Introduction

The Kitaev model, describing a spin-1/2 two-dimensional (2D) honeycomb lattice as an exactly solvable 2D spin model that achieves a quantum-spin-liquid (QSL) ground state1, has attracted considerable attention. Numerous theoretical and experimental attempts have been made in search for a QSL candidate with dominant bond-dependent anisotropic-exchange interactions K, namely, the Kitaev interactions2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19. To date, the Kitaev model has been successfully realized in several 3d, 4d, and 5d transition-metal families8,9,10,11,12,13,18,19,20,21,22. Unfortunately, owing to the ubiquitous presence of non-Kitaev interactions6,7,13,15, e.g., Heisenberg exchanges, or off-diagonal symmetric interactions Γ and Γ′, at low temperatures and in the absence of magnetic fields, most of these materials fail to realize a QSL state and exhibit instead a zigzag antiferromagnetic (AFM) order6,7,8,9,10,11,12,13,18,19,20,21,22.

Compared to the previously discovered 4d/5d Ru/Ir systems, characterized by a strong spin-orbit coupling (SOC)15,16,17,19, the 3d Co-based Kitaev QSL candidate materials remain highly controversial12,13,20,23,24,25,26,27,28,29. A good example of such materials is Na2Co2TeO6 (NCTO), originally thought as one of the most promising cases for studying Kitaev physics12,13,15,20,29,30,31,32,33,34,35,36. Although the evidence of a field-induced QSL is available in NCTO13,20,37,38, the magnetic structure of its ground state remains an open and intriguing question13,20,27,28,32,34,39, whose answer should help to understand the role of the competitors of K, such as the nearest-neighbor (NN) Heisenberg coupling J1 and the third NN Heisenberg coupling J3.

Recently, by using various experimental methods, such as single-crystal or powder neutron diffraction, nuclear magnetic resonance, and electrical polarization measurements, researchers have attempted to understand the magnetic structure of NCTO27,28,30,31,32,40,41, yet an undisputed conclusion has not been reached. i) Powder neutron diffraction experiments suggest a zigzag AFM order, with the magnetic moments lying in the ab plane [see Fig. 1a], accompanied by a Néel-type canting along the c axis below TN ~ 26 K40,41. However, single-crystal neutron diffraction suggests a triple-q order (or multi-k structure) in the absence of a magnetic field12,28,29,32. ii) Although the field dependence of the characteristic magnetic reflections (0.5, 0, 1) and (0, 0.5, 1) demonstrates the magnetic multi-domain effect of the zigzag AFM order28, the temperature dependence of the (0.5, 0, 0) magnetic reflection cannot rule out a triple-q order12,28,32. iii) The spin dynamics were analyzed by a generalized Heisenberg–Kitaev model with five symmetry-allowed terms: K, Γ and Γ′, J1, and J313,34,35,36,42, and two different exchange frustrations: Kitaev and J1-J3. However, such a complex model made the determination of the ground state rather difficult. To distinguish the two possible magnetic structures, i.e., multi-domain or multi-k, a good strategy consists in measuring the field- and temperature-dependent magnetic reflections and the related dynamics at the M-points lying in the ab-plane, such as M (0.5, 0, L), M1 (−0.5, 0.5, L), and M2(0, 0.5, L), where L is an arbitrary integer [see Fig. 1b].

Fig. 1: The honeycomb structure and single-crystal neutron diffraction for NCTO.
figure 1

a Magnetic structure of NCTO. b Schematic plots of the Brillouin zones showing the in-plane high symmetry points M, M1, M2, and K denoted by black, red, green, and purple dots, respectively. The high symmetry points are identical for L with different integers along the c-axis. c The temperature dependence of the zero-field intensity of two magnetic reflections at M (0.5, 0, 1) and M1 (−0.5, 0.5, 1). The solid lines represent the fits to a phenomenological model (see details in the text). d Field-dependent intensity of magnetic reflections at M and M1 points, collected at T = 2 K and with the magnetic field applied along the [−1, 1, 0] direction. Dashed lines are guides to the eyes. The error bars are 95% confidence intervals of the Gaussian fit.

