The interaction between a non-Kramers doublet in a tetragonal environment and a magnetic field is given by: H Z = g z μ B H z σ z + 1 2 g J μ B 2 b H x 2 − H y 2 σ x + 2 H x H y σ y , (2) where H _x,y is a magnetic field along the (x, y) direction, g _J = 7/6 for Tm³⁺ and g _c and b depend on the crystal field Hamiltonian [23]. These parameters have been measured for TmVO₄ to be g _c = 10.21 and b/k _B = 0.082 K ⁻¹ [16]. Note that H couples quadratically in the x and y directions, rather than linearly for a Kramers doublet. A field in the z direction splits the doublet linearly, and acts as a transverse field for the Ising interaction in Eq. 1.

The low temperature degrees of the Tm electronic degrees of freedom are thus captured by the sum H e x + H Z , which maps directly to the TFIM: H T m = ∑ l ≠ l ′ J l − l ′ σ y l σ y l ′ + g z μ B H c ∑ l σ z l , (3) where the sum is over the Tm lattice sites. Here we have ignored the small contribution from the perpendicular component of the magnetic field. Mean field theory predicts a QCP for a c-axis field of T _Q /g _c μ _B ≈ 0.3 T, which is close to the experimental value of H c * = 0.5 T. Note that if there is a perpendicular field oriented such that H _x or H _y is zero, the system can still be described by the TFIM, because H Z does not couple to the longitudinal order in pseudospin space (σ _y ). Rather, there is an effective transverse field in the x-z plane of pseudospin space leading to a different value of the critical field [24].

The decoherence of an NMR signal can often be extended by applying refocusing pulses [28]. The simplest such pulse sequence consists of a spin echo, in which a single π pulse at time t/2 reverses the direction of precession and refocuses static field inhomogeneities. Noise fluctuations at time scales shorter than t/2, however, will lead to decoherence and loss of signal. In general, the decay envelope, L(t), of a spin-echo can be related to the noise fluctuations of the environment. In TmVO₄, this quantity can be written as: log L t / L 0 = − A z z 2 ℏ 2 ∫ 0 ∞ S z z ω F ω t π ω 2 d ω , where S _zz is the dynamical structure factor for the transverse field fluctuations: S z z ω = ∫ 0 ∞ ⟨ σ z τ σ z 0 ⟩ e i ω τ d τ , and F(x) = 8 sin⁴(x/4) is a filter function for the spin echo pulse sequence, which takes into account the refocusing nature of the spin echo π pulse [29, 30]. The spectral area, shown in Figure 2C, is proportional to L(t) at fixed t corresponding to the pulse spacing in the spin echo experiment. Because the hyperfine coupling in TmVO₄ is solely along the transverse field direction, the nuclei are invisible to the longitudinal degrees of freedom. Only S _zz (ω), the noise spectrum in the transverse direction, contributes to the decoherence of the nuclear spins. This anisotropic coupling is highly unusual, but it enables us to probe the transverse fluctuations without any contamination from the longitudinal fluctuations, which diverge strongly at the QCP. The filter function acts to remove the static or low frequency (ω ≤ 10⁵ Hz) components of the fluctuations, which are dominated by thermal fluctuations [19, 31]. The remaining contributions to S _zz (ω), and hence to the decay of L(t), is from quantum fluctuations, which exist at finite frequency. This is because they arise from the intrinsic time evolution due to the many-body Hamiltonian, which has a finite gap except at the QCP. The fact that L(t) reaches a minimum at the QCP indicates that these quantum fluctuations are largest here. Importantly, these extend to finite temperature, even exceeding T _Q . These results thus imply that there is a broad region of phase space, a “quantum critical fan,” where quantum fluctuations are present.

