Green’s function theory for spin-$\frac{1}{2}$ ferromagnets with an easy-plane exchange anisotropy

The many-body Green’s function theory with the random-phase approximation is applied to the study of easy-plane spin-1/2 ferromagnets in an in-plane magnetic field. We demonstrate that the usual procedure, in which only the three Green’s functions $⟨⟨S_i^{\mu };S_j^{-}⟩⟩$ $\left(\mu =+,-,z\right)$ are used, yields unreasonable results in this case. Then the problem is discussed in more detail by considering all combinations of Green’s functions. We can derive one more equation, which cannot be obtained by using only the set of the above three Green’s functions, and point out that the two equations contradict each other if one demands that the identities of the spin operators are exactly satisfied. 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.

Figure 1

The temperature dependence of the magnetization of an easy-plane ferromagnet at $h/J=0.1$ for $\Delta =1$ (the isotropic Heisenberg model), $\Delta =0.7$, $\Delta =0.4$, and $\Delta =0$ (the $XY$ model). Comparison between the results obtained from (a) Eq. (18), (b) Eq. (33), and (c) the QMC calculations. The error bar of the QMC results is smaller than the linewidth.

Figure 2

(a) The temperature dependence of the magnetization obtained from Eq. (37) at $h/J=0.1$. (b) Comparison between the results obtained from Eq. (37), the QMC calculations, and the MFT (at $h/J=0.1$ for $\Delta =0.7$).