标题： 理解Ba$_4$NaRu$_3$O$_{12}$中磁基态的演化 显示英文标题

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

摘要： 我们报告了对四重钙钛矿 Ba$_4$NaRu$_3$O$_{12}$的全面研究，在其中我们发现了一个由长程反铁磁有序表征的强自旋-晶格耦合基态，其温度为$T_N \sim$257 K。该系统的独特结构特征是由三个对称不同的磁性离子组成的，包括由非磁性层分隔的 Ru 二聚体，这与其磁行为密切相关，这一点通过温度依赖性衍射测量和比热数据得到了证实。 在 13 K 时的粉末中子衍射图谱显示，二聚体内的自旋是反平行的，导致净磁矩贡献为零，并且在沿$c$-轴的角共享八面体中，由 Ru 磁矩形成的三角层呈现出交替排列。 低温比热显示，光学模式有一个最大振动能量为$\sim$55cm$^{-1}$的额外玻色子峰贡献。 在$T_N$以下，电荷传输表现出可变范围跳跃（VRH）行为，其能量依赖性比埃弗罗斯-什克洛夫斯基模型预期的更强，表明存在多粒子关联效应。 显示英文摘要

摘要： 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.