Gapless spin-liquid state in the structurally disorder-free triangular antiferromagnet NaYbO$_2$

TL;DR

This paper reports the discovery of a gapless spin-liquid state in the disorder-free triangular antiferromagnet NaYbO$_2$, characterized by persistent spin dynamics and continuous magnetic excitations, providing a rare experimental realization of such a phase.

Contribution

It demonstrates that NaYbO$_2$ is a structurally disorder-free material exhibiting a gapless spin-liquid state, offering direct experimental access to triangular antiferromagnet spin-liquid physics.

Findings

No structural disorder or crystal-electric-field randomness in NaYbO$_2$

Absence of magnetic order down to 70 mK

Presence of gapless, continuous magnetic excitations at the K-point

Abstract

We present the structural characterization and low-temperature magnetism of the triangular-lattice delafossite NaYbO$_2$. Synchrotron x-ray diffraction and neutron scattering exclude both structural disorder and crystal-electric-field randomness, whereas heat-capacity measurements and muon spectroscopy reveal the absence of magnetic order and persistent spin dynamics down to at least 70\,mK. Continuous magnetic excitations with the low-energy spectral weight accumulating at the $K$-point of the Brillouin zone indicate the formation of a novel spin-liquid phase in a triangular antiferromagnet. This phase is gapless and shows a non-trivial evolution of the low-temperature specific heat. Our work demonstrates that NaYbO$_2$ practically gives the most direct experimental access to the spin-liquid physics of triangular antiferromagnets.

Equations

$$A(t)=A_0[f_{1}exp(-{\lambda }_{1}t)+(1-f_1)exp(-{\lambda }_{2}t)]+B_0$$

Full text

Introduction. Quantum spin liquids (QSLs) in frustrated magnets have attracted a lot of attention because of the occurrence of unconventional ground states, where highly entangled spins and their strong fluctuations are observed in the absence of long-range order down to zero temperature. The ensuing excitations are interesting in their own right as they are distinct from magnons in systems with conventional long-range magnetic order Balents (2010); Savary and Balents (2017); Knolle and Moessner (2019). Historically, the QSL was first exemplified by the nearest-neighbor resonating-valence-bond (RVB) state on the triangular lattice Anderson (1973), which drew a great deal of interest in triangular antiferromagnets ever since, although the majority of real-world triangular materials order magnetically at low temperatures Starykh (2015).

Recently, a promising QSL candidate YbMgGaO4 with the effective spin-1/2 Yb3+ ion on triangular lattice has been proposed by Li et al. Li et al. (2015a, b). They observed persistent spin dynamics down to at least 48 mK Li et al. (2016a) and the $T^{0.7}$ power-law behavior of the specific heat Li et al. (2015a), indicative of the gapless U(1) QSL characterized by a Fermi surface of fractionalized (spinon) excitations. Although continuous spinon-like excitations were indeed observed experimentally Shen et al. (2016), their assignment to spinons Shen et al. (2018) is far from unambiguous, and alternative phenomenological explanations within the valence-bond framework were proposed as well Li et al. (2017a); Kimchi et al. (2018). Moreover, absent magnetic contribution to the thermal conductivity Xu et al. (2016), considerable broadening of spin-wave excitations in the fully polarized state Paddison et al. (2017), and acute broadening of the crystal-electric-field (CEF) excitations of Yb3+ Li et al. (2017b) reveal a significant complexity of this material. The problem appears to be related to the statistical distribution of Mg2+ and Ga3+ that randomizes the local environment of Yb3+ Li et al. (2017b) and may lead to peculiar effects like spin-liquid mimicry Zhu et al. (2017); Parker and Balents (2018), although the exact influence of the structural randomness (disorder) on the magnetic parameters remains debated Zhang et al. (2018). Whether or not these structural effects are integral to the spin-liquid formation, the complexity of YbMgGaO4 hinders its use as a reference model material for the QSL state in triangular antiferromagnets.

