Evidence for a Field-induced Quantum Spin Liquid in $\alpha$-RuCl$_3$

TL;DR

This study provides experimental evidence that $ ext{α}$-RuCl$_3$ exhibits a field-induced quantum spin liquid state characterized by a linear increase in spin gap with magnetic field, supporting theoretical predictions of a Kitaev QSL.

Contribution

The paper presents direct NMR evidence of a magnetic field-induced quantum spin liquid in $ ext{α}$-RuCl$_3$, demonstrating a field-dependent spin gap incompatible with conventional magnetic order.

Findings

Spin gap opens above 10 T magnetic field

Spin gap increases linearly with field, reaching ~50 K at 15 T

Spin gap is nearly isotropic with respect to field direction

Abstract

We report a $^{35}$Cl nuclear magnetic resonance study in the honeycomb lattice, $\alpha$-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 $\alpha$-RuCl$_3$ exhibits a magnetic field-induced QSL. For fields larger than $\sim 10$ 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 increasing magnetic field, reaching $\sim50$ 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.

Full text

When the interactions between magnetic spins are strongly frustrated, quantum fluctuations can cause spins to remain disordered even at very low temperatures Balents (2010). The quantum spin liquid (QSL) state that ensues is conceptually very interesting – for instance, new fractionalized excitations appear that are very different from the ordinary spin-wave excitations in ordered magnets Shimizu et al. (2003); Yamashita et al. (2010); Han et al. (2012); Fu et al. (2015). A QSL appears in the so-called Kitaev honeycomb model – a prototypical and mathematically well-understood model of strongly frustrated interacting spins Kitaev (2006); Pachos (2012). In an external magnetic field the topological QSL state acquires a gap that, in the generic case grows linearly with field strength Song et al. (2016).

This observation has motivated the search for the experimental realization of the Kitaev honeycomb model and its topological QSL phases. The quest was centered, until recently, mainly on honeycomb iridate materials Jackeli and Khaliullin (2009); Chaloupka et al. (2010) of the type $A_{2}$IrO3 ($A$ = Na or Li). However, in these iridates long-range magnetic order develops at low temperatures for all known different crystallographic phases Singh and Gegenwart (2010); Ye et al. (2012); Choi et al. (2012); Takayama et al. (2015); Modic et al. (2014). Their QSL regime is most likely preempted by the presence of significant residual Heisenberg-type interactions, by longer-range interactions between the spins or by crystallographically distinct Ir-Ir bonds, if not by a combination of these factors Kimchi and You (2011); Rau et al. (2014); Katukuri et al. (2014); Nishimoto et al. (2016). More promising in this respect is ruthenium trichloride $\alpha$-RuCl3 in its honeycomb crystal phase, as numerous experimental and theoretical studies pointed the significance of the anisotropic Kitaev exchange in the material Plumb et al. (2014); Rousochatzakis et al. (2015); Kim et al. (2015); Sandilands et al. (2016); Sears et al. (2015); Majumder et al. (2015); Kubota et al. (2015); Nasu et al. (2016). Neutron scattering studies have shown that the magnetic interactions in this material are closer to the Kitaev limit Banerjee et al. (2016), although at low temperatures also this quasi-2D material exhibits long-range magnetic order.

In this Letter, we show by means of nuclear magnetic resonance (NMR) that in $\alpha$-RuCl3 large magnetic fields larger than $\sim 10$ T melt the magnetic order, and a spin-gap opens that scales linearly with the magnetic field, implying that the detrimental effects of residual magnetic interactions between the Ru moments can be overcome by an external magnetic field that stabilizes a QSL state.

35Cl (nuclear spin $I=3/2$) NMR was carried out in a $\alpha$-RuCl3 single crystal as a function of external field ($H$) and temperature ($T$). (See Supplemental Material for the crystal growth and characterization.) The sample was reoriented using a goniometer for the accurate alignment along $\mathbf{H}$. The 35Cl NMR spectra were acquired by a standard spin-echo technique with a typical $\pi/2$ pulse length 2–3 $\mu$s. The nuclear spin-lattice relaxation rate $T_{1}^{-1}$ was obtained by fitting the recovery of the nuclear magnetization $M(t)$ after a saturating pulse to the following fitting function : $1-M(t)/M(\infty)=A[0.9e^{-(6t/T_{1})^{\beta}}+0.1e^{-(t/T_{1})^{\beta}}]$, where $A$ is a fitting parameter and $\beta$ is the stretching exponent.

