Direct evidence for the absence of coupling between shear strain and superconductivity in Sr_2RuO_4

Domain: physics
Published: 2025-12-16
Editorial outcome: minor_revision

Abstract
The superconducting symmetry of Sr2RuO4 has been intensely debated for many years. A crucial controversy recently emerged between shear-mode ultrasound experiments, which suggest a two-component order parameter, and some uniaxial pressure experiments that suggest a one-component order parameter. To resolve this controversy, we use a new approach to directly apply three different kinds of shear strain to single crystals of Sr2RuO4 and investigate the coupling to superconductivity. After characterising the strain by optical imaging, we observe variations of the transition temperature Tc smaller than 10 mK%−1 as measured by low-frequency magnetic susceptibility, indicating that shear strain has little to no coupling to superconductivity. Our results are consistent with a one-component order parameter model, but such a model cannot consistently explain other experimental evidence such as time-reversal symmetry breaking, superconducting domains, and horizontal line nodes, thus calling for alternative interpretations.

Introduction
Strains in a crystal have well-defined irreducible representations (irreps) under the symmetry of a lattice. Strains are precisely coupled to the superconducting (SC) order parameter (OP) according to their symmetries, and induce changes in Tc. Thus, the application of strains to a superconductor and the observation of the changes in Tc is a precise method of gaining information on the SC OP. In addition to isotropic hydrostatic pressure, it has recently become clear that uniaxial strains are a powerful tool to investigate superconducting symmetry, since they can profoundly alter crystal symmetry1–3.

The OP of unconventional superconductivity in Sr2RuO4 is still unresolved despite 30 years of extensive research efforts5,6. Spin-singlet-like behaviour was newly established from spin susceptibility measured by nuclear magnetic resonance (NMR)7–9 and polarised neutrons10. Time-reversal symmetry breaking (TRSB), evidenced by an emerging internal magnetic field below Tc11, has been reported by several groups12–14. Magneto-optic Kerr effect (MOKE) also supports the presence of TRSB in the superconducting state15. Anomalous switching behaviour in superconducting junctions16 points to the presence of SC domains characterised by a multi-component OP with 1D or 2D irreducible representations (irreps).

Uniaxial compression in Sr2RuO4 enhances Tc from 1.5 to 3.5 K, and it is associated with changes in the Fermi surface topology (Lifshitz transition)2,17. A splitting between Tc and appearance of the TRSB phase was observed by muon spin relaxation (μSR) under uniaxial strain14, pointing to a chiral superconducting OP. However, the splitting is not observed in specific heat and elastocaloric-effect measurements under uniaxial strain18,19, leaving the issue of TRSB unsettled at present. Another peculiar behaviour of Sr2RuO4 is the jump at Tc of the shear-mode ultrasound velocity related to the elastic modulus c66 20,21. This unusual coupling between SC OP and shear strain implies that the OP has two components: either chiral with 2D irreps (e.g., dyz ± idzx), nematic with 2D irreps (e.g., dyz only, dzx only, or dyz ± dzx), or a combination of 1D irreps (e.g., d+ig, s+id). Results of uniaxial stress and hydrostatic pressure experiments can be combined to compare with results from ultrasound experiments22, but such comparison leads to both qualitative and quantitative discrepancies6. Specifically, a V-shaped kink in Tc as a function of uniaxial strain along the [110] direction was not observed, in contrast with the expectations from the observed jump in the ultrasound c66. It is crucial to resolve this discrepancy in order to reach a conclusion about the superconducting state of Sr2RuO4.

In this study, we investigate the direct effect of shear strain on the Tc of Sr2RuO4. The magnitude of sample strains at low temperatures is measured with optical imaging down to 30 K. Kinks, linear slopes, and quadratic changes in Tc are evaluated by applying three different kinds of shear strain and comparing the results with the estimates of the Ehrenfest relations from the jumps and slope changes of the elastic moduli at Tc. We find that Tc does not change within the experimental precision of ∂Tc/∂εxy = ±6 mK%−1. This result seriously challenges the interpretation of SC in Sr2RuO4 in terms of a two-component OP.

