Evidence for a Field-Induced Quantum Spin Liquid in $\alpha$-${\mathrm{RuCl}}_3$

We report a ${}^{35}\mathrm{Cl}$ nuclear magnetic resonance study in the honeycomb lattice $\alpha$-${\mathrm{RuCl}}_3$, a material that has been suggested to potentially realize a Kitaev quantum spin liquid (QSL) ground state. Our results provide direct evidence that $\alpha$-${\mathrm{RuCl}}_3$ exhibits a magnetic-field-induced QSL. For fields larger than $\sim 10\mathrm{T}$, a spin gap opens up while resonance lines remain sharp, evidencing that spins are quantum disordered and locally fluctuating. The spin gap increases linearly with an increasing magnetic field, reaching $\sim 50\mathrm{K}$ at 15 T, and is nearly isotropic with respect to the field direction. The unusual rapid increase of the spin gap with increasing field and its isotropic nature are incompatible with conventional magnetic ordering and, in particular, exclude that the ground state is a fully polarized ferromagnet. The presence of such a field-induced gapped QSL phase has indeed been predicted in the Kitaev model.

Figure 1

(a) Low-$T$ specific heat $C_p/T$ at zero and chosen magnetic fields. The data at zero field taken from Refs. [26, 25] are compared. (b) Temperature dependence of the uniform magnetic susceptibility $\chi$ at $H=0.1\mathrm{T}$, obtained for the four different field orientations with respect to the $c$ axis. The inset enlarges the low-$T$ region.

Figure 2

(a) The principal axis of the EFG $V_{zz}$ at the ${}^{35}\mathrm{Cl}$ nuclei is along the shared edges of the ${\mathrm{RuCl}}_6$ octahedra, resulting in three inequivalent ${}^{35}\mathrm{Cl}$ sites in the field. The sample is mounted on the goniometer so that one of the three axes of $V_{zz}$’s lies in the rotating plane. (b) When $H\parallel c$ ($\theta =0$), the ${}^{35}\mathrm{Cl}$ spectrum is extremely complex and broad. As $H$ is either parallel or perpendicular to the direction of $V_{zz}$, very narrow ${}^{35}\mathrm{Cl}$ NMR lines were obtained. (c) ${}^{35}\mathrm{Cl}$ NMR spectrum measured at $H=15\mathrm{T}$ as a function of $T$ with cooling for two different field orientations. The first order character of the structural transition is evidenced by the gradual transfer of the ${}^{35}\mathrm{Cl}$ spectral weight below $T_S\sim 75\mathrm{K}$, as clearly shown in the inset. (d) NMR shift $\mathcal{K}$ as a function of $T$. The strong anisotropy of $\mathcal{K}$ increases rapidly with decreasing $T$, approaching a saturated value below $\sim 10\mathrm{K}$. The dotted line is the estimated $T$ dependence of ${\mathcal{K}}_{\mathrm{quad}}$ (see Supplemental Material [29]). The inset shows the $\mathcal{K}$ vs $\chi$ plot, which yields the hyperfine coupling constants $A_{\mathrm{hf}}^{\perp c^{\prime }}=17.4\mathrm{kG}/{\mu }_B$ and $A_{\mathrm{hf}}^{\parallel c^{\prime }}=12.3\mathrm{kG}/{\mu }_B$. (e) Spin-lattice relaxation rate $T_1^{-1}$ vs $T$. Whereas $T_1^{-1}$ is nearly $T$ independent above $T^{*}=160\mathrm{K}$, it increases (decreases) for $H\parallel c^{\prime }$ ($H\perp c^{\prime }$) below $T^{*}$, implying the development of in-plane spin correlations.

Figure 3

(a) $(T_{1}T{)}^{-1}$ as a function of $T$, measured at 15 T. At low $T$, $(T_{1}T{)}^{-1}$ for both field directions reaches a maximum at $\sim 25\mathrm{K}$, which is followed by a rapid drop upon further cooling. Inset enlarges the low $T$ region. (b) Semilog plot of $T_1^{-1}$ vs $1/T$ unravels a spin gap behavior $T_1^{-1}\propto \exp (-\mathrm{\Delta }/T)$. The deviation from the gap behavior takes place below $\sim 10\mathrm{K}$. (c) Strong field dependence of $(T_{1}T{)}^{-1}$ at low $T$ as a function of $H_{\parallel c^{\prime }}$. Below 9 T, AFM ordered phase was clearly detected by sharp peaks of $(T_{1}T{)}^{-1}$. (d) The spin gap $\mathrm{\Delta }$ is rapidly filled up with decreasing $H_{\parallel c^{\prime }}$, vanishing completely at 10 T.

Figure 4

(a) The dependence of $C_p/T$ at $H\parallel c^{\prime }$, oriented along $c^{\prime }$. With increasing $H$, AFM order is suppressed and completely disappears at 10 T—at 14 T, a gap appears to be present. (c) The $T-H$ phase diagram obtained by NMR and specific heat measurements. $T_N$ obtained by specific heat for $H\perp c$ is compared. In the QSL region, the field dependence of the spin gap $\mathrm{\Delta }$ is shown (right axis).