Cluster approximation forq-dependent correlations in magnetic and ferroelectric systems

A self-consistent cluster approximation is developed for the wave-vector ($\overrightarrow{\mathrm{q}}$)-dependent spin-spin correlation in Ising models describing magnetic and ferroelectric systems. The method is particularly suitable for describing systems with competing short-range interactions. The selfconsistent approximation for the $\overrightarrow{\mathrm{q}}$-dependent susceptibilities with clusters of size $N$ is found to be $x_{\nu }^{-1\mathrm{}}\left(\overrightarrow{\mathrm{q}}\right)=C^{-1\mathrm{}}T\left[M_{\nu }^{-1\mathrm{}}\right(\overrightarrow{\mathrm{q}})-(1-C\left)\right]$, $\nu =1,2,\dots ,N$, where $M_{\nu }^{-1\mathrm{}}\left(\overrightarrow{\mathrm{q}}\right)$ are the eigenvalues of the Fourier transform of ${\left(M^{-1\mathrm{}}\right)}_{\mathrm{ij}}$ where $M_{\mathrm{ij}}$ is the pair-correlation matrix of spins within the cluster calculated by the exact Hamiltonian of the cluster. The constant $C$ is the ratio of the number of nearest neighbors inside the cluster to the total number of nearest neighbors. The method is applied to calculate scattering intensities in potassium-dihydrogen-phosphate-type hydrogen-bonded ferroelectrics. We find a strong anisotropy in the $\overrightarrow{\mathrm{q}}$ dependence of the intensity, exhibiting a strong suppression of fluctuations along the easy ($z$) axis. The results are found to be in good agreement with neutron scattering data in K${\mathrm{D}}_2$P${\mathrm{O}}_4$. We also investigate the ice-rule limit of our results. In that case a singularity of the type ${\chi }^{-1\mathrm{}}\left(\overrightarrow{\mathrm{q}}\right)\simeq {\chi }^{-1\mathrm{}}\mathrm{}\left(0\right)+B\left(T\right)\mathrm{}\frac{q_z^{}{}_{}{}^2}{({2q}_{\perp }^2+q_z^2)}$ for $q\to 0$ is found, similar to that generated by long-range dipolar forces.