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

TL;DR

This work defines the spin-orbital von Neumann entropy $ΔS_{vN}^{SO}$ within relativistic crystal field theory to quantify spin–orbital entanglement in $5d^1$ materials. By extracting RCFT parameters from optical $d$-$d$ transitions and μ_eff across a family of hexachlorides and perovskites, it shows that Γ7 is SOC-invariant while Γ8 states develop entanglement that correlates with the relativistic parameter δ, not merely with the spin-orbit coupling strength. The study demonstrates that higher μ_eff aligns with larger $ΔS_{vN}^{SO}$, emphasizing the importance of the full RCFT parameter set (ξ_nd, Dq, p/q) in determining entanglement. This framework establishes a practical link between measurable magnetic properties and underlying quantum entanglement, guiding future ab initio and lattice-coupled extensions in heavy transition-metal compounds.

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 $ΔS_{\rm vN}^{\rm SO}$. 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 $Γ_7(t_{2g})\leftarrowΓ_8(t_{2g})$ and $Γ_8(e_g)\leftarrowΓ_8(t_{2g})$, and of the effective magnetic dipole moment $μ_{\rm eff}$. It is demonstrated that, regardless of the system, the Kramers doublet $Γ_7(t_{2g})$ exhibits no spin-orbital von Neumann entropy. The entropies obtained for the relativistic crystal field states $Γ_8(t_{2g})$ and $Γ_8(e_g)$ 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.

Figures (3)

• Figure 1: Energy splitting of the $\rm ^2D$ atomic state due to the spin-orbit coupling (SOC) and to the octahedral crystal field (CF) interactions. When the SOC is negligible the $\Gamma_{8-}$ and $\Gamma_7$ relativistic states, often referred to as $J_{\rm eff}=3/2$ and $J_{\rm eff}=1/2$, respectively, resemble the $^2T_{2g}$ non-relativistic crystal field state, see Ref. Stamokostas2018.
• Figure 2: von Neumann entropy $S_{\rm vN}$ of the $\Gamma_7$ and $\Gamma_8$ states of $nd^1$ ions in octahedral crystal fields as a function of $\delta$. The shadow rectangle indicates the typical values of $\delta$ for $5d^1(O_h)$ systems ($0.3\le\delta\le0.7$).
• Figure 3: Effective magnetic dipole moment $\mu_{\rm eff}$ and spin-orbital von Neumann entropies $\Delta S_{\rm vN}^{\rm SO}$ for $nd^1$ ions in octahedral crystal fields as a function of the parameter $\delta$. The shadow rectangle indicates the typical values of $\delta$ for $5d^1(O_h)$ systems ($0.3\le\delta\le0.7$).