Gapless spin-liquid state in the structurally disorder-free triangular antiferromagnet NaYbO 2
Lei Ding, Pascal Manuel, Sebastian Bachus, Franziska Grußler, Philipp Gegenwart, John Singleton, Roger D. Johnson, Helen C. Walker, Devashibhai T. Adroja, Adrian D. Hillier, Alexander A. Tsirlin
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Gapless spin-liquid state in the structurally disorder-free triangular antiferromagnet NaYbO Lei Ding, ∗ Pascal Manuel, Sebastian Bachus, Franziska Grußler, Philipp Gegenwart, John Singleton, Roger D. Johnson, Helen C. Walker, Devashibhai T. Adroja,
1, 5
Adrian D. Hillier, and Alexander A. Tsirlin ISIS Facility, Rutherford Appleton Laboratory, Harwell Oxford, Didcot OX11 0QX, United Kingdom Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86135 Augsburg, Germany National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, NM 87545, United States Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom Highly Correlated Matter Research Group, Physics Department,University of Johannesburg, Auckland Park 2006, South Africa (Dated: January 24, 2019)We present the structural characterization and low-temperature magnetism of the triangular-lattice delafossiteNaYbO . 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 ofmagnetic order and persistent spin dynamics down to at least 70 mK. Continuous magnetic excitations withthe low-energy spectral weight accumulating at the K -point of the Brillouin zone indicate the formation of anovel spin-liquid phase in a triangular antiferromagnet. This phase is gapless and shows a non-trivial evolutionof the low-temperature specific heat. Our work demonstrates that NaYbO practically gives the most directexperimental access to the spin-liquid physics of triangular antiferromagnets. Introduction.
Quantum spin liquids (QSLs) in frustratedmagnets have attracted a lot of attention because of the oc-currence of unconventional ground states, where highly en-tangled spins and their strong fluctuations are observed in theabsence of long-range order down to zero temperature. Theensuing excitations are interesting in their own right as theyare distinct from magnons in systems with conventional long-range magnetic order [1–3]. Historically, the QSL was firstexemplified by the nearest-neighbor resonating-valence-bond(RVB) state on the triangular lattice [4], which drew a greatdeal of interest in triangular antiferromagnets ever since, al-though the majority of real-world triangular materials ordermagnetically at low temperatures [5].Recently, a promising QSL candidate YbMgGaO with theeffective spin-1/2 Yb + ion on triangular lattice has been pro-posed by Li et al. [6, 7]. They observed persistent spin dy-namics down to at least 48 mK [8] and the T . power-lawbehavior of the specific heat [6], indicative of the gaplessU(1) QSL characterized by a Fermi surface of fractionalized(spinon) excitations. Although continuous spinon-like excita-tions were indeed observed experimentally [9], their assign-ment to spinons [10] is far from unambiguous, and alterna-tive phenomenological explanations within the valence-bondframework were proposed as well [11, 12]. Moreover, absentmagnetic contribution to the thermal conductivity [13], con-siderable broadening of spin-wave excitations in the fully po-larized state [14], and acute broadening of the crystal-electric-field (CEF) excitations of Yb + [15] reveal a significant com-plexity of this material. The problem appears to be relatedto the statistical distribution of Mg + and Ga + that random-izes the local environment of Yb + [15] and may lead to pe-culiar effects like spin-liquid mimicry [16, 17], although theexact influence of the structural randomness (disorder) on themagnetic parameters remains debated [18]. Whether or notthese structural effects are integral to the spin-liquid forma-tion, the complexity of YbMgGaO hinders its use as a refer-ence model material for the QSL state in triangular antiferro- magnets.On the theory side, significant efforts were made to estab-lish the parameter regime where long-range magnetic ordergives way to a QSL. Whereas nearest-neighbor Heisenberginteractions on the triangular lattice support the 120 ◦ mag-netic order [19], a weak second-neighbor coupling is suf-ficient to suppress this order and drive the system towarda QSL state [20–27]. The presence of multiple anisotropicinteractions–a characteristic of the Yb + compounds–lays outanother route to the QSL [28, 29]. These two regimes,the second-neighbor isotropic