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In this paper, we investigate the magnetic order and dynamics of NCTO single crystals via neutron scattering and muon-spin rotation and relaxation (μSR) techniques. Changes of the characteristic magnetic reflections of M (0.5, 0, 1) and M1 (−0.5, 0.5, 1) with temperature and magnetic field indicate that the magnetic ground state has a multi-domain structure, rather than a multi-k structure. Below TN, the modulation of the magnetic domains induces two other transitions at TF and T*, reflected also by the temperature-dependent weak transverse-field (wTF) μSR asymmetry. Furthermore, the estimated static magnetic volume fraction (~90%) confirms the homogeneous nature of the ordering of NCTO single crystals. The relatively large and temperature-independent muon-spin relaxation rate in the AFM state suggests the presence of strong quantum spin fluctuations in NCTO.

Results and discussion

Single-crystal neutron diffraction

To determine whether the magnetic ground state of NCTO is a multi-domain structure or a multi-k structure, both the temperature- and magnetic-field dependence of the magnetic reflections M (0.5, 0, 1) and M1 (−0.5, 0.5, 1) were measured for the wave vectors k = (0.5, 0, 0) and (−0.5, 0.5, 0), respectively [see Fig. 1c, d]. The integrated intensity was calculated by fitting the raw data with a Gaussian function, as illustrated in Fig. S1 of the Supplementary materials. The differences in the non-normalized scattering intensity at the M and M1 points may be due to the non-spherical sample morphology (see sample photo in Fig. S1 of Supplemental materials). The temperature-dependent intensity curves I(T) at both M and M1 points can be well fitted using a phenomenological model: I(T) = I0·[1-(T/TN)]2β43, yielding a magnetic ordering temperature TN ≈ 26 K and β ≈ 0.22. The I(T) curves for both M and M1 show similar behaviors, but a small bump near TF ≈ 15 K only appears for the M1 point, implying that the anomaly at TF may not be attributed to a magnetic phase transition. Interestingly, the spin-wave-excitation gap at the M-point which is ~1 meV vanishes at a temperature close to TF = 15 K ~ 1.3 meV32. Furthermore, the appearance of spin-wave-excitation gap can be attributed to the SOC inducing magnetic anisotropic-exchange interactions K and Γ terms in NCTO13. Therefore, the weak anomaly at TF can be interpreted as the modulation of anisotropy. Apparently, the M1 point is responsive to variations in anisotropy, whereas the M point is not. This implies that the ground-state magnetic structure is not a multi-k structure, as such a structure would require the various arms of the k vector to respond uniformly to the external perturbations44,45,46,47,48. Hence, the subtle differences between these two magnetic reflections indicate that the magnetic ground state is a multi-domain structure, that is, the domains corresponding to the k vectors of (0.5, 0, 0) and (−0.5, 0.5, 0) are not equally populated. In addition, the estimated β ≈ 0.22 is neither consistent with the ideal 2D Ising (β = 0.125) nor the 3D Ising (β = 0.326) system49, but it is more consistent with quasi-2D magnetic correlations13,20,32.

As shown in Fig. 1d, the multi-domain structure is further supported by the field-dependent intensity I(B) at the magnetic reflections M and M1 for B // [−1, 1, 0]. The M and M1 reflections exhibit completely opposite field dependence. When increasing the magnetic field, the intensity of the M reflection decreases, while that of M1 increases. Such opposite field dependence implies that the macroscopic symmetry is broken by the applied magnetic field and, thus, that the magnetic domains along the field direction gradually grow, while the domains in other directions are suppressed44. With the application of an external constraint (magnetic field or uniaxial stress), each arm of the k vector would exhibit a different response in the case of a multi-domain scenario, while they would show similar behavior in the case of a multi-k scheme45,46,47,48. Clearly, our neutron data support a multi-domain structure rather than a multi-k structure in NCTO single crystal. It is worth mentioning that, in the zigzag AFM state, NCTO exhibits a twofold symmetry in the angular dependence of the magnetic torque20, which might be related to the different I(B) field responses at the magnetic reflections M and M1.

