Field-induced instability of the quantum-spin-liquid ground state in the $J_{\rm eff}=\frac{1}{2}$ triangular-lattice compound NaYbO$_2$

TL;DR

This study investigates NaYbO$_2$, a triangular-lattice compound, revealing a spin-liquid ground state at zero field and a field-induced quantum phase transition to magnetic order above 2 T, using various bulk and local probes.

Contribution

It provides experimental evidence of a field-induced quantum phase transition in a $J_{eff}=1/2$ triangular-lattice compound, expanding understanding of frustrated quantum magnetism.

Findings

No long-range magnetic order down to 0.3 K at zero field

Field-induced magnetic order observed above 2 T

Persistent spin fluctuations at low fields

Abstract

Polycrystalline samples of NaYbO$_2$ are investigated by bulk magnetization and specific-heat measurements, as well as by nuclear magnetic resonance (NMR) and electron spin resonance (ESR) as local probes. No signatures of long-range magnetic order are found down to 0.3~K, evidencing a highly frustrated spin-liquid-like ground state in zero field. Above 2\,T, signatures of magnetic order are observed in thermodynamic measurements, suggesting the possibility of a field-induced quantum phase transition. The $^{23}$Na NMR relaxation rates reveal the absence of magnetic order and persistent fluctuations down to 0.3~K at very low fields and confirm the bulk magnetic order above 2~T. The $H$-$T$ phase diagram is obtained and discussed along with the existing theoretical concepts for layered spin-$\frac{1}{2}$ triangular-lattice antiferromagnets

Figures

Full text

Two-dimensional (2D) triangular-lattice antiferromagnets (TLAF) with spin-$\frac{1}{2}$ ions are a highly active field of research in current condensed-matter physics. Here, strong magnetic frustration due to the lattice geometry enhances quantum fluctuations and leads to exotic ground states. One of the most interesting examples is the quantum-spin-liquid (QSL) state, which is characterized by highly entangled spins, that remain dynamic down to zero temperature without any symmetry breaking Savary and Balents (2016); Balents (2010); Zhou et al. (2017). Typical examples include $\kappa$-(BEDT-TTF)2Cu2(CN)3, EtnMe4-nSb[Pd(DMIT)2]2, and Ba3CuSb2O9 Shimizu et al. (2003); Yamashita et al. (2010); Itou et al. (2008); Zhou et al. (2011). The presence of strong spin-orbit coupling (SOC), a local entanglement of spin and orbital degrees of freedom, further enhances the frustration and quantum fluctuations Chen et al. (2009); Jackeli and Khaliullin (2009); Ishizuka and Balents (2014); Lu et al. (2017). Therefore, triangular systems with SOC are prime candidates for emerging QSL ground states Maksimov et al. (2018); Zhu et al. (2018). Recently, YbMgGaO4 Li et al. (2015a, b) and NaYbS2 Baenitz et al. (2018) have been proposed as rare-earth triangular-lattice QSL candidates, where the 14-fold spin and orbital degeneracy of the Yb3+ ions is lifted to four Kramers doublets due to the strong SOC and associated crystal electric field (CEF) effects. Eventually, the low-temperature behavior of these materials can be understood as arising from the effective (pseudo)spin-$\frac{1}{2}$ ground state. The first excited doublet is well separated from the ground state by an energy gap of $\Delta\sim$ 420 K and $\sim$200 K for YbMgGaO4 and NaYbS2, respectively. The magnetic entropy also supports the formation of a robust pseudospin-$\frac{1}{2}$ state at low temperatures Li et al. (2015a, b); Baenitz et al. (2018).