exchange vs. nearest-neighboranisotropic exchange, may in fact produce isomorphic QSLphases [28], but their exact nature and even the presence [22,26] or absence [20, 23, 24] of a spin gap therein remain vividlydebated. Experimental input is thus highly desirable, but re-quires a disorder-free material characterized down to mK tem-peratures, as magnetic interactions in the Yb + oxides are ofthe order of 1 K, and the ground-state regime is practicallyreached below 0.4 K only [6, 8].Na-based chalcogenides NaYbX (X = O, S, Se) have re-cently come to the attention of researchers as disorder-freetriangular antiferromagnets [30, 31]. They feature layers ofedge-sharing YbO octahedra with the triangular arrangementof the magnetic Yb + ions. These layers are separated by thewell-ordered Na atoms, with the interlayer Yb–Yb distance of5.82 ˚A, which is shorter than in YbMgGaO (8.61 ˚A), but stillsignificantly longer than the nearest-neighbor Yb–Yb distanceof 3.34 ˚A within the triangular planes (see Fig. 1(a-b)).Here, we confirm the absence of structural disorder andreport persistent spin dynamics in NaYbO down to at least70 mK. We also observe continuous excitations that are qual-itatively similar to those predicted [28] for the QSL state intriangular antiferromagnets. Our results set up NaYbO as anew, disorder-free spin-liquid candidate, and shed light ontothe physics of the spin-liquid state in triangular antiferromag-nets. We demonstrate gapless nature of this state and the ab-sence of simple power-law scaling for the specific heat. FIG. 1. (a) Stacking of the Yb + triangular-layers along the c -axis.(b) Triangular layer formed by Yb + cations. (c) Inelastic neutronscattering spectra S ( Q , ¯ h ω ) at 5 K with E i =150 meV. Energy depen-dence of the INS intensity at 5 K integrated in Q over the range 3.9-4.4 ˚A − . Inset depicts the CEF transitions from the ground-stateKramers doublet. Absence of structural disorder.
Polycrystalline samples ofNaYbO were synthesized by a solid-state reaction as de-scribed in Ref. 32. The absence of structural disorder wasverified by synchrotron x-ray and neutron diffraction data [33]collected, respectively, at the ID22 beamline of the ESRF andat the WISH instrument [34] at the ISIS facility. No signaturesof site deficiency or disorder was observed in the structure re-finements performed down to 1.5 K. The high-resolution syn-chrotron data reveal very sharp peaks and exclude any ex-tended defects that may occur in a layered compound [33].At 10 K, the atomic displacement parameter of Yb is below10 − ˚A and excludes any off-center displacements that havebeen the most direct signature of structural randomness inYbMgGaO [15].In YbMgGaO , the structural disorder becomes most con-spicuous in the CEF excitations that broaden and even showfour peaks in the inelastic neutron spectrum [15], insteadof the three peaks expected for Yb + with its F / multi-plet split into four Kramers doublets by the trigonal crystalfield [35, 36]. The CEF excitations of NaYbO were measuredat 5 K using the MERLIN spectrometer at ISIS operating withthe incident energies of 90 and 150 meV [37]. As shown inFig. 1(c), three sharp, resolution-limited CEF excitations areobserved, as expected for Yb + . This ultimately proves theabsence of structural disorder in our material.From the excitation energies and line intensities we extractthe CEF parameters and the compositions of the four Kramersdoublets [33]. It is worth noting that the excitation energiesof 34.8, 58.5, and 83.1 meV are not far from those reportedfor YbMgGaO (39.4, 61.3, and 96.6 meV, respectively [15]),reflecting similar local environments of Yb + in both com-pounds. By contrast, NaYbS bears all CEF excitations below50 meV [30], likely in agreement with the less ionic nature ofthe Yb–S bonds. Similar CEF excitations in YbMgGaO andNaYbO indicate that on the level of single-ion physics the FIG. 2. (a) Temperature dependence of the magnetic susceptibility ofNaYbO down to 0.4 K. Zero-field-cooling and field-cooling curvesare shown down to 1.8 K. (b) The Curie-Weiss fit to the inverse mag-netic susceptibility in the range of 15-30 K after subtracting the VanVleck term. (c) Isothermal magnetization curve measured at 0.45K up to 60 T. The red line marks the Van Vleck contribution. (d)Zero-field specific heat of NaYbO with the nuclear contributionsubtracted [33]. latter can be seen as a close analogue of the former, but withthe structural disorder completely removed. Ground-state CEF doublet.