Inelastic neutron scattering

Another feature of quantum fluctuations in certain strong quantum magnets is a broadening of the spin-wave-excitation spectra50,51,52. To search for the quantum fluctuations in an external magnetic field, we performed single-crystal inelastic-neutron-scattering measurements by applying various magnetic fields up to 4 T along the [−1,1,0] direction. As shown in Fig. 2a, b, a spin-wave band can be identified at 0 and 4 T along the high symmetry momentum directions Γ-K-M (see the arrows in the inset of Fig. 2). As expected, in the zero-field case, the gapped magnon band at the M-point reaches the minimum energy value and has the largest intensity, thus supporting a zigzag AFM order driven by the non-Kitaev interactions12,20. The spin-wave band at 4 T shows similar features to the lowest-energy spin-wave-excitation spectra at 0 T [see Fig. 2b], but it is significantly broadened by the external field. Note that, in NCTO, we do not observe a splitting of the spin-wave band into multiple branches under an applied magnetic field. Hence, such broadening is most likely caused by the field-induced quantum fluctuations, which can be further evidenced by constant-Q scans at the K-point. As shown in Fig. 2c, at 0 T, the excitation gap is centered around 3 meV12,20. Upon increasing the magnetic field to 4 T, the zigzag AFM order is still present [see I(B) curves in Fig. 1d], but the excitation broadens continuously. The full width at half maximum (FWHM) of the excitation vs the magnetic field is shown in the inset of Fig. 2c. These results indicate that quantum fluctuations in NCTO are enhanced by an applied magnetic field. This is highly consistent with the field-induced magnetically disordered state with strong quantum fluctuations observed between 7.5 and 10 T with B // a*-axis and such strong quantum fluctuations are more intuitively reflected in our μSR data13,20,38.

Fig. 2: Single-crystal inelastic neutron scattering results for NCTO.
figure 2

a, b Lowest-energy spin-wave-excitation spectra at 2 K in the magnetic fields of 0 T and 4 T, respectively. The color bars indicate the scattering intensity in a linear scale. The inset shows the elastic neutron scattering plot which is integrated at the elastic location 00 L = [–2.5, 2.5] r.l.u. and E = [–0.075, 0.075] meV at 0 T using a fixed incident energy Ei = 60 meV. The white dash-dotted lines represent the Brillouin zone boundaries. The high symmetry points Γ, K, M, M1 and M2 are marked, and the white arrows show the high symmetry momentum directions Γ-K-M path in the inset. c The constant-Q scans of the K-point (1/3, 1/3, 0) collected at T = 1.5 K by applying magnetic fields up to 4 T along the [−1,1,0] direction. The solid lines represent the fits using a Gaussian function. The error bars are the standard deviation of the inelastic neutron scattering experimental data. The inset shows the field dependence of the FWHM of the excitation gap.

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Muon-spin rotation and relaxation

As an extremely sensitive probe of complex quantum magnetism, μSR is regularly used to investigate the magnetic order and dynamics at a microscopic level. Here, the combination of a long-range (neutron scattering) and a short-range (μSR) technique helped us to confirm the multi-domain magnetic structure and the quantum fluctuations in a NCTO single crystal.

The analysis of wTF-μSR spectra allowed us to establish the temperature evolution of the magnetic volume fraction and to determine the magnetic transition temperatures. Here, an external field of B = 2.3 mT, applied perpendicular to the initial muon-spin direction, leads to a precession of the muon spins with a frequency γµB (where γµ/2π = 135.53 MHz/T is the muon’s gyromagnetic ratio), as shown in Fig. 3a. Note that a field of 2.3 mT is much smaller than the internal fields created by the long-range magnetically ordered state in the NCTO single crystal (see Fig. 4). Therefore, the muon-spin precession reflects only the non-magnetic part of the sample, since in its long-range magnetically order state NCTO exhibits a very fast muon-spin depolarization, here, in less than a few tenths of μs. Consequently, without considering this very fast relaxation, the wTF-μSR spectra can be described by the function:

$${{{{\rm{A}}}}}_{wTF}(t)={A}_{NM}\,\cos ({\gamma }_{\mu }{B}_{{{\mathrm{int}}}}\cdot t+\varphi ){e}^{-\lambda t},$$
(1)

where ANM is the initial muon-spin asymmetry reflecting the muons implanted in the nonmagnetic (NM) or paramagnetic (PM) fraction of NCTO single crystal; Bint is the local field sensed by muons (here almost identical to the applied magnetic field); φ is an initial phase and λ is the muon-spin relaxation rate.

Fig. 3: Magnetic phase transition and magnetic volume fraction of NCTO, as measured by wTF-μSR.
figure 3

a Time-domain wTF-μSR spectra collected at different temperatures in a weak transverse field of 2.3 mT, acquired at the M20D beamline. The error bars are the standard error of the mean. Solid lines are the fit results utilizing Eq. (1). To highlight the low-amplitude oscillations below TN, the respective time-domain wTF-μSR spectra are also plotted in Fig. S7 of supplemental materials. b The temperature-dependent ANM asymmetry (left axis and red symbols) was obtained from fits of the wTF-μSR data. The error bars represent the standard deviation of the fit parameters. For a comparison, we present the zero-field cooling (ZFC) magnetic susceptibility curve χ(T) measured at B = 0.01 T (right axis and blue symbols) with B // a*-axis. The magnetic susceptibility data were taken from ref. 20. c The estimated magnetic volume fraction versus temperature. The red dashed lines in (b) and (c) are guides for the eyes.

