Normal state $^{17}$O NMR studies of Sr$_{2}$RuO$_{4}$ under uniaxial stress

The effects of uniaxial compressive stress on the normal state $^{17}% $O\ nuclear magnetic resonance properties of the unconventional superconductor Sr$_{2}$RuO$_{4}$ are reported. The paramagnetic shifts of both planar and apical oxygen sites show pronounced anomalies near the nominal $\mathbf{a}$-axis strain $\varepsilon_{aa}\equiv\varepsilon_v$, that maximizes the superconducting transition temperature, $T_{c}$. The spin susceptibility weakly increases on lowering the temperature below $T$$\simeq$10 K, consistent with an interpretation based on the passing of a van Hove singularity through the Fermi energy. Although such a Lifshitz transition occurs in the $\gamma$ band, formed by the Ru $d_{xy}$ states hybridized with in-plane O $p_{\perp}$ orbitals, the large Hund's coupling renormalizes the uniform spin susceptibilty, which, in turn, affects the hyperfine fields of all nuclei. We estimate this \textquotedblleft Stoner\textquotedblright\ renormalization, $S,$ by combining the data with first-principles calculations and conclude that this is an important part of the strain effect, with implications for superconductivity.


I. INTRODUCTION
The physics of the superconducting state of Sr 2 RuO 4 [1] remains a subject of longstanding importance, with particular focus on order-parameter symmetry [2]. There are numerous experimental results consistent with a chiral p-wave superconducting state [3][4][5][6], including evidence for time-reversal symmetry breaking for T < T c [7,8], and lack of suppression of the in-plane spin susceptibility on cooling through the superconducting critical temperature T c , as deduced from nuclear magnetic resonance (NMR) spectroscopy [9,10] and neutron scattering [11]. At the same time, there are other experimental results inconsistent with that interpretation [12][13][14][15][16], and the out-of-plane spin susceptibility also remains constant [10], in contradiction with the expectations for the chiral state [5,17].
For several reasons, the normal state physics of Sr 2 RuO 4 is equally topical. It was anticipated at a very early stage that electron-electron interactions are controlled by the Hund's rule coupling [18], and it was later shown within the dynamical mean field theory that the electrons are subject to strong Hund's rule correlations while the system remains metallic and far from the Mott insulator regime [19,20]. Mean-field Density Functional (DFT) calculations within the Generalized Gradient Approximation (GGA) are unstable against ferromagnetism [17]. Even though strong correlations lead to fluctuations suppressing this instability, there still remains a substantial Stoner renormalization of the uniform spin susceptibility. This led to the analogy with the triplet super-fluidity of 3 He [18], anticipated earlier on the grounds that a related compound, SrRuO 3 is ferromagnetic [2]. Although later it was found that the leading magnetic instability is at a non-zero momentum q 0 ≈(±0.3, ±0.3, 0)2π/a [21,22], the proximity to a ferromagnetic state dominates the debate related to the superconducting order parameter symmetry [6,23].
An additional feature is the proximity to a 2D Lifshitz point [24] associated with a van Hove singularity (vHs). Recently, striking physical property changes, including a factor of 2.5 increase in superconducting critical temperature T c , from 1.4 K to 3.5 K [25], accompanied by a pronounced non-Fermi Liquid behavior of the resistivity [26], were observed under application of in-plane strain ε aa . This was tentatively interpreted as a Fermi level crossing of the vHs when ε aa reaches a critical value ε v . Since direct experimental evidence is still lacking, it is important to test this interpretation in complementary studies of the normal state while subject to strain. Also, the vHs is expected to affect quite differently the triplet and singlet superconducting states, and this provides further motivation for physical property studies under strain. For singlet pairing, the order parameter (gap) can be large at the vHs (e.g., for the d x 2 −y 2 symmetry), and thus the local density of states (DOS) enhancement at the vHs is very beneficial. On the contrary, the triplet order parameter at precisely the Lifshitz point is zero by symmetry, and therefore a triplet state is less suited to take advantage of the vHs unless the pairing interaction itself is enhanced. The latter can be possible as the DOS enhancement may bring the system closer to ferromag-netism [27].
