Understanding the evolution of the magnetic ground state in Ba$_4$NaRu$_3$O$_{12}$

Abstract: We report a comprehensive investigation of the quadruple perovskite Ba$4$NaRu$_3$O${12}$, in which we discover a robust spin-lattice coupled ground state characterized by a long-range antiferromagnetic ordering at $T_N \sim$ 257 K. The system's unique structural motif of three symmetrically distinct magnetic ions, including Ru dimers separated by non-magnetic layers, is intimately correlated with its magnetic behavior, as evidenced by temperature-dependent diffraction measurements and specific heat data. The powder neutron diffraction patterns at 13 K showed that the spins within the dimers are antiparallel, leading to a net zero moment contribution and a staggered arrangement of the triangular layers formed by the Ru moments within the corner-shared octahedra along the $c$-axis. The low-temperature specific heat revealed an extra boson peak contribution from optical modes with a maximum vibrational energy of $\sim$55cm$^{-1}$. The charge transport exhibited variable-range hopping (VRH) behaviour below $T_N$, with a stronger energy-dependence than expected from the Efros-Shklovskii model, suggesting the presence of multiparticle correlation effects.