On the theory side, significant efforts were made to establish the parameter regime where long-range magnetic order gives way to a QSL. Whereas nearest-neighbor Heisenberg interactions on the triangular lattice support the $120^{\circ}$ magnetic order Zhitomirsky and Chernyshev (2013), a weak second-neighbor coupling is sufficient to suppress this order and drive the system toward a QSL state Kaneko et al. (2014); Hu et al. (2015); Zhu and White (2015); Bishop and Li (2015); Iqbal et al. (2016); Saadatmand and McCulloch (2016); Bauer and Fjærestad (2017); Saadatmand and McCulloch (2017). The presence of multiple anisotropic interactions–a characteristic of the Yb3+ compounds–lays out another route to the QSL Zhu et al. (2018); Maksimov et al. . These two regimes, the second-neighbor isotropic exchange vs. nearest-neighbor anisotropic exchange, may in fact produce isomorphic QSL phases Zhu et al. (2018), but their exact nature and even the presence Zhu and White (2015); Bauer and Fjærestad (2017) or absence Kaneko et al. (2014); Bishop and Li (2015); Iqbal et al. (2016) of a spin gap therein remain vividly debated. Experimental input is thus highly desirable, but requires a disorder-free material characterized down to mK temperatures, as magnetic interactions in the Yb3+ oxides are of the order of 1 K, and the ground-state regime is practically reached below 0.4 K only Li et al. (2015a, 2016a).

Na-based chalcogenides NaYbX2 (X = O, S, Se) have recently come to the attention of researchers as disorder-free triangular antiferromagnets Baenitz et al. (2018); Liu et al. (2018). They feature layers of edge-sharing YbO6 octahedra with the triangular arrangement of the magnetic Yb3+ ions. These layers are separated by the well-ordered Na atoms, with the interlayer Yb–Yb distance of 5.82 Å, which is shorter than in YbMgGaO4 (8.61 Å), but still significantly longer than the nearest-neighbor Yb–Yb distance of 3.34 Å within the triangular planes (see Fig. 1(a-b)).

Here, we confirm the absence of structural disorder and report persistent spin dynamics in NaYbO2 down to at least 70 mK. We also observe continuous excitations that are qualitatively similar to those predicted Zhu et al. (2018) for the QSL state in triangular antiferromagnets. Our results set up NaYbO2 as a new, disorder-free spin-liquid candidate, and shed light onto the physics of the spin-liquid state in triangular antiferromagnets. We demonstrate gapless nature of this state and the absence of simple power-law scaling for the specific heat.

Absence of structural disorder. Polycrystalline samples of NaYbO2 were synthesized by a solid-state reaction as described in Ref. Hashimoto et al., 2003. The absence of structural disorder was verified by synchrotron x-ray and neutron diffraction data sup collected, respectively, at the ID22 beamline of the ESRF and at the WISH instrument Chapon et al. (2011) at the ISIS facility. No signatures of site deficiency or disorder was observed in the structure refinements performed down to 1.5 K. The high-resolution synchrotron data reveal very sharp peaks and exclude any extended defects that may occur in a layered compound sup . At 10 K, the atomic displacement parameter of Yb is below $10^{-3}$ Å2 and excludes any off-center displacements that have been the most direct signature of structural randomness in YbMgGaO4 Li et al. (2017b).

In YbMgGaO4, the structural disorder becomes most conspicuous in the CEF excitations that broaden and even show four peaks in the inelastic neutron spectrum Li et al. (2017b), instead of the three peaks expected for Yb3+ with its ${}^{2}F_{7/2}$ multiplet split into four Kramers doublets by the trigonal crystal field Gaudet et al. (2015); Ross et al. (2011). The CEF excitations of NaYbO2 were measured at 5 K using the MERLIN spectrometer at ISIS operating with the incident energies of 90 and 150 meV Bewley et al. (2006). As shown in Fig. 1(c), three sharp, resolution-limited CEF excitations are observed, as expected for Yb3+. This ultimately proves the absence of structural disorder in our material.

From the excitation energies and line intensities we extract the CEF parameters and the compositions of the four Kramers doublets sup . It is worth noting that the excitation energies of 34.8, 58.5, and 83.1 meV are not far from those reported for YbMgGaO4 (39.4, 61.3, and 96.6 meV, respectively Li et al. (2017b)), reflecting similar local environments of Yb3+ in both compounds. By contrast, NaYbS2 bears all CEF excitations below 50 meV Baenitz et al. (2018), likely in agreement with the less ionic nature of the Yb–S bonds. Similar CEF excitations in YbMgGaO4 and NaYbO2 indicate that on the level of single-ion physics the latter can be seen as a close analogue of the former, but with the structural disorder completely removed.

