## Introduction

Electronic nematic order refers to a state in which low energy electronic degrees of freedom drive a crystal to spontaneously break discrete rotational symmetry without simultaneously breaking translational symmetry1. Much of the reason for the current interest in such states derives from observations that unconventional superconductivity tends to emerge in materials exhibiting competing ground states with different broken symmetries, including nematic, charge, and/or spin density wave fluctuations2,3,4,5,6. However, disentangling the effects of quantum critical nematic fluctuations from other competing order parameter fluctuations presents a significant challenge. Moreover, the emergence of superconductivity preempts the low temperature behavior where a quantum phase transition may be present. Hence there is an interest to investigate electronic nematicity on its own, without the presence of competing phases.

Ferroquadrupolar order of atomic orbitals is an important realization of electronic nematicity, in which electronic orbitals spontaneously develop quadrupole moments oriented in the same direction7,8. Coupling between the quadrupolar moments and the lattice gives rise to an effective interaction between the atomic orbitals, and leads to a cooperative Jahn-Teller distortion at a temperature, TQ9. In many cases ferroquadrupolar order develops in insulators, without the presence of competing broken symmetry phases. TmVO4 is an ideal example of such a material, which has been well characterized10,11. The Tm ions (4f12 with L = 5, S = 1, J = 6) in this material have partially filled 4f shells that are split by a tetragonal crystal field. The ground state doublet is well separated by a gap of ~77 K to the lowest excited state12. Importantly, the ground state is a non-Kramers doublet, such that the first order Zeeman interaction vanishes for in-plane fields (i.e., gc ~ 10 while ga = gb = 0). The ground state can be well-described by a pseudospin ($$\tilde{S}=1/2$$), in which one component, $${\tilde{S}}_{z}$$, corresponds to a magnetic dipole moment oriented along the c-axis, while the other two components $${\tilde{S}}_{x}$$ and $${\tilde{S}}_{y}$$ correspond to electric quadrupole moments with B2g (xy) and B1g (x2 − y2) symmetry respectively13. Quadrupole interactions dominate, and the material does not order magnetically. The two quadrupole moments couple bilinearly to objects that transform with the same symmetry, and hence such objects can act as effective fields for the quadrupole order. This could be strains εxx − εyy, εxy or it could be composite magnetic field variables that transform in the same way, such as $${B}_{x}^{2}-{B}_{y}^{2}$$ and BxBy, where x and y lie along the axes of the undistorted tetragonal unit cell (see Supplementary Information). Hence, although in-plane magnetic fields do not couple linearly to the quadrupole moments, they do couple quadratically, and (appropriately constructed objects that go as the square of the fields) can act as effective longitudinal and transverse fields for a quadrupole ordered state, respectively. In TmVO4, the quadrupole moment of the doublet couples to the lattice strain, εxy, and the material spontaneously undergoes a tetragonal to orthorhombic distortion with B2g symmetry below TQ = 2.15 K with orthorhombicity δ ≈ 0.01. The low energy degrees of freedom can be described well by the transverse field Ising model, in which Ising interactions couple the pseudospins which represent the quadrupolar moments and a magnetic field along the c-axis couples transverse to the quadrupolar direction7. This field enhances the fluctuations of the pseudospins and can tune the system to an Ising-nematic quantum phase transition at a critical field Bc ≈ 0.5 T11. This material thus offers an important platform to investigate quantum critical nematic fluctuations in an insulator.

The critical behavior of the ferroquadrupolar order has been well documented for fields along c-direction, but little is known about the behavior for in-plane magnetic fields. Although the in-plane Zeeman interaction vanishes to first order, the second order effect can become important for sufficiently high magnetic fields14. Nuclear Magnetic Resonance (NMR) is a technique that can probe nematicity locally through the magnetic and electronic charge environment to which the NMR-active nucleus couples through its spin and quadrupole moment, respectively. Quadrupolar fluctuations are a measure of the nematic susceptibility and contribute to the spin–lattice relaxation rate $${T}_{1}^{-1}$$, but have to be neglected usually when magnetic fluctuations dominate the relaxation. Recovering the quadrupolar relaxation is of particular interest in measurements of the iron-based superconductors, as their putative nematic quantum critical point appears to be correlated with optimal superconductivity and non-Fermi-liquid behavior in the normal state, indicating a possible role for nematic fluctuations in the pairing interaction15,16,17,18.

