Quasiparticle dynamics in the 4 d − 4 f Ising-like double perovskite Ba 2 DyRuO 6 studied using neutron scattering and machine-learning framework

Double perovskites containing 4d--4f interactions provide a platform to study complex magnetic phenomena in correlated systems. Here, we investigate the magnetic ground state and quasiparticle excitations of the fascinating double perovskite system, Ba$_2$DyRuO$_6$, through Time of flight (TOF) neutron diffraction (TOF), inelastic neutron scattering (INS), and theoretical modelling. The compound Ba$_2$DyRuO$_6$ is reported to exhibit a single magnetic transition, in sharp contrast to most of the other rare-earth (R) members in this family, A$_2$RRuO$_6$ (A = Ca/Sr/Ba), which typically show magnetic ordering of the Ru ions, followed by R-ion ordering. Our neutron diffraction results confirm that long-range antiferromagnetic order emerges at $T_\mathrm{N} \approx 47$~K, primarily driven by 4d--4f Ru$^{5+}$--Dy$^{3+}$ exchange interactions, where both Dy and Ru moments start to order simultaneously. The ordered ground state is a collinear antiferromagnet with Ising character, carrying ordered moments of $μ_{\mathrm{Ru}} = 1.6(1)~μ_\mathrm{B}$ and $μ_{\mathrm{Dy}} = 5.1(1)~μ_\mathrm{B}$ at 1.5~K. Low-temperature INS reveals well-defined magnon excitations below 10~meV. SpinW modelling of the INS spectra evidences complex exchange interactions and the presence of magnetic anisotropy, which governs the Ising ground state and accounts for the observed magnon spectrum. Combined INS and Raman spectroscopy reveal crystal-electric-field (CEF) excitations of Dy$^{3+}$ at 46.5 and 71.8~meV in the paramagnetic region. The observed CEF levels are reproduced by point-charge calculations consistent with the $O_h$ symmetry of Dy$^{3+}$. A complementary machine-learning approach is used to analyse the phonon spectrum and compare with INS data. Together, these results clarify the origin of phonon and magnon excitations and their role in the ground-state magnetism of Ba$_2$DyRuO$_6$.