| This paper mainly deals with quantum phase transitions and critical phe-nomena in low dimensional systems. These systems include one-dimensional two-component bosons system,one-dimensional Heisenberg model,one-dimensional Bose-Fermi mixture system and two-dimensional electron gas, etc. In low dimen-sional system, quantum phase transition controlled by the quantum fluctuation is worth to study. In this paper, the methods such as analytical exact solu-tion,numerical diagonal method are applied to study the related problems.Firstly this paper gives a brief introduction of low dimensional system in condensate matter physics, such as the background,research significance and partial results, the importance of low dimensional physics in the condensed matter physics is demonstrated.In the second chapter of the paper, the Bethe ansatz method is used to solve one-dimensional two-component bosons with aδ-function potential considering with negative coupling constant. The features of the ground state and low-lying excited states of this model are discussed explicitly by analytical and numerical methods. We conclude that the ground state is a bound state, whose quasi-particles are paired, and when one pair is destroyed, there will be an excited state. In particular we give a thorough discussion of this model to N=2, and show that the ground state is a ferromagnetic state regardless of the coupling constant.Then, in the third chapter, based on the Bethe ansatz method of the one-dimensional Heisenberg model under twisted boundary conditions, we study the spectra of the persistent current carried by the low-lying excited states. Though the energy spectra of the singlet and triplet excitations are degenerate, their transport properties are quite different. The non-vanishing behavior of the per-sistent current is the main contribution to the spin stiffness at finite temperatures, which may provide some useful physical intuitions to the transport properties in integrable models. Since the theory of the Bose-Fermi system is not complete yet, there are lots of issues to be discussed and solved. In the fourth chapter, we study a one-dimensional cold atomic system of Bose-Fermi mixtures based on the Bethe ansatz method. Corresponding to three possible choices of the reference states in the quantum inverse scattering method, three sets of Bethe ansatz equations are derived explicitly. Through the analysis of these equations, the features of the ground state and low-lying excitations are investigated, and we obtain the conclusion that the ground state should be bosonic purely. And the low-lying excitations are mixture of bosons and fermions which can be shown numerically.Since the realization of Bose-Einstein condensation, ultracold atoms in opti-cal lattice becomes a rich field of investigation for theoretical and applied issues of condensed matter physics. Theoretically, ultracold atoms will undergo a quan-tum phase transition from a superfluid state to a Mott insulator state. In the fifth chapter, we investigate the entanglement of bosonic atoms in a two-site optical lattice. Negativity is used to measure the entanglement between two sites. At extremely low temperatures, the entanglement can be controlled by the magnetic field intensity. Increasing the temperature, the entanglement increases and it is larger for the antiparallel fields than for the parallel fields. Given a proper chem-ical potential, we can also demonstrate that anti-parallel fields will enhance the entanglement of two sites.In the sixth chapter, it mainly discusses a two-dimensional electron system realized in an alloy disorder potential AlxGa1-xAs/AlyGa1-yAs heterostructure. The goal is to explain the abnormal scaling behavior in integer quantum hall effect which was observed on this material by Princeton group recently. Since this work is going on, the paper reports some elementary computational results. In particular, the appearance of alloy disorder potential open a new degree of freedom for lattice model of integer quantum hall effect.
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