Previous NMR measurements of TmVO4 identified a scaling between the spin–lattice relaxation rate and the shear elastic stiffness constant, c66, suggesting that the 51V (I = 7/2) nuclear spins couple to the Tm orbitals through the electric field gradient (EFG), giving rise to a quadrupolar relaxation channel19. However, the spectra were significantly broadened by inhomogeneous demagnetization fields and the anisotropic g-factor of the Tm ground state doublet. In order to better discern the spectra and relaxation mechanisms at play, we utilized a focused ion beam (FIB) to shape a single crystal of TmVO4 in an ellipsoidal shape, with a homogeneous demagnetization field. This innovation provides enhanced spectral resolution that enables detailed in-plane angular-dependent studies. We find that the EFG asymmetry parameter, η, which is a direct measure of the ferroquadrupolar order parameter, depends sensitively on the direction of the in-plane magnetic field. We interpret this behavior as a consequence of a second order Zeeman effect on the non-Kramers doublets, and demonstrate how an in-plane magnetic field, depending on the field angle, can act as either an effective longitudinal or transverse field for ferroquadrupole order to selectively enhance or suppress nematicity in TmVO4. A further consequence of this innovation is that we have been able to disentangle the magnetic and quadrupolar contributions to the V spin–lattice relaxation. We find these rates exhibit different temperature dependencies, and depend not only on the orientation of the applied magnetic field in the ab plane, but also vary strongly with applied field strength. Surprisingly, we find that for transverse in-plane fields, TQ is enhanced for applied fields 0≤B 5 T. This unusual behavior may reflect the influence of the excited crystal field states. For fields greater than ~5 T, the phase transition appears to exhibit a crossover rather than suppress to zero. We speculate that this behavior is related to the dynamics of isolated two-level systems once the magnetic field is sufficiently large to split the Tm ground state doublets. These unexpected observations for in-plane fields reveal rich new physics driven by nonlinear Zeeman interactions in this model system.

## Results

### Electric field gradient

51V NMR spectra and relaxation rate measurements were conducted in a single crystal shaped by a plasma focused ion beam (FIB) as a function of temperature and magnetic field oriented in the plane perpendicular to the c-axis (see Methods). Figure 1a compares spectra of an as-grown single crystal with that from a FIB-shaped crystal, and Fig. 1b displays spectra as a function of in-plane field direction below TQ at θ = 90 at 1.8 K and 2.7 T. There are seven transitions at frequencies given by νn = γB0(1 + K) + nνq, where γ = 11.193 MHz/T is the gyromagnetic ratio, K is the magnetic shift (≈−0.4%), n = −3,  ,+3, and νq is the quadrupolar shift that arises due to the electric field gradient (EFG) tensor. The ϕ dependence of νq is determined by the EFG asymmetry parameter, ηεxy, where εxy is the B2g strain in the ordered state (see Methods). As seen in Fig. 1c, νq exhibits four-fold, rather than twofold, symmetry at 1.8 K, in contrast to the expectation. Moreover, the spectra in Fig. 1 reveal only one set of peaks below 2 K, suggesting that there is a single nematic domain. These observations suggest the presence of a magnetoelastic coupling such that B0 detwins the ferroquadrupolar order causing η to depend on the field direction. A similar phenomenon affects the nematicity in the iron-based superconductors20. If we assume $$\eta ={\eta }_{0}\cos (2\phi )$$, then we are able to model the data in Fig. 1b with η0 = 1.44%, as shown in Fig. 1d and illustrated in Fig. 2a. This behavior suggests the Tm quadrupole moments rotate with B0, such that domains with positive shear strain are always lower in energy. The fact that η(ϕ) changes sign implies that the principal axes of the EFG switches, however we assign a fixed axis as the evolution of νq(ϕ) is captured well by a smoothly varying asymmetry parameter, implying weak pinning of nematic domains. At low fields and temperatures we are able to accurately fit the full spectra for all orientations with two magnetic sites but equal EFG parameters. In principle, we might expect two sites with ± η corresponding to different nematic domains, at least for some angles. However, even if there were two sites with different η in the nematic phase in zero field, the magnetoelastic coupling lowers the free energy of one domain over the other, effectively giving a single EFG environment for ϕ ≠ 45. Moreover, νq is independent of η for ϕ = 45, so even if two nematic domains were present their spectra would overlap at this angle. The equal width for all satellites indicates that there is little to no inhomogeneity of the EFG parameters. This result indicates both that any variations of η between nematic domains, and the structural disorder due to any impurities, crystal defects and their associated strains, must be small.

