Application of the Hard-Core Boson Formalism to the Heisenberg Ferromagnet

The rigorous method developed in a previous paper for the calculation of the time-dependent properties of localized spin systems is applied to the case of the arbitrary-spin isotropic Heisenberg model of ferromagnetism. A low-density approximation is used to calculate the spin Green's function $G_1(1; 1^{\prime })=-i〈T{S}^{-}\mathrm{}\left(1\right)\mathrm{}{S}^{+}\left(1^{\prime }\right)〉$ within the framework of this formalism. For the special case of $S=\frac{1}{2}$, it is explicitly demonstrated that the nonphysical states produce errors that may not be exponentally small, and that in the hard-core limit these terms disappear. The results of this work prove that for arbitrary $S$ the "truncated" version of the Holstein-Primakoff transformation written in normal product form will produce the correct low-temperature results to all orders in $\frac{1}{2S}$. The form of the space-time transform $G_1(p; z)$ of $G_1(1; 1^{\prime })$ that we obtain is different from that suggested by previous work, since it does not contain any function of $p$ or $z$ in the numerator. In fact, if our result is expanded at temperatures low enough, the first term in the expansion is exactly the result obtained for $G_1(p; z)$ by Silberglitt and Harris from the Dyson transformation by neglecting the effects of the nonphysical states.