With these issues in mind, we set out to verify experimentally that the same strain at which T c and Stoner factor S peak indeed corresponds to the maximum in DOS, and to assess, as quantitatively as possible, the change in DOS and Stoner enhancements to the susceptibility under strained conditions. To this end, NMR measurements inform on the details of the normal state, though site and orbitally specific hyperfine couplings. Indeed, the enhancement is evident in the results presented, and moreover, the inferred enhancement semi-quantitatively accounts for the transport results in Ref. [26]. Looking ahead, it is worth emphasizing that the method is considered a litmus test for the superconducting state parity [9,28], including any strain-induced order-parameter changes. The results presented in the next sections are normal state 17 O NMR spectroscopy for in-plane B b, and out-of-plane B c fields, as well as 17 O NMR relaxation rates for B b. These are interpreted by way of complementary DFT calculations.

II. EXPERIMENTAL DETAILS
Single crystalline Sr 2 RuO 4 used for these measurements was grown by the floating-zone method [1]. Smaller pieces were cut and polished along crystallographic axes with typical dimensions 3×0.3×0.15 mm 3 , and with the longest dimension aligned with the aaxis. 17 O isotope ( 17 I = 5/2, gyromagnetic ratio 17 γ = −5.7719 MHz/T [29]) spin-labelling was achieved by annealing in 50% 17 O-enriched oxygen atmosphere at 1050 • C for 2 weeks [9,30]. The sample quality was not observably changed following this procedure, with T c ≈1.44 K identified by specific heat measurements (Supplemental Materials, SM) [31]. For the NMR experiments, the sample was mounted on a piezoelectric strain cell (Razorbill, UK) with an effective (exposed) length L 0 ∼1 mm (see Fig. S1a, SM). Three samples (labeled as S1, S2 and S3) were measured in this work. A nominal compressive stress is applied along the a-axis, with corresponding strains (ε aa ≡ δL/L 0 ) estimated to be up to -0.72% using a pre-calibrated capacitive dilatometer; the accuracy is limited by the unknown deformations of the epoxy clamp [32]. For reference, the observed maximum T c (ε aa ) occurs at a quantitatively similar displacement as reported in Ref. [25], T max Most of the NMR measurements were performed at fixed temperature T =4.3 K (and variations around it) and carrier frequency f 0 =46.8 MHz (B 8.1 T), using a standard Hahn echo sequence. Spectra, including satellite transitions, were collected in field-sweep mode, whereas a close examination of the central transition (-1/2↔1/2) for both in-plane and apical sites was carried out under fixed-field conditions. Some field and temperature dependence was explored. The application of NMR in conjunction with the piezoelectric-driven in-situ strain is particularly challenging, because of the severe constraint on sample size, for which particular attention was paid to enhance the signal-to-noise ratio for all these measurements.

III. RESULTS AND DISCUSSION
The geometry of our experiment is depicted in Fig. 1(a) [33]. Each Ru ion is coordinated octahedrally by four planar O(1) and two apical O(2) oxygen sites, with a small elongation along the c-axis. While a-axis strain ε aa renders the sites O(1) and O(1 ) crystallographically inequivalent, their local symmetries are different even for the unstrained case in the case of the external field B b. The field-sweep spectra in Fig. S2 are described by parameters (shifts, electric field gradient (EFG)) similar to previous reports for 17 O NMR in the unstrained Sr 2 RuO 4 [28,33], with five NMR transitions for each of 3(2) dis- The most relevant orbitals for the 17 O couplings are Ru 4d t 2g , which hybridize with O p states to form the quasi-2D γ band, predominantly from the d xy orbital, and similarly the quasi-1D α and β bands from the d zx,yz orbitals, Fig. 1b. The spin-orbit coupling (SOC) mixes these. While mixing is strongest along the Brillouin Zone diagonal (Γ−M in momentum space, see Fig. 2b) [34,35], it is more important here that it mixes the d xy and d yz bands at Y. The latter has the effect of pushing down the lower band (d xy ) by about 20 meV, which shifts the critical strain ε v where the Lifshitz transition shows up in the calculations, from about ∼ −1.0% to ∼ −0.65%. Additional mass renormalization, not accounted for in the density functional calculations, reduces the critical strain still further, consistent with the experimentally observed maximum in T c between −0.55% and −0.60% [25,31]. is only a weak coupling of the O(2) p x,y with Ru d zx,yz orbitals, respectively) exhibits a very small Knight shift. In contrast, Knight shifts for O(1) and O(1 ) vary strongly and show clear extrema where the putative vHs appears, at ε v , defined as where T c (ε aa ) is largest. The anomaly is most pronounced for the in-plane field orientation. For larger strains, there is significant broadening, which could result from a strong strain dependence of the spin susceptibility combined with a distribution of strains within the sample. In the right-hand panel, O(1,1 ) absorption peaks appear indistinguishable at small strain, with pronounced broadening and splitting appearing for strains exceeding ε v . The NMR shifts K, defined as the percentage of the shift of resonance frequency with respect to that in D 2 17 O, are shown as a function of ε aa in Fig. 4. In metals, the NMR shift is governed by three main contributions to the local field (in addition to the applied field): (i) isotropic coupling from the Fermi contact interaction and core polarization, (ii) anisotropic coupling of the dipolar field generated by the electronic spin away from the nucleus, and (iii) fields generated by orbital currents. This partitioning of the hyperfine field contributions can be summarized as where s is the spin moment of an electron, and L its orbital moment. Note that h s has no anisotropy, while h d gives no isotropic contribution (h o has both).