### Spin–lattice relaxation

51V has both a nuclear magnetic moment and a nuclear quadrupolar moment (Q = 52 millibarn), and therefore both fluctuations of the hyperfine field and fluctuations of the EFG can give rise to spin–lattice relaxation. In a previous study we investigated the anisotropy of the nuclear spin–lattice relaxation rate and found evidence that fluctuations of the Tm quadrupoles may contribute to the nuclear spin–lattice relaxation via the EFG at the V site19. In general, there are three distinct relaxation channels: a magnetic, Wm, and two quadrupolar relaxation rates, WQ1 and WQ2, but it is difficult to disentangle the contribution of each21. These relaxation channels couple different sets of the Iz nuclear spin levels. WQ1 corresponds to transitions between nuclear spin states, $$\left|m\right\rangle$$, such that Δm = ±1 and WQ2 corresponds to transitions such that Δm = ±2. Usually WQ1 and WQ2 increase at structural transitions when ionic or electronic charges fluctuate strongly, and their relative contribution depends on the symmetry of these fluctuations and the magnetic field orientation with respect to the principal axes of the EFG tensor. The relaxation measured at a particular nuclear spin transition m ↔ (m −1) is a complicated function of Wm, WQ1 and WQ2 determined by a master equation (with an 8 × 8 dimensional matrix for spin 7/2 of 51V), as discussed the Supplementary Materials. The relaxation function for each transition is slightly different, thus by measuring the relaxation at multiple transitions one can globally fit the set of seven recovery curves to extract Wm, WQ1 and WQ2. However, in order to do so it is vital to resolve and excite each transition in the spectrum individually. The inhomogeneous broadening from the demagnetization field precluded such studies previously, but fortunately all seven transitions are clearly visible in the FIB sample. Figure 3a, b demonstrates that the relaxation measured at each of these transitions requires consideration of quadrupolar relaxation.

Figure 3c–f shows the field and temperature dependence of WQ1/T and Wm/T obtained by global fits to all seven nuclear transitions. Note that we constrain WQ2 = WQ1/4 for this orientation for these fits19. These data were acquired for at ϕ = 45, in which B0 is aligned transverse to the ferroquadrupolar distortion (see Fig. 2a). Figure 3g, h show the same data for the longitudinal direction ϕ = 0. Note that WQ1/T and Wm/T are determined by the dynamical magnetic and nematic susceptibilities of the Tm 4f electrons:

$$\frac{{W}_{m}}{T} \sim \mathop{\lim }\limits_{\omega \to 0}\mathop{\sum}\limits_{{{{\bf{q}}}},\alpha ,\beta }{{{{\mathcal{F}}}}}_{\alpha \beta }({{{\bf{q}}}})\frac{{{{\rm{Im}}}}{\chi }_{\alpha \beta }^{{\mathrm{mag}}}({{{\bf{q}}}},\omega )}{\omega }$$
(1)
$$\frac{{W}_{Q1}}{T} \sim \mathop{\sum}\limits_{{{{\bf{q}}}}}\frac{{{{\rm{Im}}}}{\chi }^{{\mathrm{nem}}}({{{\bf{q}}}},\omega )}{\omega },$$
(2)

