Application of the Green's function method to an easy-plane Heisenberg model

Abstract

Many-body Green's function method is a powerful tool for theoretical studies of spin systems. In this formalism, a certain decoupling approximation is generally required to terminate the infinite hierarchy of equations of motion for higher-order Green's functions. The so-called "Tyablikov decoupling" or random-phase approximation (RPA) is a very simple yet effective way to perform this operation. In the present work, this procedure is applied to the study of easy-plane ferromagnets in an in-plane magnetic field. We demonstrate that one should pay careful attention in using the RPA scheme for such a complicated system, which does not have rotational symmetry around the direction of the magnetization. By considering all combinations of the Green's functions, we derive two equations for the magnetization, and point out that the two equations contradict each other if one demands that all operator identities (S2 = S(S + 1), πsr=-s[Sμ – r] = 0; μ = x,y, z) are satisfied in the RPA. Then we discuss the cause of the contradiction and attempt to improve the method in a self consistent way. In our procedure, the effect of the anisotropy can be appropriately taken into account, and the results are in good agreement with the quantum Monte Carlo calculations.