Ground-state CEF doublet. At low temperatures, the magnetic behavior of NaYbO2 is fully determined by the ground-state Kramers doublet and can be described by an effective pseudospin-$\frac{1}{2}$ Hamiltonian Li et al. (2016b). Indeed, inverse magnetic susceptibility measured using a SQUID magnetometer (Quantum Design, MPMS-7T) shows a change in slope around 70 K, with the low-temperature linear part reflecting the Curie-Weiss behavior of pseudospins-$\frac{1}{2}$ (see Fig. 2(a-b)). Nevertheless, the excited CEF levels produce a sizeable Van Vleck term $\chi_{vv}$.

To determine the $\chi_{vv}$, we measured field-dependent magnetization at 0.45 K using a triply compensated extraction magnetometer within a 65 T short-pulse magnet at the National High Magnetic Field Laboratory, Los Alamos. As seen in Fig. 2(c), NaYbO2 saturates around 16 T. The magnetization increases linearly in higher fields, and its slope corresponds to the van Vleck term of $\chi_{vv}=0.0101$ $\mu_{B}$/T$=0.00564$ emu/mol/Oe sup . After subtracting this Van Vleck contribution, we arrive at the saturation magnetization of 1.75 $\mu_{B}$/f.u. Magnetic susceptibility with $\chi_{vv}$ subtracted follows the Curie-Weiss law with the effective moment of $\mu_{\rm eff}=2.84(2)$ $\mu_{B}$. Both values are comparable favorably to 1.5 $\mu_{B}$/f.u. and 2.60 $\mu_{B}$ expected from the powder-averaged $\bar{g}=3.00$ calculated based on the CEF fit.

Low-temperature thermodynamics. The susceptibility fit also yields the Curie-Weiss temperature $\Theta=-5.64(1)$ K. Its negative value confirms antiferromagnetic interactions between the Yb3+ pseudospins. The absolute value is about two times larger than in YbMgGaO4 ($\Theta_{\|}=-1.5$ K, $\Theta_{\perp}=-2.7$ K Li et al. (2015b)) and corroborates the two-fold increase in the saturation field from 8 T in YbMgGaO4 to 16 T in NaYbO2. The simple estimate $\Theta=-3\bar{J}_{1}/2$ for a triangular antiferromagnet yields $\bar{J}_{1}\simeq 3.8$ K, the energy scale of nearest-neighbor exchange couplings in NaYbO2.

To probe low-temperature thermodynamics, we measured specific heat ($C_{p}$) of NaYbO2 using the PPMS (Quantum Design) down to 0.4 K and a home-built dilution-fridge setup down to 70 mK sup . Magnetic contribution to the $C_{p}$ becomes visible below 8 K and shows a broad maximum around 1 K sup . At even lower temperatures, magnetic contribution decreases without showing any signatures of a magnetic transition (see Fig. 2(d)). This suggests the absence of long-range magnetic order and the possibility of a spin-liquid state confirmed by the muon spin rotation ($\mu$SR) measurement below.

By taking NaLuO2 as the isostructural non-magnetic compound, we confirmed that lattice contribution to the $C_{p}$ is negligible below 2 K, whereas the nuclear contribution was subtracted by systematic measurements in weak applied fields sup . The remaining, magnetic contribution $C_{m}(T)$ will usually take the exponential form, $e^{-\Delta/T}$, or the form of a power law, $T^{p}$, for gapped and gapless excitations, respectively. However, neither term accounts for our experimental data (Fig. 2(d)). We tentatively fit $C_{m}(T)$ with two power laws, $aT^{p}+bT^{q}$, where $p\simeq 2.9$ resembles the $T^{3}$ contribution of magnons in a long-range-ordered antiferromagnet, and $q\simeq 0.5$ reflects a sublinear behavior, which is remotely similar to the power-law behavior in YbMgGaO4 Li et al. (2015a). At first glance, a combination of the two different contributions could be seen as an effect of sample inhomogeneity, but a weak applied field restores the simpler $T^{2.2}$ power-law behavior sup , suggesting an intrinsic nature of the two power laws observed in zero field. We conclude that the low-energy excitations in NaYbO2 are neither gapped nor magnon-like. They are also distinct from the low-energy excitations in YbMgGaO4 that showed Li et al. (2015a) the robust $T^{0.7}$ power law characteristic of the ”spinon metal” of a U(1) quantum spin liquid Motrunich (2005).