where $${{{{\mathcal{F}}}}}_{\alpha \beta }({{{\bf{q}}}})$$ is the hyperfine form factor19,22. The dynamical magnetic and nematic susceptibilities, χmag and χnem, are likely coupled to one another and are driven by the fluctuations of the Tm quadrupolar moments. The details of the frequency dependence of the susceptibility and relaxation of the collective excitations, however, remain unclear. Qualitatively, we expect contributions to arise both from the dynamics of isolated doublets split by the Zeeman interaction and coupled to a bath of phonons23, as well as the collective dynamics associated with the ferroquadrupolar ordering in the vicinity of TQ13.

For the lowest fields (down to 1.35 T), both WQ1/T and Wm/T exhibit a relatively sharp peak at TQ and are suppressed below in the ordered state. It is remarkable that even at low fields both rates show distinct behaviors although magnetic fluctuations could be expected to result mainly from quadrupolar fluctuations of the ground state doublet. The relaxation rate above TQ can be fit to a Curie-Weiss expression:

$$\frac{{W}_{m}}{T}=\frac{{C}_{0}}{T-{{\Theta }}}.$$
(3)

with a Weiss temperature Θ/TQ = 0.55 ± 0.13, as shown by the dotted red line in Fig. 3c. This behavior likely reflects the divergence of nematic susceptibility at the phase transition15,22. The fact that Θ/TQ < 1 indicates that the elastic coupling between the Tm 4f moments and the εxy lattice strain renormalizes the exchange interaction between the moments15. Although we fit the data to only a limited temperature range limiting the precision of the fit, the value we obtain is consistent with previous measurements of the lattice elastic constant, c6624.

With increasing field, the sharp feature shifts upwards in temperature, as shown by the blue arrows in Fig. 3c, d, and a large broad peak emerges around ~3 K. The peak temperature increases by ~0.2 K by 5.3 T, and then disappears below the broad peak. The behavior up to this field suggests that the transition temperature TQ increases with field, in contrast to the expectation that transverse fields should suppress TQ to zero monotonically. The enhancement we observe may reflect the influence of higher order crystal field levels, as discussed below. At higher fields, both WQ1/T and Wm/T exhibit broad maxima around 3 K, and the feature associated with TQ disappears. The origin of this behavior is unknown, but may be related to a change in the dynamics of isolated two-level systems as the Zeeman interaction increases. It is unclear if the ferroquadrupolar ordering is suppressed, or the 3 K feature simply overwhelms the signature of the phase transition. For B0 6 T, both relaxation rates decrease with increasing field, and reach their low-field values by ~9 T. The peak at 6 T may reflect the critical in-plane field, B*, which is close to the estimated value based on the CEF parameters as discussed above. For higher magnetic fields, the splitting of the doublets will exceed kBT and the dynamics will be gapped out.

## Discussion

Although the in-plane g-factor is zero and the Zeeman interaction vanishes to first order25, the admixture of excited crystal field levels gives rise to a second order Zeeman interaction14:

$${{{{\mathcal{H}}}}}_{Z}=-\frac{{({g}_{J}{\mu }_{B}B)}^{2}}{2}[a{\hat{\sigma }}_{1}+b\cos (2\phi ){\hat{\sigma }}_{x}+c\sin (2\phi ){\hat{\sigma }}_{y}],$$
(4)