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Fig. 4: Zero-field μSR spectra of NCTO.
figure 4

a, b Zero-field (ZF) μSR time spectra measured at selected temperatures (displaced vertically for clarity) with the muon-spin direction Sμ // a and c-axis, respectively. Data were taken on the M20D channel at TRIUMF. The error bars are the standard error of the mean. The solid curves represent fits to Eq. (2). c–f Temperature dependence of the Bint, λT, λL, and λtail, respectively, as derived from the analysis of ZF-μSR. The error bars represent the standard deviation of the fit parameters. The magnetic ordering temperature is TN ≈ 26 K and is consistent with magnetic susceptibility data. Solid lines in panels (c) are fits using a phenomenological equation described in the text; dash-dotted lines in panels (d–f) are guides to the eyes.

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Figure 3b presents the temperature dependence of the asymmetry for the wTF-μSR spectra. Below TN, a static spin component leads to a fast reduction of asymmetry. Hence, ANM starts to decrease quickly as one approaches the AFM ordering temperature, consistently with the magnetic susceptibility data. Another broad and weak peak was observed in the wTF-5mT data at temperatures close to TF during an independent experiment (see Fig. S3 of Supplemental materials). Although NCTO undergoes three successive AFM transitions [indicated by dashed lines in Fig. 3b], the ANM(T) curve does not capture the possible phase transition at T*13. In the case of a fully magnetically ordered material, the magnetic volume fraction Vmag at temperatures below the magnetic ordering temperature is close to 100%53,54. Here, the temperature dependence of the magnetic volume fraction was estimated from Vmag(T) = 1 − ANM(T)/ANM (T > TN) [see Fig. 3c]. In our case, the static magnetic volume fraction is up to 90% and possible spin-glass behavior, typically revealed by ac susceptibility measurements, is absent in our NCTO single crystal27. These results indicate that, below TN, NCTO can be considered as fully magnetically ordered, ruling out heterogeneous behavior.

If the electronic magnetic moments fluctuate very fast (typically above 1012 Hz in the PM state), the muon-spin polarization would not be influenced. When the system starts to enter the magnetically ordered state, such fluctuations slow down significantly. This scenario gives rise to a fast depolarization with superimposed oscillations, reflected in the appearance of static magnetic moments and the coherent precession of the muon-spin polarization, respectively. Figure 4a, b shows the ZF-μSR time spectra for both Sμ // a and Sμ // c-axis (see also Fig. S4 of Supplementary materials). Such fast depolarization (within 50 ns) with superimposed oscillations below TN, has been observed in other antiferromagnets with strong frustration54,55, thus suggesting that the exotic frustrated state in NCTO might originate from the Kitaev-type frustration13,34,35,36,42. To track the evolution of the muon asymmetry across the whole temperature range, the ZF-µSR spectra were modeled by:

$${{{{\rm{A}}}}}_{ZF}(t)= {A}_{1}\cdot [\alpha \,\cos ({\gamma }_{\mu }{B}_{{{\mathrm{int}}}}t+\varphi )\cdot {e}^{-{\lambda }_{T}t}+(1-\alpha )\cdot {e}^{-{\lambda }_{L}t}] \\ +{A}_{2}{e}^{-{\lambda }_{tail}t}\cdot {G}_{KT}.$$
(2)

Here, α and 1 − α are the oscillating (i.e., transverse) and non-oscillating (i.e., longitudinal) fractions of the µSR signal, λT and λL represent the transverse and longitudinal muon-spin relaxation rates, A1 and A2 represent the asymmetries of two nonequivalent muon-stopping sites. The muons stopping at the second site do not undergo any precession but show only a weak relaxation, which can be described by an exponential relaxation λtail. The ZF-µSR spectra show only a weak relaxation at temperatures above TN. Here, GKT is the static Kubo-Toyabe relaxation function, normally used to describe the muon-spin relaxation due to the nuclear moments. In the magnetically ordered state, the muon-spin relaxation due to the electronic moments is significantly larger than the contribution from the nuclear moments. Therefore, below TN, the GKT term can be safely ignored. In our analysis, we set GKT to 1 at temperatures below TN (see details in Fig. S5 of Supplemental materials).