At low temperatures, the mag-netic behavior of NaYbO is fully determined by the ground-state Kramers doublet and can be described by an effec-tive pseudospin- Hamiltonian [38]. Indeed, inverse mag-netic susceptibility measured using a SQUID magnetome-ter (Quantum Design, MPMS-7T) shows a change in slopearound 70 K, with the low-temperature linear part reflectingthe Curie-Weiss behavior of pseudospins- (see Fig. 2(a-b)).Nevertheless, the excited CEF levels produce a sizeable VanVleck term χ vv .To determine the χ vv , we measured field-dependent mag-netization at 0.45 K using a triply compensated extractionmagnetometer within a 65 T short-pulse magnet at the Na-tional High Magnetic Field Laboratory, Los Alamos. Asseen in Fig. 2(c), NaYbO saturates around 16 T. The mag-netization increases linearly in higher fields, and its slopecorresponds to the van Vleck term of χ vv = . µ B /T = . µ B /f.u. Magnetic susceptibility with χ vv subtracted fol-lows the Curie-Weiss law with the effective moment of µ eff = . ( ) µ B . Both values are comparable favorably to1.5 µ B /f.u. and 2.60 µ B expected from the powder-averaged¯ g = .
00 calculated based on the CEF fit.
Low-temperature thermodynamics.
The susceptibility fitalso yields the Curie-Weiss temperature Θ = − . ( ) K.Its negative value confirms antiferromagnetic interactions be-tween the Yb + pseudospins. The absolute value is abouttwo times larger than in YbMgGaO ( Θ k = − . Θ ⊥ = − . to 16 T in NaYbO . Thesimple estimate Θ = − J / J ≃ . .To probe low-temperature thermodynamics, we measuredspecific heat ( C p ) of NaYbO using the PPMS (Quantum De-sign) down to 0.4 K and a home-built dilution-fridge setupdown to 70 mK [33]. Magnetic contribution to the C p be-comes visible below 8 K and shows a broad maximum around1 K [33]. At even lower temperatures, magnetic contributiondecreases without showing any signatures of a magnetic tran-sition (see Fig. 2(d)). This suggests the absence of long-rangemagnetic order and the possibility of a spin-liquid state con-firmed by the muon spin rotation ( µ SR) measurement below.By taking NaLuO as the isostructural non-magnetic com-pound, we confirmed that lattice contribution to the C p is negligible below 2 K, whereas the nuclear contributionwas subtracted by systematic measurements in weak appliedfields [33]. The remaining, magnetic contribution C m ( T ) willusually take the exponential form, e − ∆ / T , or the form of apower law, T p , for gapped and gapless excitations, respec-tively. However, neither term accounts for our experimentaldata (Fig. 2(d)). We tentatively fit C m ( T ) with two powerlaws, aT p + bT q , where p ≃ . T contribu-tion of magnons in a long-range-ordered antiferromagnet, and q ≃ . [6]. At firstglance, a combination of the two different contributions couldbe seen as an effect of sample inhomogeneity, but a weak ap-plied field restores the simpler T . power-law behavior [33],suggesting an intrinsic nature of the two power laws observedin zero field. We conclude that the low-energy excitations inNaYbO are neither gapped nor magnon-like. They are alsodistinct from the low-energy excitations in YbMgGaO thatshowed [6] the robust T . power law characteristic of the”spinon metal” of a U(1) quantum spin liquid [39].Before going further, we note in passing that the magneti-zation curve of NaYbO measured at 0.45 K shows a plateaubetween 4 and 5 T at about one half of the saturation magneti-zation as shown in Fig. 2(c). Such a -plateau contrasts withthe -plateau typically observed in Heisenberg and XXZ tri-angular antiferromagnets [5] and may be indicative of a morecomplex interaction regime. Moreover, field-induced phasetransitions should occur in NaYbO , but they go beyond thescope of our present manuscript and will be addressed in fu-ture studies. µ SR data.