Before going further, we note in passing that the magnetization curve of NaYbO2 measured at 0.45 K shows a plateau between 4 and 5 T at about one half of the saturation magnetization as shown in Fig. 2(c). Such a $\frac{1}{2}$-plateau contrasts with the $\frac{1}{3}$-plateau typically observed in Heisenberg and XXZ triangular antiferromagnets Starykh (2015) and may be indicative of a more complex interaction regime. Moreover, field-induced phase transitions should occur in NaYbO2, but they go beyond the scope of our present manuscript and will be addressed in future studies.

$\mu$SR data. Our heat-capacity data exclude long-range magnetic ordering above 70 mK, yet spin freezing would have no immediate effect on the specific heat. A direct probe of spin dynamics is thus essential to identify the QSL. To this end, we carried out zero-field (ZF) and longitudinal-field (LF) $\mu$SR measurement down to 100 mK at the MuSR spectrometer (ISIS).

Four representative ZF $\mu$SR spectra are shown in Fig. 3(a). They reveal neither oscillations nor a drastic drop in the initial asymmetry, indicative of the long-range magnetic order, but instead show signatures of persistent spin dynamics, in particular the lack of polarization recovery to $\frac{1}{3}$ of the initial value rules out the presence of static random fields. We further analyze these data by fitting the muon spectra with two exponential components:

[TABLE]

where A0 and B0 denote the initial asymmetry and the constant background, respectively, $\lambda_{1}$ and $\lambda_{2}$ represent the muon spin relaxation rates for muons implanted at two sites near O2-, f1 stands for the fraction of the first component. The fitted ZF $\mu$SR relaxation rate $\lambda_{1}$, $\lambda_{2}$ and f1 as a function of temperature are shown in sup . f1 shows temperature-independent behavior with its value close to 0.5, suggesting the same population at the two muon sites. Below we will discuss the electronic dynamics in terms of $\lambda_{1}$ since it is significantly larger than $\lambda_{2}$.

Temperature dependence of $\lambda_{1}$ tracks the onset of correlations between the Yb3+ pseudospins. As seen in Fig. 3(b). Above 10 K, NaYbO2 is paramagnetic with a smaller and temperature-independent $\lambda_{1}$. The increase in $\lambda_{1}$ below 10 K is accompanied by the growing magnetic contribution to the specific heat, whereas the second temperature-independent regime below 2 K indicates the onset of the spin-liquid state. To prove the dynamic nature of the relaxation in this temperature range, we performed the LF experiment at 1.5 K. Should the relaxation arise from a weak static field, the size of this field is $B_{\rm loc}=\lambda/\gamma_{\mu}\simeq 1.2$ mT, where $\gamma_{\mu}=135.5\times 2\pi$ s*-1* $\mu$T*-1* is the gyromagnetic ratio for muons. Our LF data show that the relaxation persists in much higher fields, thus proving the dynamic nature of the Yb3+ pseudospins sup .

We also note that the characteristic evolution of $\lambda_{1}$, its increase below 10 K and the saturation below 2 K, takes place at about twice higher temperatures compared to YbMgGaO4 Li et al. (2016a). This further supports our conclusion on the twice stronger exchange couplings in NaYbO2.

Low-energy excitations. The most interesting property of a spin liquid is arguably its excitation spectrum. We probed the low-energy excitations of NaYbO2 at the cold-neutron multi-chopper LET spectrometer (ISIS) Bewley et al. (2011) at 45 mK using incident neutron energies of 1.46, 3.7, and 7.52 meV. Spectral weight observed above 0.8 Å*-1* is continuously distributed in both energy ($E$) and momentum ($Q$) and extends to about 1 meV (see Fig. 4(a)). This low-energy spectral weight concentrates around $Q\simeq 1.25$ Å*-1* corresponding to the $K$-point of the Brillouin zone with the reciprocal-lattice vector ($\frac{1}{3}$, $\frac{1}{3}$, 0). A $120^{\circ}$ magnetic order will lead to a Bragg peak at the same position, but the excitation spectrum is clearly different from the spin-wave spectrum of such an ordered state sup . The energy dependence over the constant-$Q$ cut suggests that the excitations reach low energies down to the elastic line that becomes prominent below 0.1 meV. This further confirms gapless nature of the magnetic excitations in NaYbO2 (see Fig. 4(b)).