where $${\hat{\sigma }}_{\alpha }$$ are the Pauli matrices, $${\hat{\sigma }}_{1}$$ is the identity matrix, gJ = 7/6 is the Landé factor, B is the magnitude of the in-plane field, and ϕ is defined in the inset of Fig. 1a. The parameters a, b, and c can be determined by perturbation theory and depend on the crystal field levels of the Tm, as shown in10. Note that for ϕ = 0 or 90 (B0[110]T) the magnetic field should act as an effective longitudinal field to the ferroquadrupolar order, wiping out the phase transition at TQ. For ϕ = 45 (B0[100]T), the magnetic field acts as an effective transverse field, such that TQ is suppressed, but the transition should remain sharp. Using the values b = c = a/2 = 0.082 K−1 reported by Bleaney et al., we estimate that TQ should be suppressed to zero at a critical in-plane field of $${B}^{* }\approx \sqrt{2{T}_{Q}/c}/{g}_{J}{\mu }_{B}\approx 9.2$$ T along this direction. Note that for any other in-plane angles, the phase transition will be smeared out. Thus there exist two intersecting lines of quantum phase transitions as as function of field orientation, as illustrated in Fig. 2c.

The second order Zeeman interaction provides a natural explanation of the unusual magnetoelastic coupling observed in the spectra as a function of ϕ. For B = 2.7 T where the angular dependence of the EFG asymmetry parameter was measured (Fig. 1c), $$b{({g}_{J}{\mu }_{B}B)}^{2}/2\approx c{({g}_{J}{\mu }_{B}B)}^{2}/2 \sim 0.1{k}_{B}{T}_{Q}$$ thus the field should have a minimal effect on the ferroquadrupolar order. On the other hand, the Zeeman interaction should lower the energy of one domain over the other, giving rise to a field-induced detwinning in the ordered state and the observed anisotropy η. In other words, as the field is rotated from ϕ = 0 to ϕ = 90, domains with positive 〈σx〉 will convert to domains with negative 〈σx〉, in a manner similar to ramping an external magnetic field from negative to positive in a ferromagnet, and illustrated in Fig. 2b. An important difference, however, is that any field component away from ϕ = ± 45 or ± 135 will act as a longitudinal field, giving rise to a crossover rather than a sharp phase transition. Moreover, these fields will favor one domain over the other, leading to a single value of η, as discussed above.

The enhancement of TQ in a magnetic field is surprising. A similar effect was observed in TmCd, a material that also experiences a cooperative Jahn-Teller effect with a phase transition at 3.16 K26. However, the phase transition in TmCd is first order, and the effect was explained as the influence of a longitudinal field that shifts the first order discontinuity in the order parameter to higher temperatures and eventually broadens into a crossover. The phase transition in TmVO4 is second order, and the enhancement we observe occurs for transverse fields where the phase transition remains sharp. A possible explanation for this observation is that the quadrupole moment of the Tm ground state doublet is enhanced by the magnetic field. The excited crystal field states admix with the ground state doublet, such that the magnitude of the quadrupolar moment grows quadratically with field (see Supplemental Materials). As a result, the effective Hamiltonian for the Tm moments in transverse fields (ϕ = 45) becomes:

$${{{{\mathcal{H}}}}}_{eff}(B)=-\mathop{\sum}\limits_{i\ne j}{J}_{ij}{E}^{2}(B){\hat{\sigma }}_{x,i}{\hat{\sigma }}_{x,j}-c\frac{{({g}_{J}{\mu }_{B}B)}^{2}}{2}{\hat{\sigma }}_{y}$$
(5)

so B both enhances the effective coupling between the Tm moments as well as induces quantum fluctuations. The mean field phase diagram for this model is displayed in Fig. 2(d) for ξ = 0.002 T−2. As B increases, TQ initially rises by ~0.2 K around 8 T and is then suppressed to zero at a critical field ~11 T. The phase transition remains sharp for all fields. This model captures the enhancement of TQ for small B observed in Fig. 3c, d.

