Quantifying the Spin-Orbital Entanglement in $5d^1$ Quantum Materials

The spin-orbital entanglement in $5d^1$ transition metal ions embedded in double perovskites, where anomalous effective magnetic dipole moments are frequently observed, is quantified by the spin-orbital von Neumann entropy $\Delta S_{\rm vN}^{\rm SO}$. The framework is grounded on the relativistic crystal field theory, and is illustrated through a series of quantum materials: $A_2{\rm TaCl}_6$ ($A = {\rm K}, {\rm Rb}$), $A_2{\rm MgReO}_6$ ($A = {\rm Ca}, {\rm Sr}, {\rm Ba}$) and ${\rm Ba_2NaOsO_6}$, all analyzed in their paramagnetic phases, alongside the ${\rm ReF_6}$ molecular system. The entropies are derived from measurements of the optical $d$-$d$ transitions $\Gamma_7(t_{2g})\leftarrow\Gamma_8(t_{2g})$ and $\Gamma_8(e_g)\leftarrow\Gamma_8(t_{2g})$, and of the effective magnetic dipole moment $\mu_{\rm eff}$. It is demonstrated that, regardless of the system, the Kramers doublet $\Gamma_7(t_{2g})$ exhibits no spin-orbital von Neumann entropy. The entropies obtained for the relativistic crystal field states $\Gamma_8(t_{2g})$ and $\Gamma_8(e_g)$ uncover that, a larger effective magnetic dipole moment can be attributed to a grater spin-orbital entanglement, yet paradoxically not to a larger spin-orbit coupling constant.