Spin fluctuations in random magnetic-nonmagnetic two-dimensional antiferromagnets. II. Heisenberg percolation

Abstract

In this paper we report an extensive study using neutron scattering techniques of the spin fluctuations in the two-dimensional diluted near-Heisenberg antiferromagnet ${\mathrm{Rb}}_{2}{\mathrm{Mn}}_{c}{\mathrm{Mg}}_{1\ensuremath{-}c}{\mathrm{F}}_{4}$. The concentrations studied are $c=0.54$, $c=0.57$, and $c=0.60$; the site-percolation concentration for the nearest-neighbor square lattice is ${c}_{p}=0.593$ so that these experiments span the percolation threshold. The point $c={c}_{p}$, $T=0$ represents the percolation multicritical point which terminates the line of second-order transition of the infinite network. We give a detailed description of the theory of the magnetic behavior around the percolation point; as the temperaturelike scaling field we suggest $\ensuremath{\mu}(T)={\ensuremath{\kappa}}_{1}(T)$ where ${\ensuremath{\kappa}}_{1}(T)$ is the inverse correlation length for the associated one-dimensional chain; for the static structure factor we propose the formula $\mathcal{S}(|{c}_{p}\ensuremath{-}c|,\ensuremath{\mu},\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})\ensuremath{\propto}{\ensuremath{\kappa}}^{\ensuremath{\eta}}{({\ensuremath{\kappa}}^{2}+{Q}^{2})}^{\ensuremath{-}1}$, where $\ensuremath{\kappa}=\ensuremath{\kappa}(|{c}_{p}\ensuremath{-}c|,0)+\ensuremath{\kappa}(0,\ensuremath{\mu})$; that is, we assume that the geometrical and thermal inverse correlation lengths are simply additive. The $c=0.60$ sample is found to have a smeared second-order phase transition at about 8 K to a state with two-dimensional long-range order but only weak correlations in the third direction. The spin fluctuations in the precritical region are essentially identical to those in the concentrated systems, thus demonstrating that the phase transition in the $c=0.60$ infinite network is little affected by the proximity to the percolation threshold. For the $c=0.54$ and $c=0.57$ samples the correlation lengths and susceptibility increase with decreasing temperature down to about 3 K, below which temperature they saturate; for both samples the inverse correlation length is well described by the simple formula $\ensuremath{\kappa}=\ensuremath{\kappa}(\ensuremath{\Delta}C,0)+{[{\ensuremath{\kappa}}_{1}(T)]}^{0.9\ifmmode\pm\else\textpm\fi{}0.05}$; the susceptibility follows the law $\mathcal{S}(0)\ensuremath{\sim}{\ensuremath{\kappa}}^{1.5\ifmmode\pm\else\textpm\fi{}0.15}$; both results are consistent with our scaling crossover formula and the assumption of the importance of the underlying one-dimensional links in the percolation clusters. The corresponding thermal critical exponents are ${\ensuremath{\nu}}_{T}=0.9\ifmmode\pm\else\textpm\fi{}0.1$ and ${\ensuremath{\gamma}}_{T}=1.5\ifmmode\pm\else\textpm\fi{}0.15$ compared with the percolation exponents ${\ensuremath{\nu}}_{p}=1.36$ and ${\ensuremath{\gamma}}_{p}=2.43$, so that the experimental crossover exponent is $\ensuremath{\varphi}=1.56\ifmmode\pm\else\textpm\fi{}0.15$. There is currently no first-principle theory which properly accounts for this value of the crossover exponent, although the experiment includes within the errors the value $\ensuremath{\varphi}=1.7$, appropriate to a self-avoiding walk ansatz for the principal paths along which the correlations spread.