The behavior at higher fields remains unexplained. The data in Fig. 3 reveal a broad peak emerging around 3 K for B 4 T that spans several K, and the sharp feature associated with TQ at lower B disappears, so that it is not possible to track the phase transition to higher fields. It is possible that the relaxation dynamics are simply dominated by the behavior of isolated field-split doublets, such that the collective behavior of the Tm moments is not evident at these high fields. A similar broad peak appears for longitudinal fields, shown in Supplemental Fig. S4.

An alternative explanation is that there is a small misalignment from ϕ = 45. As seen in Eq. (4), such a misalignment would contribute to a longitudinal field that will smear out the transition. This effect would increase with field, such that the phase transition remains relatively sharp at low fields, but is broadened at higher fields. Note, however, that for fields in the range 2 T B 5 T, Wm/T and WQ1/T exhibit both a broad peak as well as a sharper feature at TQ, suggesting two separate phenomena. The phase diagram for a misaligned field would not have two features, but rather a single phase transition that broadens and moves to higher temperatures with increasing field. Thus a misalignment is not fully consistent with our observations.

It is interesting to note that both the quadrupolar fluctuations (WQ1/T) and the magnetic fluctuations (Wm/T) exhibit qualitatively similar trends with temperature and field in Fig. 3. This similarity suggests that the field-induced Tm moments are coupled with the Tm quadrupole moments, so that the critical fluctuations of the latter can drive fluctuations of the former. There are important differences between the two relaxation channels, however. For example, it is clear that WQ1/T is more rapidly suppressed below TQ than Wm/T. This observation suggests that WQ1/T is a more direct measure of the freezing nematic fluctuations when the order parameter 〈σx〉 steeply grows below TQ.

The exact mechanism that leads to the broad peak above TQ at high fields remains unclear. Previous measurements indicate that the spin–lattice relaxation rate is correlated with c66 up to 100 K at 11.7 T, suggesting that nematic fluctuations dominate the relaxation of the V nuclei19. Our results are consistent with the previous measurements, but we have now disentangled the two relaxation channels. Both the previous and current measurements reveal the presence of the broad peak. A qualitative understanding of this peak may be that for temperatures T > Δ(B)/kB, where Δ(B) is the field-dependent splitting of the Tm doublet, the relaxation rate is determined by the growth of the nematic susceptibility as the lattice softens. On the other hand, for T < Δ(B)/kB, the fluctuations will be suppressed with decreasing temperature23. These two effects may conspire to give rise to the broad maximum when T ~ Δ(B)/kB, which itself is field-dependent. On the other hand, the angular dependence suggests something more than a local phenomenon. Comparison of NMR relaxation rates with other low-frequency dynamic probes, such as ultrasound attenuation, can be fruitful27, however there is no report of ultrasound experiments with fields applied in the plane to allow direct comparison.

Our study proposes a conceptually easy mechanism to identify and disentangle magnetic and quadrupolar relaxation rates. Since the seminal work of Suter et al. it has been generally assumed that standard relaxation rate measurements preclude accurate determinations of Wm, WQ1 and WQ221, except in cases where νQ is sufficiently small that double-resonance experiments (SEDOR) can be performed28,29. However, as we show in TmVO4, a deviation in the magnetic relaxation rates on two transitions, e. g. the central peak and an outermost satellite (see Fig. 3a, b), is a direct indication that quadrupolar mechanisms are at play. This could be complicated if satellites are broad in frequency and possess inhomogeneous relaxation rates across the peak, but valuable information about charge ordering could be extracted from the median relaxation rates extracted from stretched magnetic fits for different transitions in inhomogeneous systems such as cuprates like La2−xSrxCuO430,31. This method is not directly applicable to the well-studied case of the nematic order in FeSe as both NMR active isotopes 57Fe and 77Se have no satellites since I = 1/2. However, FeSe1−xSx could be an interesting material with 33S having I = 3/2. Since the natural abundance is small such samples would need to be enriched with 33S. It is expected that the relaxation rate of 33S, 33WQ1,2/T will display a very different behavior than (77T1T)−1, because in the former 33S is sensitive to the divergent nematic susceptibility as the structural transition Ts is approached. Measurements of WQ1 and WQ2 above and below the superconducting Tc are a direct measure of interplay between superconductivity and nematicity as well as the importance of nematic fluctuations for the superconducting pairing. Related experiments have been performed in $${{{{\rm{BaFe}}}}}_{2}{({{{{\rm{As}}}}}_{1-x}{{{{\rm{P}}}}}_{x})}_{2}$$ by comparing relaxation rates of 75As (I = 3/2) and 31P (I = 1/2)32, but the method described here allows a direct determination of WQ2.