An open question is how does the transverse susceptibility behave in the vicinity of the quantum phase transition? In mean-field theory at T = 0, χ _zz remains constant in the ordered state, and vanishes for H c > H c * , as shown in Figure 3A. The NMR data are inconsistent with the mean field picture, since the relative area under the spectra decreases dramatically at the QCP, indicating that χ _zz must be strongly field-dependent in this range. Numerical calculations that are based on high and low field series expansions indicate that χ _zz diverges logarithmically on both sides of the QCP for various 3D lattices [32]. At T = 0 the enhancement is in a very narrow region but it should widen into a quantum critical fan at finite temperatures. Indeed we find significant differences between numerical calculations for small finite clusters and mean field theory at finite temperatures with enhancement in the general vicinity of the QCP, as seen in Figure 3B. We expect the differences to be much larger and centered at the critical point in the thermodynamic limit. These calculations, however, assume only a nearest neighbor interaction [e.g., J(l = l′) = 0 if l, l′ are not nearest neighbors in Eq. 3]. The interaction is expected to be long-range in TmVO₄, which could tend to stabilize mean-field behavior.

Understanding the mechanisms of decoherence is a key problem for quantum computing, and the behavior of a central spin coupled to a well-controlled environment is an important theoretical model that has been studied extensively [31, 33]. In the case where the central spin (or qubit) is coupled to a 1D TFIM via a hyperfine coupling along the transverse field direction, the decoherence of the qubit can be elegantly expressed in terms of the overlap of the wavefunction of the environment at different times and values of the transverse field. In fact, the ⁵¹V spins coupled to the ferroquadrupolar ordering in TmVO₄ maps well to this model, but with a 3D lattice for the environment. Although the central spin model was originally developed for a single spin coupled to an environment, it is straightforward to generalize to an ensemble of nuclear spins in a lattice, each with its own identical coupling [20]. Thus, TmVO₄ offers a unique opportunity to experimentally study this model.

Importantly, this connection offers a new approach to understanding NMR decoherence in terms of the quantum fidelity of the environment, which is defined as the modulus of the overlap between two states: F = |⟨Ψ′|Ψ⟩|. In the case of the central spin model, the two states are Ψ λ ′ ( t = 0 ) and Ψ _λ+ϵ (t), where λ corresponds to the transverse field, and ϵ corresponds to the small hyperfine field. Two ground states of the TFIM at different values of the transverse field may initially be very similar, but will evolve strongly away from one another in the vicinity of the QCP. At T = 0, the intensity of the NMR free induction decay is proportional to F ², thus the qubit experiences a strong decoherence as the transverse field approaches the critical value. This tendency can be captured by the fidelity susceptibility: χ _F = −∂ ² F/∂ϵ ². At finite temperatures, the fidelity can be expressed in terms of the density matrix [31]. A related quantity is the Quantum Fisher Information which quantifies the sensitivity of density matrices to small changes in parameters [34]. Because the fidelity susceptibility tends to diverge at a QCP, this quantity has been exploited theoretically to identify quantum and topological phase transitions [35, 36].

On the surface, this picture differs from the conventional NMR picture in which decoherence arises due to the presence of stochastic fluctuations of the hyperfine field, which can be quantitatively measured via Bloch-Wangsness-Redfield theory: T 2 − 1 = A z z 2 S z z ( ω = 0 ) / 2 ℏ 2 [28, 37, 38]. However, χ _F in fact can be related to the transverse field susceptibility, χ _zz = S _zz /k _B T [39]. This remarkable connection offers new insights and connections between NMR and quantum information theory. For example, NMR wipeout is ubiquitous in strongly correlated systems, and has been observed in the high temperature superconducting cuprates and the iron based superconductors [40–44]. In these cases, this phenomenon has been attributed to electronic inhomogeneity introduced because of the dopant atoms. However, the behavior in TmVO₄ suggests that it might be valuable to considering the wipeout in these other systems as a consequence of their proximity to a QCP.