The spectrum of NaYbO2 bears close similarities to that of YbMgGaO4, where no spectral weight was observed at low $Q$, around the zone center. All the spectral weight is concentrated in the vicinity of the zone boundary, in agreement with our observation of the spectral weight above 0.8 Å*-1* only. A further, and more subtle feature of YbMgGaO4, is the re-distribution of the spectral weight upon heating that signals the formation of distinct excitations at energies above and below $\bar{J}_{1}$ Li et al. (2017a). No such effect is seen in NaYbO2. Moreover, magnetic excitations in YbMgGaO4 extend to the much higher energy of 2 meV, despite the fact that $\bar{J}_{1}$ is twice smaller than in NaYbO2.

Discussion. NaYbO2 shows strong similarities to the widely studied triangular spin-liquid candidate YbMgGaO4 except for the absence of structural disorder. Both materials entail the trigonally distorted YbO6 octahedra. The CEF excitations of Yb3+ occur at about the same energies, and the compositions of the ground-state Kramers doublets are similar too. Exchange couplings differ by a factor of 2, though. This change is accompanied by a reduction in the Yb–O–Yb bridging angle from $99.4^{\circ}$ in YbMgGaO4 to $95.7^{\circ}$ in NaYbO2.

The similarity between the two materials gives us an interesting opportunity to explore which of the effects reported for YbMgGaO4 appear in the disorder-free case. Gapless ground state is retained in NaYbO2. On the other hand, neither the simple power-law behavior of the low-temperature specific heat Li et al. (2015a), nor the energy separation of the spin excitations Li et al. (2017a) have been observed. Moreover, the spin excitations of NaYbO2 extend to much lower energies despite the twice larger $\bar{J}_{1}$. This goes in line with the theory of Ref. Kimchi et al., 2018 that explains both effects in terms of quenched disorder in a valence-bond solid. These effects are probably not generic to the spin-liquid state of triangular antiferromagnets.

Rau and Gingras Rau and Gingras (2018) computed anisotropic nearest-neighbor exchange couplings for several triangular geometries inspired by the possible local structures of YbMgGaO4. By extrapolating their results to the Yb–O–Yb angle of $95.7^{\circ}$ in NaYbO2, we may expect that at least the nearest-neighbor coupling $\bar{J}_{1}$ in this material is very close to the Heisenberg limit. The second-neighbor interaction $\bar{J}_{2}$ should be then operative in order to stabilize a QSL. Such a scenario may be envisaged if, for example, $\bar{J}_{2}$ is less sensitive to the structural geometry than $\bar{J}_{1}$. With $\bar{J}_{2}/\bar{J}_{1}=0.18(7)$ reported for YbMgGaO4 Zhang et al. (2018), the increase in $\bar{J}_{1}$ accompanied by a small change in $\bar{J}_{2}$ will drive the system directly into the QSL phase 111One could argue that $\bar{J}_{2}$ increases as well, but $\bar{J}_{1}\simeq 3.8$ K will then lead to $\bar{J}_{2}$ of nearly 1 K, which is hard to reconcile with the second-neighbor Yb–Yb distance of 5.82 Å and the strongly localized nature of $f$-electrons in Yb3+.. Another interesting observation is that static structure factor calculated for this QSL phase peaks at the $K$-point of the Brillouin zone Zhu et al. (2018) in agreement with the accumulation of the low-energy spectral weight at $Q\simeq 1.25$ Å*-1* observed in our experiment (see Fig. 4), whereas YbMgGaO4 shows a larger low-energy spectral weight at the $M$-point with the reciprocal-lattice vector ($\frac{1}{2}$, 0, 0) Paddison et al. (2017). All these arguments give a strong envision that the spin-liquid state observed in NaYbO2 is the QSL phase of triangular antiferromagnets. Its gapless nature and unusual sensitivity to the magnetic field open prospects for future studies theoretically and experimentally.

Conclusions. The disorder-free NaYbO2 gives the most direct experimental access to the spin-liquid physics of triangular antiferromagnets. Thermodynamic measurements and muon spectroscopy indicate the absence of magnetic order and persistent spin dynamics down to at least 70 mK. An excitation continuum is observed, with the spectral weight accumulating around the $K$-point, as expected in the QSL phase(s) driven by the exchange anisotropy or second-neighbor coupling on the triangular lattice Zhu et al. (2018). The spin-liquid state of NaYbO2 is gapless with a non-trivial low-temperature evolution of the specific heat, which does not follow the spinon scenario originally proposed for YbMgGaO4.