The net spin magnetization is written as where the full uniform susceptibility χ can be approximately written in an RPA-type way [36] using the Stoner factor S : where χ 0 is the noninteracting susceptibility equal to (neglecting spin-orbit effects) the density of states. The total uniform magnetic field is the sum of the external and the screening field, the latter being enhanced compared to the noninteracting case by the factor S. Note that the orbital moment L in Eq. (1) is assumed to be generated by the spin magnetization through spin-orbit coupling; in addition, there is another orbital term (paramagnetic van Vleck), which is not enhanced in the same way as χ.
As discussed in some detail below, S can be more sensitive to the enhancement of the DOS than χ 0 itself, and further is important as a mechanism for transferring anomalous responses to orbitals other than Ru d xy and the corresponding hybridizing Op orbitals. The experimentally measured χ is enhanced by about a factor of 7 compared to the DFT DOS [23], originating with both mass renormalization (m * /m 0 ∼ 3.5 [3]), and Stoner factor (S ∼ 2). Using Eq. (4) one can then estimate IN (E F ) ≈ 0.5. Note that the calculated DOS ( Fig. 5) is enhanced by about 30% at the vHs. If m * /m 0 and I are strain independent, the maximal S (at the vHs) is S max ∼ 3 and χ/χ 0 ∼ 10, which compare favorably to the shift measurements plotted in Fig. 4. Deviations from standard Fermi Liquid behavior, as reported in relation to recent transport experiments [26] may be connected to the enhanced Stoner factor near the critical strain. Observed is a crossover temperature T * , above which the derivative δ=d(log ∆ρ)/d(log T ) changes from the Fermiliquid result δ = 2 to a value ∼1.5-1.6. This is close to what is expected for ferromagnetic spin-fluctuation behavior (δ∼4/3-5/3) [37]. T * varies strongly with strain (see Fig. S1b [31]), and further, is minimized at ε v . The interpretation of Ref. [26] reasonably associates the behavior with proximity to the vHs [38]; however, both the singular DOS and enhanced S may play a role.
For insight into the strain-induced changes, and particularly those associated with the vHs, Density Functional calculations using the Linear Augmented Plane Wave package WIEN2k [39] were performed, including spin-orbit interaction. The specific objective was to extract at least semi-quantitative information about the origin, evolution, and relative importance of the individual contributions in Eq. (1) to the net Knight shifts. A local density approximation (LDA) for the exchangecorrelation functional, a k-point mesh of 41×41×41, and the expansion parameter RK max =7 were utilized. Further, the optimized structures of Ref. [25] were used, and then interpolated to assure that the strain at which the vHs crosses the Fermi level is included. It turns out that the proximity to a quantum critical point forced some adjustments to the standard procedure. A means for Knight shift evaluation within DFT is to apply an external magnetic field H 0 , then compute the generated hyperfine fields. DFT overestimates the tendency to magnetism, because in reality the Hund's rule derived interaction I and, correspondingly, the Stoner renormalization S are reduced by quantum fluctuations that are not accounted for in a mean-field approach. In addition, the calculated peak in DOS is very narrow, with full width at half (the calculated) maximum of ∼3 meV, and holds only 0.0015 e − in each spin channel. As a result, an external field producing sufficiently strong hyperfine fields (compared to the computational noise), is too large to properly monitor the vHs peak. Still, the calculations at larger fields produce useful information about the origin of the net Knight shifts in terms of individual contributions in Eq. (1).
