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 Néel temperature $T_{\mathrm{N}}$ to determine the critical behavior of the magnetic order parameter $M$, correlation length $\xi$, susceptibility $\chi$, and the characteristic energy $\mathrm{\Gamma }$ with the corresponding critical exponents $\beta , \nu , \gamma$, 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.

• Received 22 December 2021
• Accepted 10 May 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.023181

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society