Competing spin-orbital singlet states in the $4d^4$ honeycomb ruthenate ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{O}_6$

When spin-orbit-entangled $d$ electrons reside on a honeycomb lattice, rich quantum states are anticipated to emerge, as exemplified by the $d^5$ Kitaev materials. Distinct yet equally intriguing physics may be realized with a $d$-electron count other than $d^5$. The magnetization, ${}^7\mathrm{Li}$-nuclear magnetic resonance (NMR), and inelastic neutron scattering measurements, together with the quantum chemistry calculation, indicate that the layered ruthenate ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$ with $d^4{\mathrm{Ru}}^{4+}$ ions at ambient pressure forms a honeycomb lattice of spin-orbit-entangled singlets, which is a playground for frustrated excitonic magnetism. Under pressure, the singlet state does not develop the expected excitonic magnetism, but two successive transitions to other nonmagnetic phases were found in ${}^7\mathrm{Li}$-NMR, neutron diffraction, and x-ray absorption fine structure measurements, first to an intermediate phase with moderate distortion of honeycomb lattice and eventually to a high-pressure phase with very short Ru-Ru dimer bonds. While the strong dimerization in the high-pressure phase originates from a molecular orbital formation as in the sister compound ${\mathrm{Li}}_2{\mathrm{RuO}}_3$, we argue that the intermediate phase represents a spin-orbit-coupled singlet dimer state which is stabilized by the admixture of upper-lying $J_{\mathrm{eff}}=1$-derived states via a pseudo-Jahn-Teller effect. The emergence of competing electronic phases demonstrates rich spin-orbital physics of $d^4$ honeycomb compounds, and this finding paves the way for realization of unconventional magnetism.

Figure 1

Absence of magnetic order in the honeycomb ruthenate ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$. (a) Crystal structure of ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$. The ${\mathrm{RuO}}_6$ octahedra compose edge-shared honeycomb layers, and Li ions locate at the center of the honeycomb. Ag ions form O-Ag-O dumbbell bonds between the layers. The crystal structure is visualized by using vesta software [42]. (b) Magnetic susceptibility $\chi$($T$) of ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$ powder sample as a function of temperature. The measurement was performed with magnetic field of 1 T. The inset shows the temperature-dependent resistivity. (c) Specific heat $C\left(T\right)$ at zero magnetic field. The inset shows the $C\left(T\right)$ divided by temperature at low temperatures under magnetic fields. The strong suppression of $C\left(T\right)/T$ at low temperatures by magnetic fields suggests that the low-temperature contributions originate from localized spin defects. (d) Zero-field (ZF) muon spin relaxation at various temperatures down to 2 K. The inset shows the longitudinal field (LF) measurement with an applied magnetic field of 100 Gauss at 5 K, together with the ZF data at 2 K. The solid lines delineate a fit with a stretched exponential function (see Supplemental Material [43]).

Figure 2

Spin-orbit-entangled singlet state in ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$ at ambient pressure. (a) Inelastic neutron scattering (INS) on the powder sample of ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$. The figures show the contour plot of scattering intensities where the phonon contribution was removed by subtracting the temperature-scaled 250 K dataset. The raw data are shown in Fig. S8 in the Supplemental Material [43]. (b) The imaginary part of the dynamic susceptibility ${\chi }^{″}$ as a function of energy transfer in the $\left|Q\right|$ region between 1.5 and $1.9{\AA }^{-1}$ at various temperatures. (c) Inverse of ${}^7\mathrm{Li}$-NMR spin-lattice relaxation time $T_1$ as a function of temperature. $T_1$ is measured at the larger shift peak of spectra [see Fig. 3]. The inset shows the Arrhenius plot of the data, and the red line represents the fit between 100 and 300 K. The fit gives a gap of $\sim 30$ meV. The peak of $1/T_1$ at $\sim 10$ K is likely attributed to the localized spin defects observed in $\chi \left(T\right)$ and $C\left(T\right)$. (d) Schematic energy levels of ${\mathrm{Ru}}^{4+}{t}_{2g}^4$ states obtained by the quantum chemistry (QC) calculation (from left to middle). The states expressed by the red bar are threefold degenerate because of $S$ = 1, while those of the blue bar represent a single state. Each state is labeled with a Mulliken symbol. The nine states of ${}^{3}T_1$ split by trigonal crystal field (${\mathrm{\Delta }}_{\mathrm{tri}}$) into the ${}^{3}A_2$ and ${}^{3}E$ states. Spin-orbit coupling (SOC) further splits those states and yields the $A_1$ singlet ground state. The diagram from right to middle shows the $J_{\mathrm{eff}}$ picture where SOC first splits the ${}^{3}T_1$ states into $J_{\mathrm{eff}}$ = 0, 1, and 2 states and ${\mathrm{\Delta }}_{\mathrm{tri}}$ acts on the $J_{\mathrm{eff}}$ states. The resultant energy diagram is identical in both representations.