It is known that in applied magnetic fields, the nature of the ground state is significantly modified for a spin-$\frac{1}{2}$ TLAF Chubokov and Golosov (1991). The so-called “order by disorder” mechanism in TLAF can stabilize different phases, including an oblique version of the 120° state ($Y$ phase), the collinear “up-down-up” ($uud$) phase, and a canted version of the $uud$ phase (canted 2:1 phase) Chubokov and Golosov (1991). The stabilization of the $uud$ phase is manifested by a magnetization plateau at $\frac{1}{3}M_{s}$ over a finite field range. Experimentally, this has been reported only in a few TLAFs, including Cs2CuBr4 Ono et al. (2003); Alicea et al. (2009) and Ba3$M$Sb2O9 ($M$ = Co, Ni) Fortune et al. (2009); Susuki et al. (2013); Shirata et al. (2011). Moreover, even a small interlayer interaction, which is present in most of the real materials, can generate additional frustration and quantum phase transitions at strong magnetic fields Gekht and Bondarenko (1997); Yamamoto et al. (2015). Even though the non-spin-$\frac{1}{2}$ TLAF families such as V$X_{2}(X$ = Cl, Br) and $A$CrO2 ($A$ = Li, Cu, Ag) have been widely discussed in this context, the larger interlayer couplings and/or very strong in-plane couplings make the field range inaccessible for observing such field-induced quantum phase transitions experimentally Chubokov and Golosov (1991); Gekht and Bondarenko (1997).

Recently, rare-earth delafossite-type materials $A^{1+}R^{3+}X_{2}$, where $A$ is a monovalent nonmagnetic ion, $R$ is a rare-earth ion, and $X$ is a divalent chalcogen, have been proposed as triangular QSL candidates Baenitz et al. (2018); Liu et al. (2018). Most of them crystallize in the trigonal structures with $R\bar{3}m$ symmetry, where the magnetic $R^{3+}$ ions form perfect triangular layers composed of edge-shared $R$X6 octahedra. These triangular layers are separated by $A$X6 octahedra along the crystallographic $c$ direction. In contrast to the heavily explored rare-earth TLAF QSL candidate YbMgGaO4 Li et al. (2016); Shen et al. (2016); Paddison et al. (2017); Li et al. (2017a); Zhang et al. (2018), the $A^{1+}$Yb${}^{3+}X_{2}$ delafossites are free from the intersite disorder, which may obscure the intrinsic properties of TLAFs Xu et al. (2016); Li et al. (2017b); Zhu et al. (2017); Luo et al. (2017); Ma et al. (2018); Parker and Balents (2018); Kimchi et al. (2018).

A possible QSL ground state in NaYbO2 is conjectured based on the absence of magnetic ordering and spin freezing down to 50 mK Liu et al. (2018); Ding et al. (2019). Here, we utilize a range of different macroscopic and local techniques and elaborate on the magnetic behavior of NaYbO2, including the peculiar field-induced order that appears above 2 T. We first demonstrate that at low temperatures the Yb3+ ions form effective spin-$\frac{1}{2}$ Kramers doublets due to the combined effect of CEF and SOC. The ground-state doublet is separated from the first excited state by a large energy gap of $\Delta\simeq$ 320 K (27 meV) as estimated from Yb ESR. The absence of magnetic long-range order (LRO) down to 0.3 K at zero or small applied fields, combined with the antiferromagnetic Curie-Weiss temperature of $\theta_{\rm CW}\sim$ $-6$ K, suggests a QSL-like ground state. Under external magnetic fields, NaYbO2 undergoes several magnetic phase transitions that may be consistent with theoretical results for TLAF with small interlayer exchange interactions. In particular, we argue that the unusual behavior of NaYbO2 arises from the proximity to an avoided antiferromagnetically ordered state. LRO is melted in zero field due to the combination of quantum fluctuations and exchange frustration, whereas an applied magnetic field quenches the quantum fluctuations and some of the exchange interactions and reinforces the LRO. Therefore, NaYbO2 may be seen as a critical QSL due to the competition of weak magnetic exchange interactions and quantum fluctuations, as well as a unique system for exploring ground-state properties of triangular-lattice antiferromagnets with a strong SOC.

