Dynamics of a $j=\frac{3}{2}$ quantum spin liquid

We study a spin-orbital model for $4d^1$ or $5d^1$ Mott insulators in ordered double perovskites with strong spin-orbit coupling. This model is conveniently written in terms of pseudospin and pseudo-orbital operators representing multipoles of the effective $j=\frac{3}{2}$ angular momentum. Similarities between this model and the effective theories of Kitaev materials motivate the proposal of a chiral spin-orbital liquid with Majorana fermion excitations. The thermodynamic and spectroscopic properties of this quantum spin liquid are characterized using parton mean-field theory. The heat capacity, spin-lattice relaxation rate, and dynamic structure factor for inelastic neutron scattering are calculated and compared with the experimental data for the spin liquid candidate ${\mathrm{Ba}}_2{\mathrm{YMoO}}_6$. Moreover, based on a symmetry analysis, we discuss the operators involved in resonant inelastic x-ray scattering (RIXS) amplitudes for double-perovskite compounds. In general, the RIXS cross sections allow one to selectively probe pseudospin and pseudo-orbital degrees of freedom. For the chiral spin-orbital liquid in particular, these cross sections provide information about the spectrum for different flavors of Majorana fermions.

Figure 1

(a) Crystal structure of ordered double perovskites, with chemical formula ${\mathrm{A}}_2{\mathrm{BB}}^{\prime }{\mathrm{O}}_6$. The oxygen ${\mathrm{O}}^{2-}$ anions correspond to the vertices of the octahedra. (b) Tetrahedron of the ${\mathrm{B}}^{\prime }$ species, highlighting the exchange path. Different bond colors represent different first-neighbor interactions in the $xy$, $yz$, or $xz$ planes.

Figure 2

Energy level splitting of $4d^1$ or $5d^1$ electrons in the presence of a cubic-symmetric crystal field and spin-orbit coupling. The density profiles of the $A$, $B$, and $C$ states are also illustrated.

Figure 3

(a) Diagrammatic representation of the most symmetric gauge choices on an elementary tetrahedron. The arrow pointing from site $i$ to site $j$ represents that ${\varphi }_{ij}=+1$ (and ${\varphi }_{ji}=-1$ in the opposite direction). Notice that the two Ansätze are conjugated by time reversal. (b) Representation of the $Z_2$ fluxes of the Ansatz on the fcc lattice. The flux through each face of a green (red) tetrahedron is positive (negative) when the sites are oriented counterclockwise with respect to a normal vector pointing outward.

Figure 4

(a) First Brillouin zone of the cubic lattice highlighting the Fermi lines (orange lines). (b) Dispersion for different fermion flavors.

Figure 5

(a) Absolute value of the order parameters of the chiral spin-orbital liquid as a function of temperature. (b) Magnetic specific heat per site calculated within the mean-field theory.

Figure 6

Spin lattice relaxation rate of the chiral spin liquid state as a function of the temperature.

Figure 7

(a) First Brillouin zone of the fcc lattice. (b) Dynamical structure factor (in arbitrary units) probed by inelastic neutron scattering. (c) Result after integration over 1.5 Å${}^{-1}<Q<1.8$ Å${}^{-1}$.

Figure 8

Schematic diagram of a RIXS experiment at the $L_2$ edge, featuring specifically the possibility of a pseudo-orbital flip. The absorbed photon creates a core $2p$ hole, which is subject to strong spin-orbit coupling. This highly unstable state decays before a $d$ electron can tunnel to or from the ion, generating a spin-orbital excitation and an emitted photon.

Figure 9

RIXS cross section (in arbitrary units) probing (a) $\mathbf{s}$, (b) $\mathbf{s}{\tau }_y$, (c) ${\tau }_x$, and (d) ${\tau }_z$ operators along the high-symmetry directions of the Brillouin zone of the fcc lattice.