Usually, changes in the magnitude of the magnetic moment can be detected by the temperature-dependent internal magnetic field Bint(T). However, the modulation of magnetic domains is related to the distribution of internal magnetic fields. The fit parameters of the ZF-µSR spectra, here summarized in Fig. 4c–f, help us to distinguish the origin of these complex magnetic orders. It is worth mentioning that, as the temperature changes, α was allowed to vary, to ensure a more reasonable parameter set below TN. Indeed, the complex competition between the two different exchange frustrations of Kitaev- and J1-J3-type13,34,35,36,42, can trigger significant spin fluctuations accompanied by changes in spin directions, as shown in Fig. S6c of Supplemental materials. The temperature-dependent parameter α shows weak anomalies near TF, most likely indicating that the modulation of the magnetic domains and the change of electronic moment directions are moderate.

The Bint(T) curves for Sμ // a and Sμ // c exhibit similar features [see Fig. 4c], reflecting the local ordered magnetic moment of the Co2+ ions and provide one distinct phase transition temperature TN. Both curves are consistent with the temperature evolution of the magnetic moments obtained from powder40 and single-crystal neutron diffractions (see details in Fig. 1). A phenomenological equation \({{{{\rm{B}}}}}_{{{\mathrm{int}}}}(T)={B}_{{{\mathrm{int}}}}(T=0)\cdot {[1-{(T/{T}_{N})}^{\gamma }]}^{\delta }\) describes very well the Bint(T) curves shown in Fig. 4c. Here, Bint(T = 0) is the internal magnetic field at 0 K, γ and δ are two empirical parameters. The fitted parameters are summarized in Table SI in the Supplementary materials.

Interestingly, a tiny anomaly was observed at TF in the temperature-dependent λT(T) curve with Sμ // a. We recall that the decay rate λT reflects the width of the static magnetic field distribution at the muon-stopping site [see Fig. 4d]. The observed anomalies at TF are consistent with previous magnetic susceptibility and neutron scattering results, and this might suggest a possibility of redistribution of the magnetic domains originating from three different k-vectors of the zigzag AFM order. Hence, the multi-domain structure is compatible with both the ZF-µSR results and the single-crystal neutron diffraction data.

Spin fluctuations can be traced by the longitudinal relaxation rate λL. As shown in Fig. 4e, both λL(T) curves diverge near TN, but also drop significantly below TN, indicating that spin fluctuations are the strongest close to the onset of the AFM order. Such strong spin fluctuations were further confirmed by the longitudinal-field (LF) µSR measurements [see Fig. S8 of Supplemental materials]. In NCTO, λL becomes approximately constant (0.45 μs-1) at low temperatures below TN. In YbMgGaO4, another promising QSL candidate material, similar T-independent behavior, with a low-temperature value of 0.3 μs-1, has been suggested to reflect its very strong quantum fluctuations56. Hence, the temperature independence of λL might also indicate strong quantum fluctuations in NCTO as thermal fluctuations are almost absent at extremely low temperatures. Incidentally, below TN, the temperature-dependent λtail(T) is nearly zero [see Fig. 4f], here reflecting a simple exponential correction originating from the low background.

We now discuss the zigzag AFM ground state with a multi-domain structure, accompanied by strong quantum fluctuations in NCTO. By using the same fitted parameters, TN ≈ 26 K and β ≈ 0.22, the power-law function I(T) = I0·[1-(T/TN)]2β gives similar I(T) curves at the M and M1 points [see Fig. 1c]. However, the small bump at TF, corresponding to an opening of the spin-wave-excitation gap32, was observed only in the I(T) curve measured at the M1 point. The absence of anomaly at TF in the I(T) curve of the M point is consistent with the scenario of multi-domain structure, wherein distinct domains exhibit different behaviors. The field-dependent intensities of the magnetic reflections further confirm that the magnetic domains oriented along the magnetic field direction grow, while those along other directions are suppressed. At the same time, the subtle feature observed in the λT(T) curves is also consistent with previous results and might indicate a domain reorientation occurring at TF, coinciding with the multi-domain scenario. Such consistent experimental observations support a multi-domain structure rather than a multi-k structure.