Results
Direct shear-strain quantification at low temperature
To apply shear strain, we construct piezo-sample assemblies by rigidly gluing thinly polished Sr2RuO4 crystals directly onto the surface of piezoelectric devices, as shown in Fig. 1a (see “Methods” for construction details). Deformation of a piezo-sample assembly is monitored by an optical microscope, which allows us to calculate the point-by-point in-plane displacements (ux, uy) and the in-plane shear strain εxy = (∂ux/∂y + ∂uy/∂x)/2. We note that this definition is symmetric (εxy = εyx) and that εxy = γ6/2 = ε6/2, where γ6 is the engineering strain and ε6 is the xy shear strain in Voigt notation. We show in Fig. 1b a typical optical image taken at room temperature with a voltage applied to the shear piezo Vpiezo = + 200V. By means of digital image correlation23,24, we extract the point-by-point displacements that are represented by the coloured dots. As indicated by the arrows, the bottom of the piezo-sample assembly is moving to the right (red) while the top is moving to the left (blue) due to the applied shear strain. We note that the piezoelectric device we use predominantly generates a shear displacement component ∂ux/∂y, while the complementary component ∂uy/∂x is small due to the device’s geometry and boundary constraints (Fig. S1 in the Supplementary Information). We comment on the importance of measuring both components because relaxation of shear strain could lead to insurgence of a negative ∂uy/∂x at the sample surface, which corresponds to an overall sample rotation (Fig. S2).

We calculate the average shear strain over the surface of the piezo device and the Sr2RuO4 sample as a function of Vpiezo in Fig. 1c. At room temperature, the piezo device can apply a maximum of about εxy = ± 0.05% at Vpiezo = ± 200V, corresponding to a shear strain transferred to the Sr2RuO4 of about εxy = ± 0.04%. We emphasize the advantage of this optical technique, which enables non-contact measurement of the effective shear strain directly on the sample surface, allowing us to estimate the shear strain transferred to the bulk of the sample and to neglect the uncertainty due to glue-mediated strain transfer from the piezoelectric device. The shear–voltage characteristics in Fig. 1c are nearly linear, with a small hysteresis that is typical of piezoelectric actuators25. While this hysteresis is generally negligible for the purposes of this study, it introduces an uncertainty in the shear strain that is most significant around zero applied voltage, where the upper bound estimated from the vertical width of the hysteresis is δεxy ≈ 0.02%. Importantly, this uncertainty is minimal at the largest applied voltages, where the strain response is well defined and reproducible. As shown in Fig. 1d, the shear applied at a certain voltage becomes gradually smaller at low temperature due to the decreasing capacitance of the piezoelectric device (Fig. S3). We compensate for this effect by applying a larger voltage to the piezo device at low temperature, thus achieving similar values of maximum shear strain at all temperatures.

We show in Fig. 1e that the strain transferred from the piezoelectric device to the Sr2RuO4 sample is as large as 75% at lower temperature, where a weak enhancement is possibly caused by a stiffening of the glue used to attach the sample to the piezo surface. This strain transfer is consistent with the value predicted by our finite element simulations (Fig. S2) for samples of 30 μm thickness. A direct correlation between the travel distance of the piezoelectric device and its capacitance (Fig. S3) allows us to calculate a conversion factor εxy/Vpiezo = 0.007%/100V at 2 K. Because the samples investigated in this study have similar thicknesses, we found this conversion factor to be approximately constant.

Response of Tc to shear strain
We now consider changes in Tc by shear strain. To detect the superconducting transition, we insert the piezo-sample assembly of Fig. 1a into the mutual-inductance coil set shown in Fig. 2a. The mutual-inductance technique allows us to measure the imaginary χ′′ and real χ0 parts of the alternating current (AC) susceptibility extracted from the in-phase and out-of-phase lock-in voltages Vx, Vy, respectively. These are measured as a function of temperature for different applied voltages Vpiezo in Fig. 2b, c. We use Sr2RuO4 samples with a sharp superconducting transition that appears as a dissipation peak in χ′′, with full width at half maximum of about 20 mK, and as a drop in χ0 that indicates the onset of diamagnetic susceptibility, with a transition width smaller than 40 mK. We compare in Fig. 2d two possible criteria for defining Tc: from the position of the peak in χ′′ (squares), and from the mid-point of the drop in χ0 (dots). We comment that both criteria allow us to overcome time-dependent changes in the electronic signal, which suffers from small voltage offsets over time. The values of Tc resulting from the two criteria show consistent trends, with a minor difference in absolute value. Since the peak in χ′′ is about an order of magnitude smaller than the drop in χ0, the latter provides a more reliable criterion across different samples and will be used to determine Tc throughout this work.