TmVO4 is a model system for investigating nematic quantum criticality in an insulator, in which ferroquadrupolar ordering of Tm 4f orbitals is well-described by the transverse field Ising model for magnetic fields oriented along the crystalline c-axis11. The response to in-plane fields has not been previously investigated, because the Zeeman interaction for the Tm non-Kramers doublets vanishes to first order in this configuration. We have studied the ferroquadrupolar ordering for in-plane fields using NMR of a single crystal shaped by a FIB to eliminate inhomogeneous broadening effects, and uncovered pronounced in-plane anisotropy in the EFG and spin–lattice relaxation rates. By measuring the relaxation curves at multiple nuclear transitions of the 51V nucleus, we disentangle magnetic and quadrupolar relaxation channels. We find that the phase diagram for in-plane fields is dominated by the second order Zeeman interaction, and that the in-plane magnetic field can act as an effective transverse or longitudinal field to the Ising-nematic order, depending on direction, and leads to a marked in-plane anisotropy in both relaxation channels. Also, we find that even for the case where the in-plane field corresponds to an effective transverse field, nevertheless small values of the magnetic field initially enhance the ferroquadrupolar ordering temperature before eventually suppressing the long-range order. This effect is tentatively ascribed to competing effects of field-induced mixing of higher energy crystal field eigenstates and the destabilizing effects of field-induced quantum fluctuations. For higher fields, the spin–lattice relaxation is dominated by a broad peak in temperature and field, which may be due to the dynamics of isolated doublets split by magnetic field. The second order Zeeman interaction also gives rise to a magnetoelastic coupling, so that as the magnetic field rotates in the plane, the nematic domains switch. Our results establish in-plane magnetic fields as a novel approach to tune TmVO4 to a quantum phase transition, and also establishes an important new method to disentangle magnetic and quadrupolar relaxation channels for high spin nuclei in a variety of condensed matter systems. Our results also indicate that TmVO4 should exhibit rich nonlinear magnetization effects for in-plane fields.

## Methods

### Sample preparation

Single crystals of TmVO4 were grown from a Pb2V2O7 flux using 4 mole percent of Tm2O3, following the methods described in refs. 33,34. The crystals grow as rectangular needle-like prisms, with the long-axis along the c-direction (see Supplemental Materials). The NMR spectrum depends sensitively on the shape of the crystal and the direction of the applied magnetic field, B0, which polarizes the magnetic moments and creates a magnetization M in the sample. Continuity of the flux-density B = μ0M + B0 across the sample surface requires a demagnetizing field μ0Hd in addition to B035. In an arbitrarily shaped sample μ0Hd(r) and hence B(r) will be inhomogeneous and the NMR spectrum will be broadened because the nuclei each resonate at the local field36. In most cases this broadening is not sufficient to cause any significant problems for NMR, however in TmVO4 the non-Kramers doublet has gc ≈ 10 and gab = 0. Although B0 can be oriented perpendicular to c, parts of the crystal near edges and corners tend to have components of B parallel to c, which exacerbates the broadening effect due to the large anisotropy of the susceptibility, χ. The resulting broadening is sufficient to wash out the quadrupolar splitting (~150 kHz) between the Tm satellites at low temperatures and high fields19.

