Spatially Random Heisenberg Spins at Very Low Temperatures. II. Dilute Antiferromagnet with Nearest-Neighbor Substitutional Short-Range Order

Abstract

A formulation for the computation of the single-site dynamics of a dilute Heisenberg antiferromagnet with isotropic nearest-neighbor exchange and arbitrary nearest-neighbor substitutional short-range order (SRO) is given. In the random limit, i.e., when the SRO is zero, the resultant density of states is shown to exactly conserve the first four frequency moments (of the spin-wave density of states) for all lattice structures which admit two nearest-neighbor interpenetrating sublattices and correctly give the leading two terms in the $z^{-1\mathrm{}}$ expansion for all frequency moments. In the presence of the SRO, the theory is shown to lead to correct results in the limit of perfect spatial correlation and to give cumulantlike decoupling results for the three-and higher-site substitutional correlations when these correlations are short ranged. The minimum relative concentration of the magnetic atoms for which the antiferromagnetic long-range order does not obtain is found to be $\frac{2Q}{z}$, where the renormalization factor $Q$ depends both on the actual magnetic concentration $m$ and the SRO parameter. For perfect clustering among the magnetic atoms, $Q=m$ and hence for all nonzero magnetic concentration the system orders, whereas for complete substitutional randomness, $Q=1\mathrm{}$. Hence antiferromagnetic long-range order obtains only when the relative magnetic concentration is higher than $\frac{2}{z}$.