Using several piezo-sample assemblies with Sr2RuO4 glued along different crystalline axes, we apply continuous shear strains along the three directions shown in Fig. 2e to g: a pure εxy shear along the [100], a diagonal shear ε1x100y0 = εxx − εyy along the [110], and a c-axis shear εzx along the [001]. As shown in the insets above the panels, these correspond to B2g, B1g, and Eg strain modes, respectively. We use several samples from at least two different batches of crystals for the measurements. In contrast to ultrasound experiments where the response of Sr2RuO4 to shear modes was measured with strains oscillating at high frequencies, our technique allows us to apply static shear strains across the whole dynamic range of the piezoelectric device. Due to the C4z rotational symmetry of the lattice, a positive and a negative shear are expected to yield identical results [i.e., ΔTc(εxy) = ΔTc(−εxy)], hence a symmetric trend is expected around εxy = 0. The data in Fig. 2e to g show no change of Tc within our experimental resolution for all three kinds of applied shear strain, with neither a kink nor a quadratic trend. We estimate upper limits of a purely linear slope of the kind ∂Tc/∂ε by drawing grey shaded regions that correspond to Tc variations smaller than ±6, ±9, and ±6 mK%−1, respectively. Based on the same shaded regions, upper limits for purely quadratic slopes ∂2Tc/∂ε2 are estimated to be ±60, ±90, and ±60 mK%−2. These values will be further discussed in the following and compared with thermodynamic predictions from ultrasound experiments in Table 1.

We now consider the possible effect of thermally-induced pre-strains in Sr2RuO4 due to the difference in thermal contraction with the piezoelectric device. As we show in Fig. S4, an Sr2RuO4 sample glued along the [100] piezo-sample assembly is affected by a biaxial tensile pre-strain up to about +0.1 to +0.3%. For a [100] piezo-sample assembly, this pre-strain can be seen as a combination of compressive A1g and shear B1g strain modes. In such configuration, the voltage-driven shear-piezo device applies a pure B2g strain, a shear mode that is not present in the pre-strain. Therefore, the Tc variations that we investigated provide direct information of the SC coupling to a specific shear mode.

Tensile control experiment
To validate our technique of detecting small Tc changes with strain, we perform a similar experiment with a uniaxial tensile piezo. As shown in Fig. 3, tensile strain applied to Sr2RuO4 induces a clear Tc change in the range of several mK (as stated). As indicated by the fit to the strain curve (black line), the strain-dependence of Tc follows a quadratic behaviour. This quadratic trend is consistent with that reported by Watson et al.26, although our minimum is not at Δεxx = 0. A likely cause of this shift is the presence of pre-strains caused by the differential thermal contraction of Sr2RuO4 and the piezoelectric device27. We discuss in Fig. S4 that for an assembly of Sr2RuO4 glued over the active surface of a tensile piezo there is a biaxial pre-strain up to about +0.1 to +0.3%, which is reasonably consistent with the value of +0.1% extracted from the fit in Fig. 3. Although it is difficult to extract quantitative results from this control experiment with tensile strain, it proves that the technique used in this work is suitable to capture the smallest variations of Tc.

Comparison with uniaxial stress and ultrasound experiments
In Table 1 (left columns), we give some of the irreducible representations (irrep) of the D4h point group that apply to space group I4/mmm of Sr2RuO4 crystal structure. The couplings between different strains and superconductivity are dictated by symmetry. Due to the C4z rotational symmetry, the Tc variation under shear is expected to be symmetric and show either a linear kink for a two-component SC or a quadratic dependence for a one-component SC. For the two-component order parameter η=(ηx, ηy) that belongs to Eg irrep with components {dxz, dyz}, the Ginzburg-Landau (GL) free energy density that describes the coupling to lattice strain is given by (more details in the Supplementary Information)
fηε = r1εA1g,1|η|2 + r2εA1g,2|η|2 + r3εB2g(η*xηy + ηxη*y) + r4εB1g(|ηx|2 − |ηy|2) (1)
up to second order in ηx, ηy and first order in strain, with constant coupling parameters r. The D4h symmetry does not allow linear coupling to strains εyz and εzx of irrep Eg and modulus c44, so to lowest order these strains should not affect Tc. For a nematic or chiral OP, we have
∂Tc/∂εB2g = ±|r3| (±sign for εxy ≠ 0), ∂Tc/∂εB1g = ±|r4| (±sign for εxy ≠ 0), (2)
indicating that application of pure shear strains B2g or B1g is expected to lead to a kink in Tc around ε=0. We show again with a grey confidence interval in Fig. 4 that our experiment proves little to no coupling to εxy.

We now compare our direct measurement of shear strain with estimates from other techniques where shear strain was indirectly applied along other strains. By combining the results of other stress–strain experiments (details in the Supplementary Information and Table S1) as
∂Tc/∂εxy = 2c66(dσ1x100y0/dp hydro + ∂σz/c33), (3)
where σ1x100y0 is the uniaxial compression along the [110] direction, σzz is uniaxial compression along the z axis, and the hydrostatic pressure phydro is positive for compression (opposite sign compared to the σ). By using c66 = 65.5 GPa21 and values of the three slopes +71 mk GPa−1 22, −210 mk GPa−1 29, and +76 mk GPa−1 30, respectively, we obtain ∂Tc/∂εxy ~ 10.5 mK%−1, which is shown by the green V-shaped line in Fig. 4. This estimate is consistent with the one provided by Jerzembeck et al. |∂Tc/∂εxy| < 13 mK%−1 22.

