Singlet-Triplet Excitations and Long-Range Entanglement in the Spin-Orbital Liquid Candidate ${\mathrm{FeSc}}_2{\mathrm{S}}_4$

Theoretical models of the spin-orbital liquid (SOL) ${\mathrm{FeSc}}_2{\mathrm{S}}_4$ have predicted it to be in close proximity to a quantum critical point separating a spin-orbital liquid phase from a long-range ordered magnetic phase. Here, we examine the magnetic excitations of ${\mathrm{FeSc}}_2{\mathrm{S}}_4$ through time-domain terahertz spectroscopy under an applied magnetic field. At low temperatures an excitation emerges that we attribute to a singlet-triplet excitation from the SOL ground state. A threefold splitting of this excitation is observed as a function of applied magnetic field. As singlet-triplet excitations are typically not allowed in pure spin systems, our results demonstrate the entangled spin and orbital character of singlet ground and triplet excited states. Using experimentally obtained parameters we compare to existing theoretical models to determine ${\mathrm{FeSc}}_2{\mathrm{S}}_4$’s proximity to the quantum critical point. In the context of these models, we estimate the characteristic length of the singlet correlations to be $\xi /(\mathbf{a}/2)\approx 8.2$ (where $\mathbf{a}/2$ is the nearest neighbor lattice constant), which establishes ${\mathrm{FeSc}}_2{\mathrm{S}}_4$ as a SOL with long-range entanglement.

Figure 1

Energy levels of the ${\mathrm{Fe}}^{2+}$ ion in an $S=2$ configuration after tetrahedral crystal field splitting and first and second order SOC, without including lattice effects. Numbers in parenthesis represent the degeneracy of each level. Second order SOC splits the lower ${}^{5}E$ doublet into 5 equally spaced levels separated by $\lambda =6({{\lambda }_0}^2/{\mathrm{\Delta }}_{CF})$.

Figure 2

(a)–(b) Field dependence of the transmission magnitude of ${\mathrm{FeSc}}_2{\mathrm{S}}_4$ taken at $T=5\mathrm{K}$ for the longitudinal (a) configuration (THz magnetic field $h_{\text{ac}}\parallel H_{\text{dc}}$) and transverse (b) configuration ($h_{\text{ac}}\perp H_{\text{dc}}$). Offsets of 0.05 are included between the curves for clarity.

Figure 3

Field dependence of the imaginary ${\chi }^{\prime \prime }$ and real ${\chi }^{\prime }$ parts of the complex ac susceptibility in the transverse (a)–(b) and longitudinal (c)–(d) configurations. All spectra were taken at $T=5\mathrm{K}$. The susceptibility is shown in SI units, given by the ratio of the magnetization to the applied field. Dashed lines are guides to the eye. Offsets of 0.1 are included for clarity.

Figure 4

Excitation energies for both configurations. Error bars are based on the quality of the fits. (b) Midinfrared reflectivity data taken at $T=5\mathrm{K}$ (black, left axis) along with the dielectric loss (red, right axis). Three features are seen with energies of 63.52 (0.262), 70.32 (0.290), and 80.16 THz (0.331 eV), corresponding to crystal field excitations to the ${}^5{T}_2$ energy levels plus coupling to Jahn-Teller modes of ${\mathrm{Fe}}^{2+}$ in a tetrahedral environment.