Our heat-capacity data exclude long-range mag-netic ordering above 70 mK, yet spin freezing would haveno immediate effect on the specific heat. A direct probe ofspin dynamics is thus essential to identify the QSL. To thisend, we carried out zero-field (ZF) and longitudinal-field (LF) µ SR measurement down to 100 mK at the MuSR spectrome-ter (ISIS).Four representative ZF µ SR spectra are shown in Fig. 3(a).They reveal neither oscillations nor a drastic drop in the ini-tial asymmetry, indicative of the long-range magnetic order,but instead show signatures of persistent spin dynamics, in
FIG. 3. (a) ZF µ SR spectra collected at various temperatures. Thesolid lines represent the fit using function (1). (b) Temperature de-pendence of the muon spin relaxation rate for the ZF µ SR spectra. particular the lack of polarization recovery to of the initialvalue rules out the presence of static random fields. We fur-ther analyze these data by fitting the muon spectra with twoexponential components: A ( t ) = A [ f exp ( − λ t ) + ( − f ) exp ( − λ t )] + B (1)where A and B denote the initial asymmetry and the con-stant background, respectively, λ and λ represent the muonspin relaxation rates for muons implanted at two sites nearO − , f stands for the fraction of the first component. The fit-ted ZF µ SR relaxation rate λ , λ and f as a function of tem-perature are shown in [33]. f shows temperature-independentbehavior with its value close to 0.5, suggesting the same pop-ulation at the two muon sites. Below we will discuss the elec-tronic dynamics in terms of λ since it is significantly largerthan λ .Temperature dependence of λ tracks the onset of correla-tions between the Yb + pseudospins. As seen in Fig. 3(b).Above 10 K, NaYbO is paramagnetic with a smaller andtemperature-independent λ . The increase in λ below 10 Kis accompanied by the growing magnetic contribution to thespecific heat, whereas the second temperature-independentregime below 2 K indicates the onset of the spin-liquid state.To prove the dynamic nature of the relaxation in this tem-perature range, we performed the LF experiment at 1.5 K.Should the relaxation arise from a weak static field, thesize of this field is B loc = λ / γ µ ≃ . γ µ = . × π s − µ T − is the gyromagnetic ratio for muons.Our LF data show that the relaxation persists in much higherfields, thus proving the dynamic nature of the Yb + pseu-dospins [33].We also note that the characteristic evolution of λ , its in-crease below 10 K and the saturation below 2 K, takes place atabout twice higher temperatures compared to YbMgGaO [8].This further supports our conclusion on the twice stronger ex-change couplings in NaYbO . Low-energy excitations.
The most interesting property ofa spin liquid is arguably its excitation spectrum. We probedthe low-energy excitations of NaYbO at the cold-neutronmulti-chopper LET spectrometer (ISIS) [40] at 45 mK usingincident neutron energies of 1.46, 3.7, and 7.52 meV. Spec-tral weight observed above 0.8 ˚A − is continuously distributed FIG. 4. (a) INS spectra measured at 45 mK with the incident energies E i =3.7 meV. (b) Energy dependence of the integrated cut along Q inthe range of 1.2-1.5 ˚A − . in both energy ( E ) and momentum ( Q ) and extends to about1 meV (see Fig. 4(a)). This low-energy spectral weight con-centrates around Q ≃ .