### Focused ion beam

μ0Hd can be made homogeneous by cutting the sample in either a spherical or ellipsoidal shape37. For these studies we utilized a Xe2+ plasma focused ion beam (FIB) to cut our sample to an ellipsoid with the long-axis along the c-axis of the crystal38. The sample was cut from a carefully aligned cuboid from which calculated and programmed concentric circles are removed using by a xenon plasma FIB by Thermo Fisher Scientific with a 30 kV, 1 μA beam. Sample damage from the beam is only expected on the surface within a depth of 30–40 nm39,40,41 and energy dispersive X-ray analysis (EDX) of a test surface verifies the unchanged composition of TmVO4 below. The final sample diameter is 0.4 mm and the length of 1.3 mm require a total cutting time in excess of 25 h of each side. In the FIB process Al and C are deposited on the sample surface layer, however these do not affect the NMR signal from the bulk of the sample. The magnetic broadening was dramatically reduced in the FIB crystal, such that each of the seven peaks separated by the quadrupolar splitting are clearly resolved. The ability to resolve all seven peaks is important because it enables us to extract details of the magnetic and quadrupolar contributions to the spin–lattice relaxation rate that would otherwise be inaccessible, as discussed above in the section on Spin-lattice relation. For fields B0 ≤ 1.35 T the broadening from demagnetizing fields is sufficiently reduced, so that a larger crystal whose sharp corners were carefully polished away by hand was utilized.

### Crystal alignment

The sample was secured with superglue inside the coil to prevent motion due to torque arising from the anisotropic susceptibility. The coil was attached to a platform that was in turn inserted in a dual-axis goniometer (see Supplemental Materials). The crystal was aligned with B0c by minimizing $${T}_{1}^{-1}$$, and the in-plane orientation was controlled with a precision <0.119.

### Electric field gradient

To first order in the nuclear quadrupolar interaction, νq is given by:

$${\nu }_{q}=\frac{{\nu }_{zz}}{2}\left[3{\cos }^{2}\theta -1+\eta {\sin }^{2}\theta \cos (2\phi )\right],$$
(6)

where νzz = eQV33/12h, η = (V22 − V11)/(V11 + V22), and Vii (i = 1, 2, 3) are the eigenvalues of the principal directions of the EFG tensor42. In the disordered tetragonal phase, we find that νq is independent of ϕ, reflecting the axial symmetry of the EFG where V11 = V22 and η = 0. Below TQ there is a small but discernible variation in νq as a function of ϕ. In this case V11 ≠ V22, and the principal directions of the EFG tensor lie along [001]T with eigenvalue V33, along [110]T with eigenvalue V11, and along $${[1\overline{1}0]}_{T}$$ with eigenvalue V22. Note that these axes lie at 45 with respect the high temperature tetragonal axes, as illustrated in Fig. 2a. Moreover, ηεxy, where εxy is the B2g strain in the ordered state. Fig. 1c displays νq as a function of ϕ above (blue) and below (red) TQ. The spectra are fitted with Gaussian peaks and νq is determined from the average separation of the satellite peaks. Note that the separation of adjacent peaks is not exactly νq due to second order quadrupole shifts, however, the separation of corresponding high (n > 0) and low-frequency satellites (n < 0) are equal to 2nνq, as shown in Fig. 1b for illustration, so the average separation νq with the smallest error bars is determined based on the separation of all three pairs of satellites with ± n: νq ≡ (ν1 − ν−1 + ν2 − ν−2 + ν3 − ν−3)/12. It can be seen that the T dependence of νq is weak at a particular orientation (namely ϕ = 45) for which νq is nearly identical at 1.8 K and 15 K. This behavior arises because the temperature dependence of νq comes primarily from η(T), and $$\cos (2\phi )=0$$ for ϕ = 45, allowing to identify the directions of the principal axes in the plane. The four-fold angular dependence observed for νq makes it impossible to distinguish between the V11 and V22 directions and consequently the angles ϕ = 0 and 90 are assigned arbitrarily.