Experimentally, in $\alpha$-RuCl3, a very peculiar strongly anisotropic magnetism has been reported Kubota et al. (2015); Majumder et al. (2015); Sears et al. (2015) based on measurements of the uniform magnetic susceptibility $\chi$ and the specific heat $C_{p}/T$. From the data it is clear that the antiferromagnetic (AFM) state observed at low $T$ is hardly affected by external fields along the $c$ direction whereas the signatures of the long-range magnetic order seen in $C_{p}/T$ and $\chi(T)$ disappear for moderate fields of about 8 T applied along the $ab$ plane. This pronounced anisotropy of the magnetism is also found in our crystals (see Fig. 1a and b). Note that whereas earlier studies Kubota et al. (2015); Majumder et al. (2015); Johnson et al. (2015) reported either two magnetic transitions at $T_{N1}\sim 8$ K and $T_{N2}\sim 14$ K or a single transition at $T_{N}\sim 13$ K, our measurements show, essentially, a single transition occurring at a considerably lower temperature, $T_{N1}\sim 6.2$ K. This evidences that our sample is of high quality with a (nearly) uniform stacking pattern Banerjee et al. (2016); Cao et al. (2016).

We now turn to the 35Cl NMR measurements on $\alpha$-RuCl3. Since the 35Cl nuclei possess a large quadrupole moment, the NMR spectra are strongly affected by‘ the electric field gradient (EFG). In $\alpha$-RuCl3, the principal axis of the largest eigenvalue of the EFG tensor $V_{zz}$ at 35Cl is expected to point along the shared edges of the RuCl6 octahedra which are tilted $\sim 35^{\circ}$ away from the $c$ axis as illustrated in Fig. 2a.

As a result, there exist three inequivalent 35Cl sites, yielding a very complex and broad 35Cl spectrum in a magnetic field, as shown in Fig. 2b, which would make further NMR studies extremely difficult. However, taking advantage of the fact that the influence of the quadrupole interaction is very sensitive to the angle between the direction of $V_{zz}$ ($\equiv\hat{V}_{zz}$) and $H$, it is possible to separate one 35Cl spectrum from other two spectra by applying $H$ along one of the three local directions of $V_{zz}$ at 35Cl. Moreover, when $H\parallel\hat{V}_{zz}$, the quadrupole line broadening should be significantly reduced, allowing further narrowing of the line.

Indeed, by rotating the sample in the $ac$ plane, we achieved a very narrow single 35Cl line with the linewidth of 10 kHz at $\theta\sim 30^{\circ}$ (see Fig. 2b). When the sample is reversely rotated by 90∘ (i.e, $H\perp\hat{V}_{zz}$), we also detected a narrow 35Cl line. These observations confirm that $V_{zz}$ is directed $\sim 30^{\circ}$ from the $c$ axis. Therefore, it is very convenient to define $\hat{V}_{zz}\equiv c^{\prime}$, and we, in the following, will present our NMR results with respect to the $c^{\prime}$ axis.

The $T$ dependence of the 35Cl NMR spectrum at 15 T is presented in Fig. 2c. Clearly, there is no signature of long-range magnetic order, which would cause a large broadening or splitting of the 35Cl line. Another feature is the appearance of a new NMR peak that replaces the original one below $\sim$75 K. This is due to a first order structural phase transition Kubota et al. (2015); Sears et al. (2015); details are provided in the Supplemental Material.

Figure 2d presents the $T$-dependence of the resonance frequency $\nu$ in terms of the NMR shift $\mathcal{K}=(\nu-\nu_{0})/\nu_{0}$ where $\nu_{0}$ is the unshifted Larmor frequency. $\mathcal{K}$ is composed, mainly, of the three terms: $\mathcal{K}=A_{\text{hf}}\chi_{\text{spin}}+\mathcal{K}_{\text{chem}}+\mathcal{K}_{\text{quad}}$ where $A_{\text{hf}}$ is the hyperfine (hf) coupling constant, $\chi_{\text{spin}}$ the local spin susceptibility, $\mathcal{K}_{\text{chem}}$ the $T$ independent chemical shift, and $\mathcal{K}_{\text{quad}}$ the second order quadrupole shift. Since $\mathcal{K}_{\text{quad}}$ which is determined by the charge distribution around the 35Cl nucleus weakly changes with $T$ Kaufmann and Vianden (1979), the strong upturn of $\mathcal{K}$ observed at low $T$ has to be attributed to $\chi_{\text{spin}}$ which is consistent with the macroscopic susceptibility (see Fig. 1c).

