Random Phrase Approximation In The Low-dimensional Quantum Anisotropy Heisenberg Model

The study of magnetism materials has attracted increasing attention. So persons develop many experimental and theoretical methods to discuss magnetism of magnet. The double-time Green function is just one of them. This method provides a simple enough way for giving results in good agreement with other approaches and experiments in a wide range of temperatures and magnetic fields. So it is good method to investigate magnetism of magnet. In this paper, we applied the double-time Green function method to investigate ferromagnetic material in a magnetic field or in coexisting transverse and longitudinal magnetic fields, which can be described by low-dimensional quantum Heisenberg model.In Chapter 1, the research background, contents and methods of this dissertation are introduced.In Chapter 2, the low-dimensional anisotropic ferromagnetic Heisenberg model, in a magnetic field, is studied. By use of the double-time Green's Function method within the random phrase approximation, the expression of the magnetization is obtained through the solution of the motion equation. It is found that the magnetization, the susceptibility and the correlation function are dependent upon the spatially anisotropy. Meanwhile we also observe the crossover from the anisotropic system to XY-kind behavior in low temperature. The crossover temperature attained is well fitted to the results obtained by other theoretic techniques. When the relationship between the magnetic field and maximum in the temperature dependence of the susceptibility is investigated, our results show that the field dependence of maximum in the temperature dependence of the susceptibility yields the power laws behavior, but doesn't support the 2/3 power laws which are obtained from the mean-field Landau's theory. For the isotropic case, our results are in agreement with the results obtained by other theoretic techniques. And values of power-low are dependent on the spatially anisotropy. However, the 2/3 of power-low is only fitted to the mean-field Landau's theory and three dimension.In Chapter 3, two-dimensional anisotropic Heisenberg ferromagnet is studied in the coexisting transverse and longitudinal magnetic fields. By use of the double-time Green's Function method within the random phrase approximation, the expression of the transverse and longitudinal magnetization are found through the solution of the motion equation. The relations between the magnetization, the susceptibility and temperature, the transverse field and anisotropic parameter are discussed. We have found that the transverse magnetic field plays an important role in the magnetic properties of the system. The transverse magnetic field has a tendency to decrease the longitudinal magnetization, and increases the transverse magnetization. And the magnetic properties of this model are found to be dependent on the anisotropy. The stronger the anisotropy (i.e. smaller a), the weaker the fluctuation, the smaller the maximum values of the susceptibilities, the more slowly the magnetization components change with the temperature and transverse magnetic field. For any temperature T>0 and anisotropic parameter a<1, the magnetic anisotropy tends to suppress part of the fluctuation along the hard axis.In the last Chapter, the conclusions of the whole dissertation are given, as well as prospect for the future work.