Figure 3

Pressure-induced change of magnetism in ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$. (a) Magnetic susceptibility $\chi \left(T\right)$ under pressure. The measurements were performed at magnetic field of 1 T. (b) ${}^7\mathrm{Li}$-nuclear magnetic resonance (NMR) spectra at various pressures. The two-peak structure at ambient pressure likely originates from the strong magnetic anisotropy (see Figs. S5 and S6 in the Supplemental Material [43]). At 0.9 GPa, a change of spectrum from an asymmetric shape to a symmetric and lower shift peak was seen on cooling <200 K. The spectra at 0.9 and 3.1 GPa at low temperatures remain sharp, pointing to a nonmagnetic nature of the intermediate phase. At 4.5 GPa, the spectra become much sharper than the lower pressure data. (c) Inverse of spin-lattice relaxation time $T_1$ under pressure obtained from ${}^7\mathrm{Li}$-NMR. The $1/T_1$ at 0.9 GPa was obtained at the smaller shift peak <200 K. The inset shows the Arrhenius plot of $1/T_1$ where the similar gapped behavior as in the ambient pressure phase is seen. The dotted line shows a fit for the 3.1 GPa data between 300 and 100 K. The estimated gap sizes at 0.9 and 3.1 GPa are nearly the same ($\sim 30$ meV), while the one at 4.5 GPa is larger ($\sim 48$ meV), implying a different origin of excitation gap.

Figure 4

Pressure-induced structural transitions in ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$. (a) Pressure-dependent lattice parameters at room temperature obtained from neutron diffraction. The values of lattice constants are evaluated using the monoclinic unit cell of the ambient pressure phase (space group $C2/m$) and normalized by those at ambient pressure. (b) Ru honeycomb lattices at ambient pressure (left) and in the intermediate phase (right). The structure at ambient pressure is taken from Ref. [19], and the one in the intermediate phase was obtained from the Rietveld refinement of neutron diffraction at 200 K and at 3.1 GPa. (c) The magnitude of the complex Fourier transform (FT) of x-ray absorption fine structure (XAFS) data for various pressures at $T$ = 50 K. Note that the FT is not corrected for scattering phase shifts, and the $R$ values do not exactly correspond to the bond lengths.

Figure 5

Pressure-temperature phase diagram of ${\mathrm{Ag}}_3{\mathrm{LiRu}}_2{\mathrm{O}}_6$. The blue and red open squares represent the phase boundaries determined by the neutron diffraction, whereas the green squares show the transition temperatures determined by $d\chi \left(T\right)$/$dT$. The filled (open) black squares indicate the points where the strong dimerization of Ru atoms was found in the x-ray absorption fine structure (XAFS) spectra in the high-pressure (intermediate) phases, respectively. The inset depicts the $J$-dimer model of the intermediate phase. On each site, we consider a singlet $|s\rangle$ and upper doublet $|T_{\alpha }\rangle$ ($\alpha =x,y$) which is split from the $J_{\mathrm{eff}}$ = 1 triplet by trigonal crystal field. These states form the ground state $|\mathrm{G}\rangle$, bonding states $|\mathrm{B}\rangle$, and antibonding states $|\mathrm{A}\rangle$ by exchange interactions. In the full $J$-dimer model including the upper singlet $|T_z\rangle$, there are 16 states per dimer in total, but we show here only the lower five states for brevity. See Supplemental Material [43] for more discussions about the $J$-dimer model.