The magnetic instability was avoided in two ways: (1) the less magnetic LDA was utilized, rather than the GGA functional, and (2) an ad hoc scaling factor was applied, which effectively reduced the Hund's coupling, somewhat arbitrarily, by a factor of two [40]. This comes at some cost, however. While it allows monitoring of the changes with strain, the overall scale of the calculated Knight shifts is not very accurate. Also, the LDA functional is less accurate than GGA in term of addressing the structural effects, and therefore the critical strain at which the vHs crosses the Fermi level is overestimated by about 50%. Therefore, all computational results below are presented as a function of the reduced strain, ε aa /ε v . Note that important by-products of these calculations are the full uniform susceptibility χ = M/H, where M is the total magnetization induced by the field H, and the oneelectron susceptibility χ 0 = µ 2 B N (E F ), and therefore also the ratio, S = χ/χ 0 .
The resulting partial densities of states and characterprojected bands in Sr 2 RuO 4 are shown in Figs. 5 and 2(a). Only O(1)p x states are sensitive to the vHs, as expected from symmetry considerations (Fig. 1b). Indeed, at Y, only O(1)p x orbitals couple by symmetry with Rud xy states, and then, one might infer that only the O(1) Knight shift should be affected by the DOS peak at the vHs. This is borne out by the DOS calculations. However, on general grounds, all sites are sensitive through the Stoner enhancement S. Indeed, all measured Knight shifts are affected by strain (Fig. 4), with K 1 more so, presumably because of the direct influence of increased γ DOS. The last point is also important because it associates the enhanced DOS and S at the critical strain, with the vHs at Y, where the O(1)p x orbital is the only O orbital with considerable weight. Further evidence for the narrow vHs and its influence is shown in Fig. 6, which depicts shifts with strikingly strong field and temperature dependences. The observations are qualitatively consistent with comparable energy scales for Zeeman, thermal, and vHs terms, where, for instance, the broadening of the Fermi distribution progressively weakens the sensitivity of thermodynamic properties to the vHs, even when it is situated precisely at the chemical potential [41]. Similar observations for the magnetization were previously reported in a doping study, in which the effects of substitution of La for Sr in Sr 2−y La y RuO 4 were interpreted as evidence for moving γ-band Fermi energy to the X and Y points of the Brillouin Zone [42].
For a semiquantitative evaluation of the Stoner enhancement and impact on the observable quantities, the data were contrasted to the results of the DFT calculations. As discussed in the previous section, the inherent deficiency of the DFT calculations for such a strongly correlated material as Sr 2 RuO 4 forced deviations from the usual routine. The standard calculations, as used, for instance, in Ref. [25], are unstable against spontaneous formation of a ferromagnetic state. The tendency toward this instability was reduced, somewhat arbitrar-  ily, by scaling the Hund's coupling by half. This ensured numerically stable calculations in external fields up to at least 5 T, even at ε aa = ε v . The impact of the reduced Hunds' coupling appears to produce systematic errors in related absolute parameters, but less so for the relative changes induced by strain. For example, for the selected scaling, Fig. 7 indicates that the calculated χ(ε aa = 0) renormalization is ∼ 60%, whereas the known correlation-induced mass enhancement is 3.5 [3]. Therefore, the downscaling is too strong. Given this caveat, at the critical strain, χ is enhanced over χ(0) by about 70%, while S(ε v ) is about 30% larger than S(0). Thus, expected enhancements are ∼1.3 for K 1 ⊥ and K 2b , and ∼1.7 for K 1 , all consistent with the calculated shown in Fig. 7b.