Figure 2e shows the $T$ dependence of $T_{1}^{-1}$ at $H=15$ T. At high $T>T^{*}\sim 160$ K, $T_{1}^{-1}$ follows roughly the behavior expected for simple paramagnets; $T_{1}^{-1}$ is nearly independent of $T$. The different absolute values of $T_{1}^{-1}$ for the two orientations of $H$ are ascribed to the anisotropic hf couplings (see Fig. 2d).

As $T$ is lowered below $T^{*}$, $T_{1}^{-1}$ increases for $H\parallel c^{\prime}$ but it decreases for $H\perp c^{\prime}$. Since the spin-lattice relaxation process is induced by the transverse components of spin fluctuations (SFs) with respect to the nuclear quantization axis, it is clear that $T_{1}^{-1}$ for $H\parallel c^{\prime}$ experiences stronger in-plane and weaker out-of-plane SFs than for $H\perp c^{\prime}$. Hence, the increase of the $T_{1}^{-1}$ anisotropy with lowering $T$ is an indication of the development of strong in-plane SFs below $T^{*}$.

At low temperatures, roughly below 50 K, $T_{1}^{-1}$ starts to decrease. For the study of spin dynamics at low $T$, it is convenient to consider the quantity $(T_{1}T)^{-1}$, which is proportional to the $\mathbf{q}$-average of the imaginary part of the dynamical susceptibility, $\sum_{\mathbf{q}}A_{\text{hf}}^{2}(\mathbf{q})\chi^{\prime\prime}(\mathbf{q},\omega_{0})/\omega_{0}$, where $\omega_{0}$ is the Larmor resonance frequency. As shown in Fig. 3a, a broad maximum of $(T_{1}T)^{-1}$ occurs near 30 K, being followed by a rapid drop towards low $T$ in an identical manner for both field orientations. The rapid decrease of $(T_{1}T)^{-1}$ implies a pronounced depletion of spectral weight in the spin excitation spectrum. The semilog plot of $T_{1}^{-1}$ against $1/T$ drawn in Fig. 3b unambiguously reveals a spin gap behavior, $T_{1}^{-1}\propto\exp(-\Delta/T)$, with the gap $\Delta\sim 44$ and 50 K for $H\parallel c^{\prime}$ and $\perp c^{\prime}$, respectively.

An explanation of the observed spin gap in terms of static magnetic order can be ruled out. For example, the 35Cl spectra measured at $H=15$ T do not show any signature of magnetic order down to 4.2 K (see Fig. 2c). Moreover, it is difficult to attribute the extracted large spin gap to some kind of anisotropy gap occurring in the spin wave spectrum in magnetically ordered systems. As displayed in Fig. 3a, the low temperature behavior of $(T_{1}T)^{-1}$ is very similar for both field orientations, indicating that spin dynamics is nearly isotropic at least in the range of field orientations ($30^{\circ}$ – $60^{\circ}$ off the $ab$ plane). Therefore, not only the measured large gap size, but also the isotropic gap behavior, contradicts any interpretation in terms of anisotropy gaps. The findings are also incompatible with the gap being due to a saturating ferromagnetic (FM) polarization of spins. The magnetization near 10 T is far less than the saturated value Johnson et al. (2015), particualarly for $H\parallel c^{\prime}$. For this field orientation, the $g$ factor is also very small, estimated to be $\sim 1\mu_{B}$/Ru3+ Kubota et al. (2015), which is an order of magnitude smaller than the required value 13.4 for the slope between the gap and the field shown in Fig. 4b. This clear-cut conclusion from the bare experimental findings is further supported by a detailed theoretical analysis (see Supplemental Material).

