Long range order in strongly correlated systems

Long range order (LRO) is one of the most important properties of physics systems, especially for a strongly correlated system. In this thesis, the long range order in a few strongly correlated systems is investigated both rigorously and numerically.; The Neel long range order in various quantum spin models was studied. A spatially anisotropy antiferromagnetic Heisenberg system was studied and a critical point Jup⊥S was obtained. When J⊥ is larger than Jup⊥S , the Neel long range order was proved to exist in the ground state of the system. Then an onsite single-ion anisotropy D-term was imposed on the above system and its effect on the existence of the long range order was studied. We obtained a critical line on the DJ⊥ plane such that above which the long range order exists. To get insight into the long range order in the two-dimensional isotropic Heisenberg model, the XXZ model and the Heisenberg system with next nearest neighbors interaction were studied. For the XXZ model, two critical couplings DeltaXY and DeltaIsing were obtained such that when 0 ≤ Delta ≤XY or Delta ≥ DeltaIsing, the Neel order appears. For the antiferromagnetic Heisenberg model with next nearest neighbors interaction both spin S = 1/2 and S = 1 were studied. For S = 1/2, a critical next nearest neighbors ferro-coupling Jc2 was obtained such that when J2 ≤ Jc2 , the Neel long range order appears, while for S = 1 an improved Jc2 was obtained such that when J2 ≤ Jc2 the Neel order still exists even with frustration.; Magnetic orders in the two-dimensional periodic Anderson model (PAM) were investigated in the project. Several numerical methods including exact diagonalization, mean field methods and the constrained path Monte Carlo (CPMC) method were used here. We studied the effect of the dispersion of the impurity band on the magnetism and gave estimated phase diagrams on the band filling and impurity chemical potential plane, by comparing the ground state energies and by studying the Fourier transformation of the spin-spin correlations.