Relatively strain-independent orbital terms were calculated for the shifts of all three sites, as would occur for a van Vleck-dominated response. Since this is notoriously difficult to calculate in DFT, these orbital contributions were adjusted to ensure a good agreement with the experiment. The constants were, respectively, −0.07%, 0.35% and 0.06% for K 1 , K 1 ⊥ and K 2b . We emphasize that the described procedure is too approximate to assign particular importance to the exact numbers; however the resulting, indeed rather good agreement between the calculated and measured hyperfine contributions, lends some validity to the approach and interpretation. Therefore, the qualitative conclusions are as follows: (1) there are two mechanisms for enhancing the Knight shifts near the critical strain, one applicable to all sites and field directions, and the other only to K 1 . Both are directly related to the DOS enhancement and show unambiguously that the maximum in T c indeed coincides with that in DOS; (2) ferromagnetic spin fluctuations intensify substantially at the same strain, and may play a key role in boosting T c . Furthermore, it provides a natural explanation to the recent resistivity measurements on strained samples. Some comments on the data collected for field aligned parallel the c-axis are in order. In principle, one would expect similar behavior to that for the in-plane field, however, it appears that K c behaves in a way difficult to rationalize in total. For strain ε aa ≤ −0.63%, a single absorption peak at ∼ 0.28% shift is observed for O(1,1 ), with only a small increase in the range of ε v . For larger strain, ε aa = −0.72%, the peak broadens considerably, and could be construed as exhibiting two components, but with drastically reduced first moment. The drop in intensity is likely a T 1 effect, a consequence of a too rapid pulse repetition rate (see Fig. S5, SM). The apparent "splitting" and distorted lineshape are consistent with what could result from a strain gradient along with a nonlinear variation of shift with strain. The main challenge, however, is to explain the observed evolution on approaching ε v from smaller strain, where the DFT calculations indicate larger shifts for O(1) than for O(1 ).
It is possible that the orbital contributions play an important role in this case. Interestingly, for the orbital part of K 1c , and to some extent, of K 1 c , the calculations predict a sizeable enhancement, suggesting that the van Vleck contribution is not dominant, or, at least, less prominent here than for the in-plane fields, and, conversely, the SOC induces sizeable orbital Knight shifts. Moreover, the sign of this orbital contribution is opposite to the spin shifts, so there is a tendency toward cancellation. It is believed that correlation effects enhance the SOC in Sr 2 RuO 4 by about a factor of two [43]. Empirically, if the O(1) and O(1 ) shifts are assumed to be entirely generated by SOC, while the O(2) shift is entirely van Vleck, a reasonable agreement with experiment is obtained, but with small but not negligible peak splittings for strains near ε v (Fig. S6). Clearly, the NMR spectra for the field parallel to c require further investigation.

IV. FURTHER CONSIDERATIONS REGARDING STONER RENORMALIZATION
The experiments and calculations clearly demonstrate the importance of Stoner renormalization near the critical strain, but this is evaluated only semiquantitatively.  Fig. 7. The RPA Eq. (4) works reasonably well, provided the Stoner exchange parameter I as an adjustable parameter. However, this agreement may be somewhat misleading, since S varies substantially over the Fermi surface (depicted in Fig. 8). Note also that S, for the same external field, is most strongly enhanced for the α and β bands, and less so for the γ band, and that this enhancement is indeed about twice as strong at the critical strain than for the unstrained structure. Overall, in the former the Stoner factor S varies between 3.2 and 4.7, and in the latter between 5.7 and 10.0, about a factor of two larger than for the unstrained structure. Consequently, it is entirely possible that this variation will weight differently the dipole and the spin-contact contributions. This is consistent with the fact that the temperature dependence of the in-plane, and only in-plane Knight shifts is opposite to that of the uniform susceptibility at T 100 K, and only these are affected by the vHs in our experiment [28].
Given the proximity to magnetism, it is natural to assume that spin fluctuations play a major role in transport scattering. Moriya et al worked out a theory of paramagnon scattering in the uniform Stoner approximation [37], but as shown here, it is not accurate for Sr 2 RuO 4 . Moreover, Sr 2 RuO 4 combines both antiferromagnetic (not enhanced by the vHs) and ferromagnetic (enhanced) fluctuations. In favor of the applicability of Moriya's theory to Sr 2 RuO 4 is that the full susceptibility as evaluated from neutron scattering results [23] and interpreted in the framework of Moriya's theory, generates mass renormalization in agreement with specific heat data. Within the same framework, the resistivity is expected to change from the usual (Fermi Liquid) low-temperature T -quadratic behavior to ≈ T 1.6 at a crossover temperature T * ∝ S −1 . The transport measurements in Ref. [26] revealed such behavior, where moreover, the crossover temperature T * exhibits a pronounced minimum near ε v (Fig. S1b in SM). This is fully consistent with the present conclusion that S peaks at the same strain.

