Theory of the spin excitations of ${\mathrm{Rb}}_2{\mathrm{Mn}}_x{\mathrm{Ni}}_{1-x}{\mathrm{F}}_4$

We give a systematic treatment of the spin excitations of a family of disordered quasi-two-dimensional Heisenberg antiferromagnets, ${\mathrm{Rb}}_2{\mathrm{Mn}}_x{\mathrm{Ni}}_{1-x}{\mathrm{F}}_4$, for arbitrary values of $x$. The density of states, static response functions, and the dynamic susceptibility are calculated numerically. Results at several concentrations are presented graphically. We derive simple analytic theories which give an adequate description of the calculated quantities. The static response functions characterize excitations at low energy and long wave length, and enable us to predict the concentration dependence of the anisotropy gap frequency and the temperature dependence of the magnetization. The dynamic susceptibility is in good agreement with recent neutron-scattering experiments on ${\mathrm{Rb}}_2$ ${\mathrm{Mn}}_{0.5}$ ${\mathrm{Ni}}_{0.5}$ ${\mathrm{F}}_4$. The spin-wave density of states for $0.1<x<\mathrm{}1\mathrm{}$ is found to split into two separated subbands, consisting essentially of Mn and Ni spin excitations, respectively. The evolution of the two subbands as a function of concentration is studied through the dynamic susceptibility, which resolves both $q$ and $\omega$, and by numerical tests for localization of the normal modes. The two-mode character of this system persists even after the gap between subbands has closed to become a pseudogap.