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

Abstract

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 $\hat{I}”S_{\mathrm{vN}}^{\mathrm{SO}}$. The framework is grounded on the relativistic crystal field theory, and is illustrated through a series of quantum materials: $A_2{\mathrm{TaCl}}_6$ ($A=\mathrm{K},\mathrm{Rb}$), $A_2{\mathrm{MgReO}}_6$ ($A=\mathrm{Ca},\mathrm{Sr},\mathrm{Ba}$) and $\mathrm{B}{\mathrm{a}}_2\mathrm{NaOs}{\mathrm{O}}_6$, all analyzed in their paramagnetic phases, alongside the $\mathrm{Re}{\mathrm{F}}_6$ molecular system. The entropies are derived from measurements of the optical $d$-$d$ transitions $\hat{I}{“}_7(t_{2g})\leftarrow \hat{I}{“}_8(t_{2g})$ and $\hat{I}{“}_8(e_g)\leftarrow \hat{I}{“}_8(t_{2g})$, and of the effective magnetic dipole moment $\hat{I}{¼}_{\mathrm{eff}}$. It is demonstrated that, regardless of the system, the Kramers doublet $\hat{I}{“}_7(t_{2g})$ exhibits no spin-orbital von Neumann entropy. The entropies obtained for the relativistic crystal field states $\hat{I}{“}_8(t_{2g})$ and $\hat{I}{“}_8(e_g)$ 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.