Based on the generalized Heisenberg–Kitaev model13,20,29,34,35,36,42, many previous studies have revealed a complex competition between the two types of exchange frustrations: Kitaev and J1-J3, which may lead to strong quantum fluctuations in the ground state of NCTO. Such quantum fluctuations are clearly evidenced by our ZF-μSR measurements: (1) NCTO exhibits a very fast depolarization (within 50 ns), typical of strongly frustrated antiferromagnets54,55 [Fig. 4a, b]; (2) The temperature independence of λL, here close to 0.45 μs−1 [Fig. 4e] at low temperatures, is similar to that observed in FeTe2O5Br, where strong spin fluctuations coexist with a fully magnetically ordered state57. Taken together, our µSR data indicate that the spin dynamics at low temperatures is strongly influenced by quantum effects57, with the opening of a spin-wave-excitation gap below TF further eliminating the contribution of thermal fluctuations32. The zero-field quantum fluctuations inevitably remind us of the recently reported field-induced Kitaev QSL between 7.5 T and 10 T in NCTO13,20,27,38. Further, the broadening at the K-point, where the FWHM gradually increases in a magnetic field [see inset in Fig. 2c], suggests that quantum fluctuations are enhanced by the applied magnetic fields. These results indicate that the applied magnetic fields could quickly suppress the magnetically ordered states and highlight the contribution of Kitaev interactions to quantum fluctuations.

Conclusions

In summary, we investigated both the magnetic structure and the ground-state dynamics in NCTO. Our most significant finding is that NCTO hosts a multi-domain zigzag AFM order with strong quantum fluctuations. Our results provide experimental evidence in favor of the Heisenberg–Kitaev model, where the coexistence of static magnetic order (from the non-Kitaev interactions) with dynamic quantum fluctuations (from the frustrated K term) suggests a highly frustrated magnetic structure.

Note added. While preparing the present manuscript, we noticed that µSR experiments, with a lower early-time resolution (here up to 50 ns), were presented independently58.

Methods

Sample preparation and characterization

The measurements were performed on high-quality single crystals grown by the flux method, as described elsewhere in ref. 20.

Single-crystal neutron diffraction and inelastic neutron scattering

A piece of NCTO single crystal was used for the neutron diffraction experiments [see the inset in Fig. S1a of Supplemental materials], performed at the thermal single-crystal diffractometer ZEBRA at the Swiss Spallation Neutron Source SINQ, Paul Scherrer Institut (PSI), Switzerland. A neutron wavelength of 1.383 Å, obtained by a Ge monochromator, was used for all the measurements at ZEBRA. In-field data were collected using a lifting arm normal-beam geometry, where the crystal was inserted in a 10-T vertical magnet. The magnetic states of NCTO were investigated with B // [−1 1 0] (equivalent to a*-axis). Spin-wave excitation spectra were measured using the SEQUOIA time-of-flight spectrometer at the Spallation Neutron Source, Oak Ridge National Laboratory, USA59,60. Measurements at 2 K with applied field B = 0 T and 4 T were performed by rotating the sample around the vertical axis (sample a*-axis) in steps of 1° with Ei = 18 meV with high energy resolution (∆E) of 0.41 meV (full width at half maximum of the elastic peak). In order to subtract the background, the INS data were collected at 90 K. The constant-Q scans of the K-point (1/3, 1/3, 0) at 1.5 K with applied field B up to 4 T and B // [-1,1,0] direction were measured on the Cold Triple Axis Spectrometer SIKA at the ANSTO, Australia61. Data on SIKA were collected using a fixed final-energy mode with Ef = 5.0 meV, achieving a resolution of 0.13 meV.

μSR

The ZF-, LF-, and wTF-μSR experiments were performed on the M20D surface muon beamline at TRIUMF in Vancouver, Canada using the LAMPF spectrometer. In addition, wTF-μSR experiments were also carried out at the general-purpose surface-muon (GPS) instrument at the πM3 beamline of the Swiss muon source (SμS) at PSI in Villigen, Switzerland. For the experiments on the M20D surface muon beamline at TRIUMF, four layers of aligned NCTO crystals were positioned on aluminum backed mylar tape, Fig. S2a, b of Supplemental materials, with their c axis parallel to the incident muon spin direction. We aimed to study the temperature evolution of the magnetically ordered phase and the dynamics of spin fluctuations. For the experiments on the GPS instrument, since NCTO is thin and flat, we simply used two layers of crystals and then wrapped them with Kapton foil, Fig. S2c, d of Supplemental materials. The incident muon spin direction was always parallel to the c-axis of the crystal. The muon spin could be rotated into the ab-plane (TRAN mode) or be left along the c-axis (LONG mode). Based on the wTF-μSR data taken at TRIUMF, we could determine the temperature evolution of the magnetic volume fraction. For the wTF-μSR measurements, the applied magnetic field was perpendicular to the incident muon spin direction. All the μSR spectra were analyzed using the musrfit software package62.