to be a critical phase. We here also define dimerized and trimerized order parameters to describe the exotic phases. By using the scattering lengths in literature, we point out the possible phases of related spin-2 elements in cold atom experiments. Finally we propose an easy-implemented method to verify the ternary phase diagram experimentally. Secondly, We carry out a systematic study in entanglement entropy scaling of the XXZ spin-1/2 chain. We evaluate the scaling constant, central charge $ c $, from the scaling of both entanglement entropy and finite-size ground state energy by three different numerical methods: finite-size DMRG, infinite-size DMRG (iDMRG) and exact diagonalization (ED). Our calculation shows that all the estimations of $ c $ in the critical XY phase are influenced by the neighboring ferromagnetic phase and result in crossover behavior. In the end, we certify the universality class of the critical trimerized phase as a $ SU(3)_{1} $ WZW model by evaluating central charge $ c $ and scaling dimension $ x $. In addition, unusual scaling behavior of entanglement and R\'enyi entropy is observed.