NMR spectra in doped systems are generally broader than in undoped materials because the dopants often give rise to inhomogeneity. As seen in Figure 4A, the spectra of the pure TmVO₄ and YVO₄ consist of seven clear resonances with small linewidths, but these resonances grow progressively broader with doping. Each of the seven resonances broadens equally between 0 ≤ x ≤ 0.1. This behavior indicates that the broadening mechanism is not quadrupolar inhomogeneity, but rather a Knight shift inhomogeneity. The red dotted lines in Figure 4A are fits to the spectra, and the data in panel (e) show how the Gaussian width, σ, varies with doping for the spectra that can be clearly fit. It is surprising that even though random strain fields are clearly present and rapidly suppressing T _Q , they apparently do not significantly alter the local EFG at the V sites. In many other strongly-correlated systems, doping usually causes significant quadrupolar broadening [47–50]. In Tm_1−x Y _x VO₄, the larger Y atoms slightly displace the O and V in their vicinity [14]. On the other hand, it is possible that the VO₄ tetrahedra may not be significantly distorted upon Y substitution. Also, there are two main contributions to the EFG: a lattice term arising from the arrangement of charges, and an on-site term that is determined by the electronic configuration of the local electronic orbitals [28]. It is reasonable that the latter term dominates the EFG at the V, and that the electronic configuration of the V and O orbitals remain relatively unperturbed by Y doping.

Figure 6A displays ( T 1 T ) − 1 versus temperature for several different doping levels, measured for field perpendicular to the c-axis. Note that for this field orientation the resonance frequencies of sites A, B, and C overlap, and thus we are unable to discern if these spin fluctuations are spatially inhomogeneous. We do not see any evidence for stretched relaxation for x < 0.1, which would indicate the presence of inhomogeneity. In this range the different quadrupolar satellites are clearly resolved, and the relaxation was measured at all transitions to extract both the magnetic and quadrupolar relaxation channels, although just the magnetic contribution is shown [24]. For higher doping levels where the spectra no longer show any structure, we are unable to determine if there is any stretched relaxation behavior. There is a clear peak for the pure TmVO₄ at T _Q reflecting the critical slowing down at the thermal phase transition. As the doping level increases this peak is suppressed to lower temperatures, yet ( T 1 T ) − 1 increases and reaches a broad maximum around x ≈ 0.10. In fact, the spin fluctuations appear to be enhanced near the vicinity of the critical doping level, x _c , possibly reflecting quantum critical fluctuations at this doping. At higher doping levels, the fluctuations gradually are suppressed and eventually disappear. For the pure YVO₄, there are no magnetic moments present anymore, and ( T 1 T ) − 1 is several orders of magnitude smaller.

Sites A, B and C can be discerned when there is a finite H _c component present. Figure 6B compares ( T 1 T ) − 1 versus H _c in pure TmVO₄ with Tm_0.6Y_0.4VO₄ for the A and B sites. The strong field dependence of the pure system reflects the growth of the gap as the system is tuned away from the QCP at H c * : ( T 1 T ) − 1 ∼ exp ( − Δ ( H c ) / T ) [20]. As H _c is tuned beyond the QCP, the gap increases and ( T 1 T ) − 1 decreases. It is surprising that in the x = 0.40 sample, which has no long range order, the A and B sites exhibit behavior that is qualitatively similar to that in the pure system. In other words, they each increase with decreasing field as H _c approaches the critical value. This behavior suggests that there are still localized clusters of Tm which continue to exhibit behavior reminiscent of the undoped lattice. Statistically there are regions of the disordered lattice with connected Tm atoms, and these may continue to exhibit correlations despite the absence of long-range order, giving rise to Griffiths phases [52]. An interesting open question is how such disconnected clusters may be affected by the presence of random strain fields.

Inhomogeneous dynamics in the disordered lattice may also explain the fact that the spectra in Figure 5A appear to exhibit an increasing intensity for H _c ≳ 1.5 T once the A and B peaks emerge. If local clusters of Tm continue to exhibit quantum critical fluctuations at these sites, then T 2 − 1 will be large, suppressing the signal from these sites. In other words, the A and B sites may experience partial wipeout in the vicinity of H c * . Overall these sites contribute 84% of the total area, and the relative area under the spectra decreases by approximately the same value near H c * in Figure 2C. These observations further support the argument that the A and B sites are locally unperturbed by the Y dopants, and may exhibit behavior consistent with quantum Griffiths phases.