Magnetic Field Dependence of Excitations Near Spin-Orbital Quantum Criticality

The spinel ${\mathrm{FeSc}}_2{\mathrm{S}}_4$ has been proposed to realize a near-critical spin-orbital singlet (SOS) state, where entangled spin and orbital moments fluctuate in a global singlet state on the verge of spin and orbital order. Here we report powder inelastic neutron scattering measurements that observe the full bandwidth of magnetic excitations and we find that spin-orbital triplon excitations of an SOS state can capture well key aspects of the spectrum in both zero and applied magnetic fields up to 8.5 T. The observed shift of low-energy spectral weight to higher energies upon increasing applied field is naturally explained by the entangled spin-orbital character of the magnetic states, a behavior that is in strong contrast to spin-only singlet ground state systems, where the spin gap decreases upon increasing applied field.

Figure 1

Generic phase diagram as a function of $x=J_2/\lambda$ [1] with location of ${\mathrm{FeSc}}_2{\mathrm{S}}_4$ indicated by the vertical arrow. Labels SOS, QC, and Ordered stand for spin-orbital singlet, quantum critical, and spin and orbitally ordered, respectively. Solid (dashed) line indicates a phase transition boundary (crossover). Inset shows ${\mathrm{Fe}}^{2+}$ ions in ${\mathrm{FeSc}}_2{\mathrm{S}}_4$ are locally coordinated by a regular tetrahedron (red shaded polyhedron) of ${\mathrm{S}}^{2-}$ atoms (yellow balls) and are arranged in two FCC sublattices (light and dark red balls) with intrasublattice NN AFM exchange $J_2$.

Figure 2

(a) Background subtracted (see Ref. [12]) zero field INS data observing highly dispersive magnetic excitations (MERLIN, $E_i=15\mathrm{meV}$, 12 h counting time, energy resolution 0.79 meV FWHM on the elastic line). (b) Corresponding one-triplon spectrum including convolution with the instrumental resolution. (c) Intensity along the tilted dashed line direction in (b), compared to the model calculation (solid line). Horizontal bar indicates expected peak FWHM due to instrumental resolution and spherical averaging.

Figure 3

(a)–(c) Background-subtracted INS data at 0, 4, and 8.5 T magnetic field compared with model calculations (d)–(f), respectively. The data was collected using FOCUS with $E_i=3.27\mathrm{meV}$, elastic line energy resolution 0.18 meV FWHM and 11 h counting per setting.

Figure 4

(a) Energy scan at the softening wave vector $Q_S$ through the background subtracted data in Figs. 3: filled circles, triangles, and squares denote fields of 0, 4, and 8.5 T, respectively. Dashed lines are the model calculations shown in Figs. 3. Data at 0 T in the paramagnetic phase at high temperatures (80 K) are also included (red open circles). (b) Schematic of the ground and excited states’ energies as a function of applied field in the case of a SOS and (c) a spin-singlet system.