Spin-Orbital Singlet and Quantum Critical Point on the Diamond Lattice: ${\mathrm{FeSc}}_2{\mathbf{S}}_4$

We present a theory of spin and orbital physics in the $A$-site spinel compound ${\mathrm{FeSc}}_2{\mathrm{S}}_4$, which experimentally exhibits a broad “spin-orbital liquid” regime. A spin-orbital Hamiltonian is derived from a combination of microscopic consideration and symmetry analysis. We demonstrate a keen competition between spin-orbit interactions, which favor formation of a local “spin-orbital singlet,” and exchange, which favors magnetic and orbital ordering. Separating the spin-orbital singlet from the ordered state is a quantum critical point. We argue that ${\mathrm{FeSc}}_2{\mathrm{S}}_4$ is close to this quantum critical point on the spin-orbital singlet side. The full phase diagram includes a commensurate-incommensurate transition within the ordered phase. A variety of comparisons to and suggestions for experiments are discussed.

Figure 1

The schematic phase diagram versus $T$, the temperature, and $x$, the ratio of exchange to spin-orbit interaction (see text). Here the labels for the phases are “SOS,” spin-orbital singlet; “$\mathrm{CAF}+\mathrm{OO}$,” commensurate antiferromagnet with orbital order; and “IC,” incommensurate spin and orbital order. “QC” stands for the quantum critical regime.

Figure 2

The low-lying spectrum along the $[100]$ direction. The solid curves are calculations from the small $x$ expansion, with $\lambda =25(J_2+K_2)$. The blue (dark) and green (light) curves have $J_1+K_1=0$ and $J_1+K_1=J_2+K_2$, respectively. The red (dashed) curve is a schematic spectrum close to the quantum critical point, where it shows a Dirac-type structure at low energy.