V. CONCLUSION
It is demonstrated, by means of the NMR spectroscopy under uniaxial stress, and corresponding density functional calculations, that there are two different effects associated with the strain-induced vHs, which both need to be taken into account, namely the enhancement of the DOS associated with the γ-band Fermi energy passing through the vHs at the Y point of the Brillouin Zone, and a substantial Stoner enhancement S. This finding has immediate ramifications for superconductivity. Namely, first, the DOS is enhanced near the vHs point. In the first approximation, this effect strongly favors some singlet pairings, such as extended s, d zx ± id yz or d x 2 −y 2 , mildly favors the d xy pairing, and less so any triplet pairing. However, this enhancement of the DOS, through the Stoner factor, boosts FM spin fluctuations, which favors a triplet states and would seem to disfavor singlet pairing. Experimentally and theoretically, these two effects are comparable, and therefore it is unclear which is stronger. More information will be gained by studying NMR in the superconducting state as a function of strain; measurements are currently underway. In this Supplemental Material (SM), we provide the experimental setup, sample characterization, field-swept spectra, spin-lattice relaxation rate and electric field gradient (EFG) results of Sr 2 RuO 4 under uniaxial stress, as well as additional theoretical details that further support the discussions in the main text. Figure S1a is a photograph of the set-up for our NMR measurements under strain. The compressive uniaxial pressure is generated by a set of piezoelectric actuators [45]. A Sr 2 RuO 4 sample is glued between two pairs of titanium plates with stycast 2850 (black). To get a perfect filling factor, a small NMR coil (about 25 turns) is made in-situ surrounding the sample after the stycast hardens with 25 µm Cu wire.  S1. (a) A photograph of the uniaxial stress apparatus. The stress/strain effect is applied through a set of piezoelectric actuators. The forces are applied uniaxially, and the strain response is measured using a capacitive dilatometer. A small coil is made in-situ surrounding the sample that is bonded between two pairs of titanium plates. (b) Electronic specific heat divided by temperature of the sample before and after 17  as well as the normal state Sommerfeld coefficient are also in good agreement with previous findings [47]. All these guarantee the high quality of the sample studied in this work.

SM I: Sample characterizations
In Fig. S2, we present the field-sweep 17 O NMR spectra of Sr 2 RuO 4 for both B b (left panel) and B c (right panel). All the peaks in this field region can be assigned to signals from O(1), O(1 ) and O(2) sites, and no extra peaks can be identified. This excludes the impurity phases from other members in Sr n+1 Ru n O 3n+1 family. For each O site, it shows one central peak ( 1 2 ↔ − 1 2 ) and four satellite peaks corresponding to ± 1 2 ↔ ± 3 2 and ± 3 2 ↔ ± 5 2 , respectively. These satellite peaks arise from nuclear quadrupole interaction with the electric field gradient (EFG) at the nuclear site, as described by whereÎ=(Î x ,Î y ,Î z ) is nuclear spin operator, Q is nuclear quadrupole moment, and η=(V xx −V yy )/V zz is the asymmetry parameter with V xx , V yy and V zz being the components of the EFG tensor. Here we adopt the convention V zz ≥V xx ≥V yy , and V xx +V yy +V zz =0. In Sr 2 RuO 4 , V zz is along Ru-O bonding [33]. This allows us to determine principle-axis nuclear quadrupole resonance (NQR) frequency ν Q =ν z from the spectra shown in Fig. S2. Note that ν z is related to V zz by Other components of NQR frequencies conform to the formula: where θ and φ are respectively polar and azimuthal angles as defined in regular xyz-frames, see Fig. S3a. Eq. (S3) also enables us to verify the sample orientation with respect to magnetic field. In fact, for B c and ε aa =0, we should expect the NQR peaks of O(1) and O(1 ) to merge. The splitting of them seen in Fig. S2b is a consequence of small angular misalignment which we estimate to be θ∼5 o according to Eq. (S3). Table S1 summarizes all the physical parameters of O(1), O(1 ) and O(2) sites after the correction of angular misalignment. The results at ambient pressure are in good agreement with that reported by Mukuda et al [33].

