Phonon-Modulated Magnetic Interactions and Spin Tomonaga-Luttinger Liquid in the $p$-Orbital Antiferromagnet ${\mathrm{CsO}}_2$

The magnetic response of antiferromagnetic ${\mathrm{CsO}}_2$, coming from the $p$-orbital $S=1/2$ spins of anionic ${\mathrm{O}}_2^{-}$ molecules, is followed by ${}^{133}\mathrm{Cs}$ nuclear magnetic resonance across the structural phase transition occurring at $T_{s1}=61\mathrm{K}$ on cooling. Above $T_{s1}$, where spins form a square magnetic lattice, we observe a huge, nonmonotonic temperature dependence of the exchange coupling originating from thermal librations of ${\mathrm{O}}_2^{-}$ molecules. Below $T_{s1}$, where antiferromagnetic spin chains are formed as a result of $p$-orbital ordering, we observe a spin Tomonaga-Luttinger-liquid behavior of spin dynamics. These two interesting phenomena, which provide rare simple manifestations of the coupling between spin, lattice, and orbital degrees of freedom, establish ${\mathrm{CsO}}_2$ as a model system for molecular solids.

Figure 1

Structural and magnetic properties of ${\mathrm{CsO}}_2$. (a) Schematic crystal structure with representative ${\mathrm{O}}_2^{-}$ ${\pi }_{x,y}^{*}$ orbitals (yellow) and ${\mathrm{Cs}}^{+}$ $p_z$ orbitals (green). Above $T_{s2}$, the average direction of ${\mathrm{O}}_2^{-}$ dumbbells is along the $c$ axis (top $ab$ layer) resulting in degenerate ${\pi }_{x,y}^{*}$ orbitals and frustrated-square magnetic lattice with exchange couplings $J$ ($-J$) between nearest (next-nearest) neighboring spins (top). Below $T_{s1}$, the tilt of ${\mathrm{O}}_2^{-}$ dumbbells is staggered along the $b$ axis (bottom $ab$ layer) resulting in ${\pi }_{x,y}^{*}$ orbital ordering and formation of magnetic chains with exchange coupling $J$ along $b$ (bottom) [8]. (b) Hysteretic evolution of ${}^{133}\mathrm{Cs}$ NMR spectrum across the structural phase transition taken on warming (solid lines, $T_{s2}=75\mathrm{K}$) and cooling (dashed lines, $T_{s1}=61\mathrm{K}$). The presence of the high-$T$ (low-$T$) phase is indicated by red (blue) spectra. (c) ${}^{133}\mathrm{Cs}$ shift (open circles) follows static susceptibility $\chi$ (red line) down to 40 K. Both are taken on warming. A shift in the low-$T$ phase is fitted with the chain model (green line) exhibiting a maximum as a signature of 1D spin correlations (blue background). The dotted vertical line indicates an anomaly at $T_{s2}$ due to the structural phase transition. The upper inset shows a linear dependence of the relative shift on $\chi$ above 40 K, yielding a hyperfine coupling constant $A=-1.16\mathrm{T}$. The lower inset shows the field dependence of the ${\mathrm{O}}_2^{-}$ magnetic moment $\mu$ calculated from the measured magnetization. The red line is a linear fit with $\partial \mu /\partial B=0.0142$ ${\mu }_B/\mathrm{T}$.

Figure 2

Spin dynamics in ${\mathrm{CsO}}_2$. $T_1^{-1}$ as a function of temperature $T$ taken on warming in three different magnetic fields $B$. Solid red, blue, and green lines are power-law fits characteristic of the TLL behavior (blue background) valid in the range from 15 K up to $T_{\mathrm{TLL}}$ (indicated by dashed line). Solid gray lines are the joint fit to the high-$T$ behavior (red background) for three magnetic field values. Arrows indicate the divergence in $T_1^{-1}(T)$ at $T_N$ due to the magnetic phase transition into the ordered Néel state. The dotted vertical line indicates the jump in $T_1^{-1}(T)$ at $T_{s2}=75\mathrm{K}$ due to the structural phase transition. The inset outlines the low-$T$ phase diagram obtained from the data in the plot.

Figure 3

Details of the TLL and high-$T$ behaviors extracted from Fig. 2. Field dependence of (a) the TLL exponent $K$ (solid line is guide to the eye) and (b) the ratio $u/K$ compared to the prediction of Eq. (1) using the measured $\mu (B)$ [lower inset of Fig. 1] (solid line). (c) Temperature dependence of the exchange coupling $J(T)$ extracted from the data in Fig. 2 using Eq. (2). The best overlap of the three data sets is obtained for $|A|=0.82\mathrm{T}$. The upper horizontal scale is calculated using Eq. (3). The solid gray line is fit with the function $J(\vartheta )=J_0+J_2(\vartheta /{\vartheta }_0{)}^2+J_4(\vartheta /{\vartheta }_0{)}^4$ where $J_0/k_B=5.9\mathrm{K}$, $J_2/k_B=19\mathrm{K}$, and $J_4/k_B=-28\mathrm{K}$. The inset shows a schematic dependence of the overlap between the ${\mathrm{O}}_2^{-}$ ${\pi }_{x,y}^{*}$ orbital (yellow) and the ${\mathrm{Cs}}^{+}$ $p_z$ orbital (green) on the tilt $\vartheta$ of the ${\mathrm{O}}_2^{-}$ dumbbell from the $c$ axis. An optimal overlap (shaded yellow) is obtained for ${\vartheta }_{\max }=5.2°$ reached at 225 K where $J(T)$ exhibits a maximum.