Polycrystalline samples of NaYbO2 were synthesized using a solid-state reaction method, and the sample quality was confirmed by powder x-ray diffraction (for more details, see the Supplemental Material 111see the Supplemental Material). Magnetization ($M$) data were collected using a SQUID magnetometer (Quantum Design, VSM and MPMS) and an ac/dc susceptometer (PPMS). Studies below 2 K were performed using a 3He cooling stage (down to 500 mK) in the MPMS. Magnetization measurements in pulsed magnetic fields up to 35 T and at 500 mK were performed at the Dresden High Magnetic Field Laboratory (HLD). Specific-heat ($C_{p}$) measurements were performed on a pellet by using a relaxation technique (PPMS; Quantum Design) down to 350 mK. Electron spin resonance (ESR) experiments were performed at the $X$-band frequency (9.4 GHz). $T$-dependent nuclear magnetic resonance (NMR) experiments were carried out by applying a pulsed NMR technique on the 23Na nucleus (nuclear spin $I$ = $\frac{3}{2}$) using a commercially available NMR spectrometer (Tecmag).

Figure 1(a) shows the $T$ dependence of the magnetic susceptibility $\chi(T)$ of NaYbO2, measured at different applied fields. Towards low temperatures, $\chi(T)$ increases in a Curie-Weiss (CW) manner. At low fields ($H\leq 2$ T), the $\chi(T)$ data do not show any indication of magnetic LRO down to 0.5 K. Above 200 K, 1/$\chi(T)$ can be well described using $\chi(T)=\chi_{0}+\frac{C}{T-\theta_{CW}}$, where $\chi_{0}$ is a $T$-independent contribution and the remaining term represents the CW law. Our fit yields $\chi_{0}\simeq$ 2$\times$10*-4* cm3/mol, a Curie-Weiss temperature $\theta_{CW}=-100$ K, and an effective moment $\mu_{\rm eff}=4.5$ $\mu_{\rm B}$. The obtained parameters are consistent with the literature Liu et al. (2018); Hashimoto et al. (2003), and $\mu_{\rm eff}$ agrees with the theoretical prediction (4.54 $\mu_{\rm B}$) for the Yb3+ ion having a ${}^{2}F_{7/2}$ multiplet with $g$ = 8/7.

A change of slope was observed in 1/$\chi(T)$ below 100 K, indicating an evolution toward the ground-state CEF doublet with only a small intersite coupling. Below 25 K, 1/$\chi(T)$ could again be described by a Curie-Weiss law after subtracting a constant term $\chi_{0}(\simeq$0.0065 cm3/mol), yielding $\mu_{\rm eff}=2.6\mu_{\rm B}$ and $\theta_{CW}=-6$ K. Moreover, no indication of LRO is detected down to 0.35 mK, which yields a frustration parameter $f(\equiv\theta_{\rm CW}/T_{\rm N})\geq$ 17. The obtained moment $\mu_{\rm eff}(\equiv g\sqrt{J(J+1)}~{}\mu_{\rm B})$ is reminiscent of an effective spin-$\frac{1}{2}$ state with an average $g$-value of 3. Similar effective spin-$\frac{1}{2}$ states are also reported for the Yb3+-based TLAFs NaYbS2 Baenitz et al. (2018) and YbMgGaO4 Li et al. (2015a). At low temperatures, we observed a large $\chi_{0}$ value, which includes the Van-Vleck susceptibility $\chi_{\rm VV}$, arising from CEF excitations Shirata et al. (2012). Such a large low-temperature $\chi_{\rm VV}$ contribution is typical for Yb-based compounds Baenitz et al. (2018); Miyasaka et al. (2009).

