Critical magnetic fluctuations in the layered ruthenates Ca2RuO4 and Ca3Ru2O

Materials realizing the $XY$ model in two dimensions are sparse. Here we use neutron triple-axis spectroscopy to investigate the critical static and dynamical magnetic fluctuations in the square-lattice antiferromagnets ${\mathrm{Ca}}_{2}{\mathrm{RuO}}_{4}$ and ${\mathrm{Ca}}_{3}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$. We probe the temperature dependence of the antiferromagnetic Bragg intensity, the $Q$ width, the amplitude, and the energy width of the magnetic diffuse scattering in the vicinity of the N\'eel temperature ${T}_{\mathrm{N}}$ to determine the critical behavior of the magnetic order parameter $M$, correlation length $\ensuremath{\xi}$, susceptibility $\ensuremath{\chi}$, and the characteristic energy $\mathrm{\ensuremath{\Gamma}}$ with the corresponding critical exponents $\ensuremath{\beta}, \ensuremath{\nu}, \ensuremath{\gamma}$, and $z$, respectively. We find that the critical behaviors of the single-layer compound ${\mathrm{Ca}}_{2}{\mathrm{RuO}}_{4}$ follow universal scaling laws that are compatible with predictions of the two-dimensional (2D) $XY$ model. The bilayer compound ${\mathrm{Ca}}_{3}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$ is only partly consistent with the 2D $XY$ theory and best described by the three-dimensional (3D) Ising model, which is likely a consequence of the intrabilayer exchange interactions in combination with an orthorhombic single-ion anisotropy. Hence, our results suggest that layered ruthenates are promising solid-state platforms for research on the 2D $XY$ model and the effects of 3D interactions and additional spin-space anisotropies on the magnetic fluctuations.