Correlation Effects In Low Dimensional Quantum Many-particle Systems

In this thesis we study correlation effects in low dimensional quantum many-particle systems with the eigenfunctional theory.First, with the Falicov-Kimball model,we give the general exact expression of the local electron Green function, and apply it to study the correlation effects of the local electrons in one dimension.For the two local electrons case, the correlation ex-ponent of the local electron Green function has weak even-odd oscillation with the dis-tance between these two local electrons,and it approaches zero in the strong coupling limit. While, at half filling of the local electrons,the ground-state phase is complicated. When the conduct electron concentration is near 0.5,the ground state configuration is the chessboard phase for a considerable range of U/t, and the correlation exponent increases from zero to a finite value as U/t increases.When the conduct electron con-centration is far away from 0.5,the ground state configuration is the segregated phase, and the correlation exponent first increases and then jumps to zero when U/t is larger than a finite value.Second, with the eigenfunctional theory, we study a general interacting electron system, and give a rigorous expression of its ground state energy which is composed of two parts, one part is contributed by the non-interacting electrons,and another one is represented by the correlation functions that are controlled by the electron correlation. Moreover, according to the rigorous expression of the ground state energy, an effective scheme beyond the local density approximation of the density functional theory is pro-posed. Then, we apply this general theory to study two one-dimensional models:the spin-1/2 XXZ model and the Hubbard model.Numerical calculations show that the re-sults obtained in the scheme proposed above and its extensions according to the special nature of the quantum system in hand is quite close to that obtained from Bethe ansatz.