Zero-field specific heat, measured as a function of temperature, is shown in Fig. 1(b). No signatures of magnetic ordering were observed down to 0.35 K. The magnetic contribution to the specific heat is obtained by subtracting the lattice contribution, which is estimated by the specific heat of the isostructural reference compound NaLuO2. The magnetic specific heat $C_{\rm mag}$ features a broad maximum at around 1.3 K with $C_{\rm mag}^{\rm max}/R\simeq 0.21$ Note (1). Below 1 K, $C_{\rm mag}(T)$ follows a linear temperature dependence down to the lowest measured temperatures. Such a linear low-temperature dependence is rather unusual for 2D antiferromagnets, where magnon excitations would give a quadratic $T$-dependence Nakatsuji et al. (2005). On the other hand, a linear low-$T$ dependence is common among various types of spin liquids Zhou et al. (2011); Yamashita et al. (2008); Helton et al. (2007). As shown in the inset of Fig. 1(b), the magnetic entropy from lowest temperature to 40 K is nearly $~{}R\ln 2$, as expected for an isolated Kramers doublet.

ESR provides information about the local magnetic properties of Yb3+ in NaYbO2. The anisotropic $g$-factor of a polycrystalline sample was determined by fitting a Lorentzian ESR line with powder averaging and a uniaxial $g$-factor anisotropy. As shown in Fig. 2(a), a narrow and well-resolved Yb-ESR line could be observed, very similar to the line observed for NaYbS2 but quite different from the one reported for YbMgGaO4 Li et al. (2015b); Baenitz et al. (2018). In the latter compound, the averaged values of $g_{\parallel}$ = 3.06 and $g_{\perp}$ = 3.72 have been reported Li et al. (2015b). However, the Mg-Ga site disorder triggers off-center displacements of the Yb3+ ions and leads to a distribution of the $g$-values that manifests itself in the strong broadening of both the ESR line Li et al. (2015b) and CEF excitations as probed by inelastic neutron scattering Li et al. (2017b).

In general, the ESR $g$-factor accounts for the Zeeman splitting of the lowest Kramers doublet of the Yb3+ ion, and depends on the local site symmetry. The $g$-values for NaYbO2 are found to be almost temperature-independent, but strongly anisotropic with $g_{\parallel}$ = 1.75(3) and $g_{\perp}$ = 3.28(8). The average $g$-value, $g_{\rm av}=\sqrt{(g_{\parallel}^{2}+2g_{\perp}^{2})/3}\simeq$ 2.86, yields an effective moment of $\mu_{\rm ESR}\simeq$ 2.4 $\mu_{\rm B}$, which is in good agreement with the value obtained from the low-temperature CW fit (2.6 $\mu_{\rm B}$). The integrated intensity of the ESR line is proportional to the local susceptibility. At low temperatures, $1/\chi_{\rm ESR}(T)$ yields $\theta_{CW}\simeq-9$ K, which is close to the low-temperature value extracted from the $1/\chi(T)$ analysis.

Figure 2(b) shows the ESR linewidth, $\Delta B(T)$, which for temperatures above 40 K is dominated by a broadening due to both spin-orbit coupling and modulation of the ligand field by lattice vibrations. The red-dashed line is the fit to the data with $\Delta B\propto 1/\exp(\Delta/T)-1$, which corresponds to an Orbach process, namely, the excited CEF states at an energy $\Delta$ above the ground state are involved in the relaxation Orbach (1961); Abragam and Bleaney (1970). With a residual line width $\Delta B_{0}=15$ mT we obtain $\Delta\simeq 320$ K ($\equiv 27$ meV).

The isothermal magnetization measured at $T=$ 0.5 K for magnetic fields up to 35 T, is shown in Fig. 2(c). It shows two plateau-like anomalies at about $\sim$2 and $\sim$5 T before reaching full saturation. This is rather unusual for a 2D TLAF, where one would expect a $\frac{M_{s}}{3}$ plateau only. In NaYbO2, such multiple magnetization plateaus may indicate possible field-induced quantum phase transitions at low temperatures. $M(H)$ saturates nearly around 12 T and it linearly increases further in higher fields due to the Van-Vleck contribution. The estimated Van-Vleck contribution, $\chi_{\rm VV}\simeq 0.01~{}\mu_{\rm B}$/T $\simeq$ 0.0056 cm3/mol is comparable with the value obtained in the low-$T$ $\chi(T)$ analysis. The value of the saturation magnetization, $M_{s}\simeq$ 1.36 $\mu_{\rm B}$, is in good agreement with calculations based on the ESR $g$-value of $g_{\rm av}\simeq$ 2.86 and $J_{\rm eff}=\frac{1}{2}$, which leads to $M_{s}=gJ_{\rm eff}\mu_{\rm B}\sim$ 1.43 $\mu_{\rm B}$.

