The materials such as iron, and cobalt, nickel, have the spontaneous magnetizationwhen temperature is not too high(negation external magnetic feld has intrinsic magneticmoment). When the temperature rises to above the Curie point, the spontaneous magne-tization disappear. These are called ferromagnet material. Since Quantum mechanics setup we understand ferromagnetism is the quantum efects. Heisenberg exchange energy canput forward to explain the mechanism of ferromagnetic, called Heisenberg model. In theprocess of the research, diferent theory was used to deal with diferent range of temper-ature of magnetic iron magnet. Under low temperature using spin wave theory, near theCurie temperature is with the molecular feld theory, and Thermodynamic perturbationtheory are used when above the Curie temperature. Is there a feasible method to unifedthe diferent range temperature theories? The answer is yes. The thermodynamic green’sfunction method proposed in1959can be used for ferromagnetic theory, commonly usedapproximate “cut of”method, which is applicable both for zero-temperature and fnite-temperature regions. The main points of this method is from the motion of the green’sfunction equations, and then using the feld operator to calculate the time derivative ofGreen’s functions for single particle. So you get the motion of green’s function whichcontains double particles. Repeated derivative of the motion of green’s function, we getmotion of Green’s function chain. And then some approximation methods are used todecoupling the green’s function into a lower order ones, the equation of motion chain willbe closed.In this paper, Green’s function method is applied to investigate the anisotropiesferromagnetic Heisenberg model. For the antiferromagnetic that with exchange anisotropyand frustration model, spin wave method is applied. Some properties of the ground stateare get. This thesis can be divided into four parts as follows:The fst chapter is introduction. In this part, we introduced general concept of low-dimensional ferromagnetic Heisenberg model with anisotropies, and the antiferromagneticmodel with anisotropy and frustration. Also, we have introduced the research course ofthese ferromagnetic systems. At last, the theory research methods used in this paper ismainly introduced.In the second chapter, Green’s function method is used to investigate the two-dimensional spin-1ferromagnetic Heisenberg model with the exchange and single-ionanisotropies. And the system are treated by the random phase approximation (RPA)for the exchange term, and Anderson-callen approximation (ACA)for the single-ion term,respectively. Calculation results show the relationship of susceptibility, magnetization,specifc heat, internal energy between the external magnetic feld, temperature and theanisotropic parameters. In addition, we also analyzed the temperatures at the point ofsusceptibility and specifc heat maximum which present the power lows and linear rulewith external magnetic feld in the diferent scopes, respectively. Our results are deviatiedfrom the ones Landau theory. In the last, something attention is pay to the properties ofmagnetization and internal energy at the point of susceptibility and specifc heat maxi-mum.In the third chapter, spin wave method is applied to investigate the two-dimensionalspin-1/2antiferromagnetic J1J2model with the exchange anisotropy.We present theground-state properties of this model from our calculations, in particular the ground-stateenergy and zero-temperature magnetization as functions of the anisotropy and frustrationparameter. Spin wave dispersion relation is discussed in this chapter.In the fourth chapter, the main results of our research are summarized. And also,some defciencies are pointed out, and the further research are mentioned. |