25 ˚A − corresponding to the K -pointof the Brillouin zone with the reciprocal-lattice vector ( , ,0). A 120 ◦ magnetic order will lead to a Bragg peak at thesame position, but the excitation spectrum is clearly differentfrom the spin-wave spectrum of such an ordered state [33].The energy dependence over the constant- Q cut suggests thatthe excitations reach low energies down to the elastic linethat becomes prominent below 0.1 meV. This further confirmsgapless nature of the magnetic excitations in NaYbO (seeFig. 4(b)).The spectrum of NaYbO bears close similarities to that ofYbMgGaO , where no spectral weight was observed at low Q , around the zone center. All the spectral weight is con-centrated in the vicinity of the zone boundary, in agreementwith our observation of the spectral weight above 0.8 ˚A − only. A further, and more subtle feature of YbMgGaO , isthe re-distribution of the spectral weight upon heating thatsignals the formation of distinct excitations at energies aboveand below ¯ J [11]. No such effect is seen in NaYbO . More-over, magnetic excitations in YbMgGaO extend to the muchhigher energy of 2 meV, despite the fact that ¯ J is twice smallerthan in NaYbO . Discussion.
NaYbO shows strong similarities to thewidely studied triangular spin-liquid candidate YbMgGaO except for the absence of structural disorder. Both materi-als entail the trigonally distorted YbO octahedra. The CEFexcitations of Yb + occur at about the same energies, and thecompositions of the ground-state Kramers doublets are similartoo. Exchange couplings differ by a factor of 2, though. Thischange is accompanied by a reduction in the Yb–O–Yb bridg-ing angle from 99 . ◦ in YbMgGaO to 95 . ◦ in NaYbO .The similarity between the two materials gives us an in-teresting opportunity to explore which of the effects reportedfor YbMgGaO appear in the disorder-free case. Gaplessground state is retained in NaYbO . On the other hand, nei-ther the simple power-law behavior of the low-temperaturespecific heat [6], nor the energy separation of the spin exci-tations [11] have been observed. Moreover, the spin excita-tions of NaYbO extend to much lower energies despite thetwice larger ¯ J . This goes in line with the theory of Ref. 12that explains both effects in terms of quenched disorder in a valence-bond solid. These effects are probably not generic tothe spin-liquid state of triangular antiferromagnets.Rau and Gingras [41] computed anisotropic nearest-neighbor exchange couplings for several triangular geome-tries inspired by the possible local structures of YbMgGaO .By extrapolating their results to the Yb–O–Yb angle of 95 . ◦ in NaYbO , we may expect that at least the nearest-neighborcoupling ¯ J in this material is very close to the Heisenberglimit. The second-neighbor interaction ¯ J should be then op-erative in order to stabilize a QSL. Such a scenario may beenvisaged if, for example, ¯ J is less sensitive to the struc-tural geometry than ¯ J . With ¯ J / ¯ J = . ( ) reported forYbMgGaO [18], the increase in ¯ J accompanied by a smallchange in ¯ J will drive the system directly into the QSLphase [42]. Another interesting observation is that static struc-ture factor calculated for this QSL phase peaks at the K -pointof the Brillouin zone [28] in agreement with the accumulationof the low-energy spectral weight at Q ≃ .
25 ˚A − observedin our experiment (see Fig. 4), whereas YbMgGaO showsa larger low-energy spectral weight at the M -point with thereciprocal-lattice vector ( , 0, 0) [14]. All these argumentsgive a strong envision that the spin-liquid state observed inNaYbO is the QSL phase of triangular antiferromagnets. Itsgapless nature and unusual sensitivity to the magnetic fieldopen prospects for future studies theoretically and experimen-tally. Conclusions.
The disorder-free NaYbO gives the most di-rect experimental access to the spin-liquid physics of trian-gular antiferromagnets. Thermodynamic measurements andmuon spectroscopy indicate the absence of magnetic orderand persistent spin dynamics down to at least 70 mK. An ex-citation continuum is observed, with the spectral weight accu-mulating around the K -point, as expected in the QSL phase(s)driven by the exchange anisotropy or second-neighbor cou-pling on the triangular lattice [28]. The spin-liquid state ofNaYbO is gapless with a non-trivial low-temperature evo-lution of the specific heat, which does not follow the spinonscenario originally proposed for YbMgGaO .LD acknowledges support from the Rutherford Inter-national Fellowship Programme (RIFP). This project hasreceived funding from the European Union’s Horizon2020 research and innovation programme under the MarieSkłodowska-Curie grant agreements No.665593 awarded tothe Science and Technology Facilities Council. 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