To gain more insight into the field-induced phases, we measured $\chi(T)$ and $C_{\rm p}(T)$ at different magnetic fields. The $\chi(T)$ data measured at higher fields ($H\geq 2$ T) show a kink at low temperatures, which confirms the field-induced magnetic transitions (see Fig. S3 in Note (1)). They are much better visible in the $C_{\rm p}(T)$ data shown in Fig. 3. The application of an external field of 2 T causes a magnetic transition at around 0.6 K, whereas no such transition was observed below 2 T. As shown in Fig. 3(a), this field-induced magnetic transition shifts towards higher temperatures with increasing magnetic field up to 4 T, where the transition temperature reaches its maximum at $T_{\rm N1}\simeq$ 1.03 K. Further increase in the field results in a decrease of $T_{\rm N1}$ and triggers an additional feature, related to a second transition at $T_{\rm N2}$ above 6 T. This kind of multiple magnetic transition is not uncommon in TLAF with easy-axis anisotropy Ranjith et al. (2016, 2017). As shown in Fig. 3(b), both transition temperatures, $T_{\rm N1}$ and $T_{\rm N2}$, decrease with further increase of the field.

We further used 23Na NMR to probe the local static and dynamic magnetism of NaYbO2. The 23Na NMR powder spectra, measured at 35 MHz, are shown in Fig. 4(a). At high temperatures, the spectra exhibit a powder pattern typical for the $I=\frac{3}{2}$ nuclei with an anisotropic central line and two symmetric singularities, stemming from the first-order quadrupole interaction. The spectra can be well described by the isotropic and axially anisotropic NMR shift components $K_{\rm iso}$ and $K_{\rm ax}$, respectively, a quadrupole frequency $\nu_{Q}\simeq$ 456 kHz, and an electric-field gradient (EFG) asymmetry parameter $\eta=0$. The value of $\nu_{Q}$ is much larger than that observed in the isostructural NaYbS2, which might be due to the smaller $c$ parameter of the oxide compound. The line is shifted and monotonously broadened with decreasing temperature, reflecting the anisotropic coupling of the nuclear moments to the electron moments of Yb3+.

The $T$ dependence of the NMR shifts $K_{\rm iso}$ and $K_{\rm ax}$ could be extracted by consistently fitting these parameters to the spectra, taking broadening of the central and satellite lines into account [see the red solid line in Fig. 4(a)]. The $T$ dependencies of the NMR shifts are remarkably different from those of $\chi(T)$ Note (1). In order to extract the transferred hyperfine coupling constant $A_{\rm hf}$, $K_{\rm iso}$, and $K_{\rm ax}$ are plotted against the $\chi_{\rm 3T}$ as shown in Fig. 4(b). The nonlinear dependence of $K$ vs $\chi$ indicates a $T$-dependent transferred hyperfine coupling in NaYbO2. $K_{\rm iso}$ and $K_{\rm ax}$ show linear dependencies with different slopes below about 60 and above 140 K. The obtained high- and low-temperature hyperfine coupling constants are 456 Oe/$\mu_{\rm B}$ and $-127$ Oe/$\mu_{\rm B}$ for $K_{\rm iso}$, and 684 Oe/$\mu_{\rm B}$ and 296 Oe/$\mu_{\rm B}$ for $K_{\rm ax}$, respectively. The change in $A_{\rm hf}$ around 100 K might originate from the CEF splitting. At about 100 K, a depopulation of the excited CEF states may change the effective moment and hyperfine coupling. Similar anomalous $T$-dependent hyperfine coupling constants were also reported for other 4$f$-based spin systems Ohama et al. (1995); Myers and Narath (1973). The NMR line is significantly broadened below 0.95 K, indicating the development of an internal field that confirms the onset of magnetic order, in agreement with our thermodynamic measurements.

