Magnon band splitting without altermagnetism in CuF2

Conclusive identification of an altermagnetic state requires going beyond mere symmetry arguments. We illustrate this in a combined computational and experimental study of the rutile-like material CuF${}_2$, which is on the list of predicted altermagnets. Using ab initio and linear spin-wave calculations supplied by magnetization measurements, we show that CuF${}_2$ in its experimental monoclinic structure can be described by a spin-$\frac{1}{2}$ model of weakly coupled square-lattice layers with the in-plane coupling $J_1\simeq 115$ K and two synergistic antiferromagnetic interplane couplings amounting to 4% and 8% of $J_1$, respectively. Driven by long-range superexchange, these interlayer couplings are oblique to the square planes, resulting in the unit-cell doubling in the magnetically ordered state, thus effectively suppressing any altermagnetic band splitting. Concurrently, we identify unusually strong Dzyaloshinskii-Moriya interactions, $\mathrm{\mid }\mathbf{D}\mathrm{\mid }\mathrm{/}{J}_1\simeq 0.3$, that produce spin canting and, together with order-by-disorder effect, pin the Néel vector to the crystallographic $b$-axis. Additionally, DM anisotropy promotes magnon band splitting, but these bands remain non-chiral. Our results highlight the importance of relativistic effects even in $3d$ magnets with altermagnetic symmetries.