Low-energy excitation spectrum for ferromagnetic spin waves on a percolating network: An effective-medium approximation

We develop a theory for the spin waves in a dilute Heisenberg ferromagnet. The $d$-dimensional bond percolating network of ferromagnetic spins is solved within the effective-medium approximation. It is shown that these excitations obey a classical equation of motion. We have found that for $2<d<\mathrm{}4\mathrm{}$, there is a critical frequency ${\omega }_c$, consistent with the mobility edge, such that ${\omega }_c$ scales as ${(p-p_c)}^2$. When $\omega <{\omega }_c$, the excitations are extended, while when $\omega >{\omega }_c$, they are localized. We also found that there is a rapid increase in the density of states at the mobility edge. The jump near the mobility edge is found to scale as ${(p-p_c)}^{1-\left(\frac{d}{2}\right)}$.