The spin-lattice relaxation rate $1/T_{1}\propto[\sum_{\vec{q}}\mid A(\vec{q})\mid^{2}\chi^{\prime\prime}(\vec{q},\omega_{0})]$ probes low-energy spin excitations, where $A(\vec{q})$ is the form factor of the hyperfine interactions as a function of the wave vector $\vec{q}$, and $\chi^{\prime\prime}(\vec{q},\omega_{0})$ is the imaginary part of the dynamic susceptibility at the nuclear Larmor frequency $\omega_{0}$. Figure 4(c) shows the $T$ dependence of the 23Na-spin-lattice relaxation rate $1/T_{1}$, measured at 0.8 and 3.1 T. At high temperatures, the overall temperature dependence of 1/$T_{1}$ is not affected by the field. The increase in $1/T_{1}$ upon cooling is followed by a broad maximum at around 70 K, instead of a $T$-independent behavior expected in the paramagnetic regime for a system with local magnetic moments. This anomalous behavior might be again attributed to the depopulation of the excited CEF levels. After passing through the maximum at 70 K, 1/$T_{1}$ decreases monotonously down to 3 K. At a small field of 0.8 T, 1/$T_{1}$ passes through a minimum at 1 K, followed by a strong increase toward low temperatures. This indicates a divergence of the imaginary part of the dynamic susceptibility ($\vec{q}\neq 0$) on cooling and suggests some sort of critical fluctuations. In strong contrast, at 3.1 T the 1/$T_{1}$ shows a sharp anomaly at about $T_{\rm N1}\simeq$ 0.95 K, thus confirming the field-induced magnetic transition. Below $T_{\rm N}$, 1/$T_{1}$ decreases smoothly toward zero.

In the $1/T_{1}$ measurements, the magnetization recovery is fitted with a recovery function, which can provide information about the nature of the underlying relaxation mechanism Note (1). The coefficient $\beta$ from a stretched exponential fit function is constant above 2 K, indicating a homogeneous distribution of fluctuating moments Note (1). Below 2 K and at 3.1 T, $\beta$ changes and shows a dip at $T_{\rm N1}$, which is typical for strongly frustrated low-dimensional spin systems Itou et al. (2010); Shockley et al. (2015); Ranjith et al. (2018). In contrast, at 0.8 T there is only a smooth change in $\beta$, which might indicate the onset of correlations among the Yb3+ ions. At this point, it is unclear whether these are antiferromagnetic correlations in the vicinity of LRO, or if they are associated with a critical precursor state before entering the QSL.

A quantitative estimate of the in-plane exchange interaction $J$ can be made using the Heisenberg Hamiltonian $J\sum_{\langle i,j\rangle}{\mathbf{S}}_{i}{\mathbf{S}}_{j}$, where the summation is over bonds, and positive $J$ stands for an antiferromagnetic coupling. On the mean-field level, one expects the Curie-Weiss temperature of $\theta_{\rm CW}=-zJS(S+1)/3k_{\rm B}$, where $z$ is the number of nearest-neighbor spins and $k_{\rm B}$ is the Boltzmann constant. The number of nearest neighbors in a TLAF is $z=6$, so $\theta_{\rm CW}\simeq-6$ K results in an antiferromagnetic exchange interaction of $J/k_{\rm B}\simeq$ 4 K. The saturation for a purely 2D ($J^{\prime}$ = 0) TLAF is expected at $H_{s}=4.5J\frac{k_{B}}{g\mu_{B}}\simeq$ 9.7 T, which is slightly below the observed value of 12 T. This indicates the presence of an interlayer interaction and/or of a second-neighbor interaction within the plane, both contributing to $H_{s}$.

