Spin Waves and Long-Range Order in the Diluted Heisenberg Ferromagnet at Zero Temperature

The variational treatment of spin waves in the randomly diluted Heisenberg ferromagnet at zero temperature is studied. It is argued that a convenient variational principle for the excitation spectrum of an alloy is difficult to formulate rigorously. However, at extremely long wavelengths a simple variational calculatation probably has approximate validity providing localized excitations in isolated clusters of magnetic sites are excluded. Within the approximation which ignores this exclusion we obtain a second-order variational bound for the spin-wave energy which, unlike that found by Murray, is positive at all concentrations. Similar results for concentrations above the critical percolation concentration $x_p$ are obtained when the localized excitations are excluded in which case the restricted configuration averages can only be evaluated approximately. We point out that the critical concentration $x_c$ for the occurance of long-range order depends only on the properties of the infinite cluster. Thus the thermal stability of spin waves depends on the dimensionality of the infinite cluster. An argument is given to show that the infinite cluster is not one dimensional. The range of concentrations for which the infinite cluster is two dimensional is either nonexistent or small. We conclude, then, that $x_c$ is close, if not exactly equal, to $x_p$. The condition for a discontinuity in $T_C\mathrm{}\left(x\right)\mathrm{}$ for $x\to x_c$ is discussed in terms of a simple periodic model for dilution.