SM II: Strain dependent νQ -experimental and theoretical
Under strain, the peaks of O(2) sites remain essentially unchanged, while both O(1) and O(1 ) change drastically. In particular, for B c, the satellite peaks of O(1) and O(1 ) merge "coincidentally" when ε aa =−0.55%, implying that the two move at different rates under strain. The strain dependencies of ν Q and η are displayed in Fig. S3b-c. Evidently, the changes of ν Q in O(1) and O(1 ) are of opposite signs. This is because an expansive strain is induced along b-axis, i.e. ε bb >0, which is characterized by the Poisson's ratio −ε bb /ε aa =0.40 for Sr 2 RuO 4 [48]. We should note that the ratio of the slopes in ν 1Q (ε aa ) and ν 1 Q (ε aa ) is very close to Poisson's ratio.
The on-site hole contribution ν hole Q is proportional to the hole content (n) in each O orbitals, and the latter can be obtained from DFT calculations by integrating the partial density of states up to Fermi energy, viz.
The variation of n for each O orbitals are displayed in Fig. S4.  Taking O(1) p x orbital as an example, the yielded quadrupoar frequencies are (ν a , ν b , ν c ) 1,px =n 1,px (q xa , q xb , q xc ), where the ratios q xa =−2q xb =−2q xc =2.452 MHz for 17 O according to previous reports on cupartes [51]. The total quadrupolar frequency should be the sum of the contributions from all the three p orbitals for each O site.
The calculated quadrupolar frequencies ν Q and the associated asymmetry pramaeter η are shown in Fig. S4b and c, respectively.
Comparison can be made for ν Q and η between measured (Fig. S3) and calculated (Fig. S4) results. Regardless of some difference in magnitude, agreement between experiment and theory in the evolution trend upon strain effect is striking at both ν Q and η.

SM III: Spin-lattice relaxation rate
Additional evidence for vHs comes from the measurements of the spin-lattice relaxation rate [T 1 T ] −1 as shown in Fig. S5. The [T 1 T ] −1 is recorded for the central transition of the O(1) site and for field B b. As a means to extract the strain dependence of the relaxation rate in a minimum of measurement time, the recovery curves at high (ε aa = ε v ) and low strain (ε aa = 0) were established to follow the appropriate form for spin I=5/2, and dominantly magnetic relaxation governing selective irradiation of the central transition. Between these endpoints, a single recovery was recorded, with short delay time selected prior to application of the echo read sequence, so that the relaxation rate could be inferred from the recorded signal amplitude.
As shown in Fig. S5, the relaxation is maximum at the strain where the shifts are extremal, consistent with the vHs-tuning scenario. Although only a narrow temperature range is covered, a temperature dependence is clearly evident in the inset, where the behavior is contrasted to the zero strain results of Ref. [33]. The variation could originate partially or entirely from proximity to the vHs, with the remainder related to correlations. Note that the singularity in two dimensions scales as ln(t/T ), with t the relevant hopping integral, and its effect on thermodynamic properties is rapidly diminished due to thermal broadening of the Fermi function. In the main text, the NMR Knight shifts for B c were not closely examined. In part, this is because of an apparently reduced sensitivity to the vHs. In particular, for all strains |ε aa | < |ε v |, the central transition for the O(1,1 ) sites are only weakly changing and remain unresolved, indicating cancellation effects of contributions to the total shifts. Large changes are observed for |ε aa | > |ε v |, where large drops in spin susceptibility and severe line-broadening are qualitatively consistent with inhomogeneous strain within the measured sample volume and an accompanying amplified sensitivity to the inequivalent environments. In the DFT calculations for the same quantities, K 2c does show essentially full cancellation of the DOS effects, as shown in Fig. S6. On the other hand, both K 1c and K 1 c appear quite sensitive to the vHs, and, interestingly, in both Fermi and orbital terms. As discussed in the main text, there are two mechanisms by which O electrons can acquire an orbital moment: directly induced by the external field, and via spin-orbit coupling to the induced spin moment. Our calculations show the former effect in K 1c and K 1 c to be strong, and opposite in sign to the spin mechanism. In the raw calculations the amplitude of the orbital shifts is too small to ensure a full cancellation, but, as discussed in the main text, spin-orbit effects may be considerably enhanced by correlation (Ref. [43]). Assuming a semiphenomenological approach, we plot in Fig. S6 the sum of all contributions to K 1c and K 1 c , multiplying the orbital part by a factor of four, without adding any van Vleck constant. The result still show a small split between K 1c and K 1 c (albeit smaller than the measured peak widths) and an overall good agreement with the measurements. As we declared in the main text, the NMR spectra for the field parallel to c require further investigation.