Spatially Random Heisenberg Spins at Very Low Temperatures. I. Dilute Ferromagnet

Abstract

A formulation is given for calculating magnetic-single-site density of states for a completely random dilute Heisenberg ferromagnet, with isotropic nearest-neighbor exchange, in the limit of very low temperatures. In conformity with Kohn's suggestion that the dynamics of sufficiently random many-body systems may be approximated by that of typical small neighborhoods, a consistent hierarchy of truncation schemes for the spatial matrix elements of the $T$ matrix is described. The case of a drastically truncated Kohn neighborhood, consisting only of two neighboring sites, is worked out in detail. It is shown that for lattices without nearest-neighbor triangles, the given density of states exactly preserves the first four frequency moments. Moreover, for Bravais lattices with $z$ nearest neighbors, all frequency moments of the density of states are given exactly to the two leading orders in $z^{-1\mathrm{}}$. By analyzing the renormalization of the $K\to 0$ spin-wave energy, estimates for the critical temperature are obtained. In the present approximation, the magnetic long-range order cannot occur for magnetic concentrations which are $\le \frac{2}{z}$. For the simple-cubic lattice, numerical computations of the magnetic-single-site density of states and the real and imaginary parts of the coherent exchange are given for several concentrations.