Furthermore, the magnetic specific heat ($C_{\rm mag}$) of NaYbO2 shows a rather broad maximum at around 2.1 K reaching $C_{\rm mag}^{\rm max}/R\simeq 0.21$. The magnetic specific heat is a direct measure of quantum fluctuations. For example, in 2D antiferromagnets $C_{\rm mag}^{\rm max}/R$ is reduced from 0.44 in the nonfrustrated square lattice Hofmann et al. (2003); Johnston et al. (2000) to 0.22 in the geometrically frustrated triangular lattice Elstner et al. (1993); Bernu and Misguich (2001). Our result is thus very similar to the theoretical one for the triangular geometry. Moreover, the anticipated position of the maximum at $T/J=0.55$ (2.1 K in NaYbO2) perfectly fits our data. It is worth noting that $C_{\rm mag}$ of YbMgGaO4 also shows a maximum around the same temperature, but its magnitude is higher, whereas the average nearest-neighbor coupling $\bar{J}\simeq 1.4$ K is lower than in NaYbO2. Therefore, NaYbO2 seems to be much closer to the Heisenberg TLAF than YbMgGaO4.

Despite this proximity, NaYbO2 does not show the transition toward the $120^{\circ}$ ordered state expected in a Heisenberg TLAF. Instead, we detect signatures of field-induced magnetic order that have not been observed in other Yb3+-based triangular-lattice magnets reported to date. In Fig. 3(c), we present a comprehensive $H$-$T$ phase diagram, where four different phases (I-IV) may occur. Phase IV is the paramagnetic phase, whereas phase I represents a fluctuating QSL-like state, as reported by Liu $et~{}al.$ Liu et al. (2018). Phases II and III represent the magnetically ordered phases. This phase diagram suggests that NaYbO2 may be at the verge of a field-induced quantum critical point, and indeed we observed signatures of critical fluctuations in $1/T_{1}$. Further on, the NMR spectra measured at $\sim$7 T show different shapes within phase II and phase III suggesting that these are two different ordered states. Neutron-diffraction work under external magnetic fields would be highly desirable to understand the exact nature of these ordered phases.

In conclusion, we have studied the rare-earth triangular-lattice QSL candidate NaYbO2 in detail. The low-temperature behavior of the Yb3+ ions can be attributed to a spin-$\frac{1}{2}$ Kramers doublet, well separated from the first excited doublet. The results of the 23Na NMR reveal increasing magnetic fluctuations toward low temperatures and suggest that NaYbO2 may be a critical spin liquid in analogy to the critical Fermi liquid. The instability of this tentative QSL state is observed in higher fields, with two distinct ordered states forming above 2 T. Upon applying magnetic fields, the fluctuations are suppressed, and the system undergoes a quantum phase transition. The observed anomalous temperature dependence of the NMR shifts and spin-lattice relaxation rates are ascribed to CEF effects. Furthermore, thermodynamic properties are in agreement with theoretical predictions for spin-$\frac{1}{2}$ triangular-lattice Heisenberg antiferromagnets. Therefore, NaYbO2 could be a perfect model system for understanding the spin-$\frac{1}{2}$ magnetism with spin-orbit coupling in the triangular geometry Maksimov et al. (2018); Zhu et al. (2018). In addition, it features critical fluctuations that deserve detailed investigations with local probes such as muon spin rotation/relaxation or inelastic neutron scattering.

Note added. Recently, we became aware of Ref. Bordelon et al. (2019), where NaYbO2 was further investigated by neutron-diffraction experiments.

We thank H. Rosner, B. Schmidt, A. Mackenzie, C. Geibel, L. Hozoi, and D. Inosov for fruitful discussions. We thank C. Klausnitzer and Y. Skourski for technical support. We acknowledge the support from the Deutsche Forschungsgemeinschaft (DFG) through SFB 1143, and support by the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL). A.A.T. acknowledges financial support by the Federal Ministry for Education and Research through the Sofja Kovalevskaya Award of Alexander von Humboldt Foundation.