In order to study the $H$ dependence of $\Delta$, we measured $T_{1}^{-1}$ as a function of $H\parallel c^{\prime}$ at low $T$. The results are shown in Fig. 3c and 3d. A spin gap is only seen for $H>10$ T and $\Delta$ increases with increasing $H$. At $H=10$ T our data show a Curie-like upturn of the SFs, i.e., $(T_{1}T)^{-1}$ diverges for low $T$. Upon further lowering $H$ below 10 T, a sharp peak in $(T_{1}T)^{-1}$ signals static magnetic order below $T_{N}$ which decreases with increasing $H$. Below $T_{N}$, the 35Cl spectrum progressively spreads out with decreasing $T$, indicating the incommensurate character of AFM order Majumder et al. (2015); the spectra in the paramagnetic and magnetically ordered states are compared in the Supplemental Material. Thus, our data for $(T_{1}T)^{-1}$ clearly show a qualitative change of the behavior as a function of $H$: the peak due to static order occurring at low field is replaced by a spin gap behavior at $H\geq 10$ T. At the border the spin dynamics suggests quantum criticality, i.e. a divergence of $(T_{1}T)^{-1}$ for $T=0$. To back our NMR findings, we measured $C_{p}/T$ for $H\parallel c^{\prime}$ (Fig. 4a). The anomaly associated with AFM order is rapidly suppressed toward 10 T, which perfectly agrees with the $T_{1}^{-1}$ results. Further, we confirmed that at 14 T $C_{p}/T$ is significantly suppressed at low $T$, evidencing the opening of a spin gap at $H>10$ T 111Our analysis of the data at low $T$ yields $\Delta\sim 20$ K at 13.9 T (see SM for details) in rough agreement with the NMR findings..

The data thus indicate a field-induced crossover from a magnetically ordered state at low fields to a disordered state showing gapped spin excitations in large fields. Moreover, as evident from Fig. 3d, the field dependence of $T_{1}^{-1}(T)$ reveals that $\Delta$ increases linearly with $H$ above 10 T 222It is interesting to note that for the pure Kitaev model a magnetic field generates in lowest-order perturbation theory an excitation gap proportional to the field cubed Kitaev (2006). In $\alpha$-RuCl3 certainly magnetic interactions beyond the pure Kitaev exchange are of relevance Kim et al. (2015); Sears et al. (2015); Banerjee et al. (2016); Yadav et al. (2016)—how the perturbation results for the pure model are affected by the significant residual interactions in the presence of a magnetic field is, at the moment, an open theoretical question.. Extrapolating the curve to lower fields yields a threshold value of $H_{c}\sim 10$ T, i.e. the same field where the $T$-dependence of the $T_{1}^{-1}$ changes its qualitative behavior from an upturn to a downturn at low $T$. Fig. 3d also shows a low $T$ flattening out of $T_{1}^{-1}$ indicating the presence of another very low energy scale for spin dynamics. This feature is likely related to inhomogeneous states, for instance, due to magnetic defects. It becoming suppressed with increasing $H$ is consistent with competition between partially defect-induced magnetism and a spin gap that increases with $H$. Our findings are summarized in the $H$-$T$ phase diagram, see Fig. 4b 333Note that the suppression of AFM order for $H\parallel c^{\prime}$ occurs at $\sim$10 T, which is only slightly higher than $\sim$9 T observed for $H\parallel ab$. This observation is striking because for $H\parallel c^{\prime}$ the field strength projected to the honeycomb plane is only half of the applied one. This suggests that the AFM order is robust only when $H$ is nearly parallel to the normal direction of the plane..

Previously it was established on theoretical and experimental grounds that the magnetic interactions between the quantum spins in $\alpha$-RuCl3 are well described by the Kitaev model, however in the presence of residual interactions which ultimately preempt the QSL state in a zero magnetic field Plumb et al. (2014); Rousochatzakis et al. (2015); Kim et al. (2015); Sandilands et al. (2016); Sears et al. (2015); Majumder et al. (2015); Kubota et al. (2015); Sandilands et al. (2015); Banerjee et al. (2016). In the pure Kitaev model at zero field, the ground state is an Abelian QSL that is gapless Kitaev (2006), and the present observations suggest that the absence of a gap leaves this Abelian QSL very susceptible to the perturbing residual interactions that drive the formation of long-range magnetic order. In a finite field, however, a non-Abelian QSL forms in the pure honeycomb Kitaev model, which is protected by a spin gap Kitaev (2006). The data suggest that when the magnetic field and gap become large enough, it can overcome the energy scale related to the residual magnetic interactions so that a QSL emerges.

Note Added.— Recent thermal transport Hentrich et al. and specific heat Sears et al. (2017) measurements verified the field-induced gapped phase in $\alpha$-RuCl3, in great support of our work. Interestingly, we find some detailed quantitative differences, e.g., for the critical field, the slope of the gap vs field, and the anisotropy of the gap. These are ascribed to the fact that the temperature and field regime considered in these studies are quite lower than ours. This suggests that the spin gap behavior critically changes when approaching the quantum critical point—which is an interesting subject for future study.