Interestingly, we observe no measurable change in Tc under ε1x100y0, which corresponds to pure B1g shear strain (Fig. 2f). This seems in contrast to previous uniaxial stress studies along [100], where an increase in Tc was reported and attributed to the B1g symmetry channel31. In those works, the strain state includes a combination of symmetry components due to the Poisson effect, with variation of both B1g and A1g strains. By contrast, our setup applies a pure B1g shear strain, with constant A1g strain. Furthermore, the strain magnitude in our experiment is smaller, such that the Fermi surface remains far from the van Hove singularity. At present, the absence of a Tc response under pure B1g strain remains an open question, but it highlights the complexity of how different strain symmetries couple to the superconducting state.

In addition, we compare our results with estimates from the thermodynamic Ehrenfest relations
∂Tc/∂εij = −Tc(Δc ij/ΔC) (4)
where Δcij is the jump in elastic moduli at Tc and ΔC ~ 570 J m−3 K−1 21 is the jump in specific heat, with f(β) a function of the 4th order GL free energy coefficients β28. Here, the coefficients f(β) are parameters that can be estimated in the weak-coupling limit under the isotropic-band approximation28, yielding the equations and estimates presented in the central columns of Table 1. As shown in Fig. 4, the V-shaped kinks estimated from Ehrenfest relations have a much larger slope than our direct measurement. It should be noted that discrepancies are also serious for the A1g compressional modes (c11 + c12)/2 and c33: the Tc variations under constant stress are both about a factor five smaller than the expectation from the compressional-mode ultrasound jumps6. Thus, we find here an inconsistency with the Ehrenfest relations, which has to be resolved.

Finally, we consider the upper limit for quadratic variation in Tc found in this study (grey regions of Fig. 2e to g) and compare it with estimates from second order Ehrenfest relations that depend on the slope jump of elastic moduli at Tc (right columns of Table 1)28. While the quadratic term estimated from c66 is rather small, the one from (c11 − c12)/2 is very large. It is rather puzzling that a second-order coupling is so large in the absence of the first-order coupling. Such discrepancy in both shear and compressional modes suggest that the ultrasound response may contain additional contributions not given by the coupling to the SC OP. In the normal state of Sr2RuO4, the elastic moduli c11 and (c11 − c12)/2 are known to exhibit a peculiar temperature dependence compared to the other modes32. These modes involve stretching and shrinking of the in-plane Ru-O bond lengths that may cause additional coupling to the quasiparticles, although it is not clear how such coupling changes below Tc.

Comparison with thermal expansion
Another thermodynamic comparison can be made with jumps in thermal expansion coefficient αi = ∂εi/∂T using the different Ehrenfest relation
∂Tc/∂σii = −Tc(Δαi/ΔC), (5)
where C is the specific heat, and Δ refers to the jumps at Tc. Note that, unlike Eq. (4) for elastic moduli, which involves elastic couplings to the superconducting order-parameter in the free energy, this Ehrenfest relation is a more direct thermodynamic requirement. From the negative jumps Δαa = − 0.42 × 10−7 K−1 and Δαc = − 0.52 × 10−7 K−1 observed by Grube et al.33, we obtain ∂Tc/∂phydro = −360 mk GPa−1 and ∂Tc/∂σzz = + 140 mk GPa−1. These estimates are within a factor two in agreement with the results of hydrostatic pressure −210 mk GPa−1 29 and c-axis uniaxial-stress +76 mk GPa−1 30. Thus, the strong disagreements of ultrasound jumps in both shear and compression modes suggest that the ultrasound jumps may contain additional contributions which are not properly included in the Ehrenfest relation of Eq. (4).

To conclude, we investigated the coupling of Sr2RuO4 superconductivity to shear strain. For this purpose, we developed a technique where thin sample crystals were glued on shear-strain piezoelectric devices that allowed us to directly apply static shear strains along three symmetry channels. The magnitudes of strains were evaluated from optical imaging down to 30 K. We found no detectable change of Tc, with a resolution limit smaller than 10 mK%−1. This poses strict limits on the possible nature of the superconducting order parameter, pointing at ruling out a two-component order parameter. We note, however, that there remains an unresolved discrepancy between the results of ultrasound and static deformation (stress/strain) experiments, which requires further investigation. Our analysis suggests that the Sr2RuO4 superconductivity requires a one-component OP, in contrast to the ultrasound results. The new method developed in this study can be readily applied to other superconductors for which a two-component OP is suggested, such as UPt3, or to other materials showing different kinds of phase transitions where a two-component order parameter may emerge.

Methods
Piezo-sample assemblies and sample crystals
In order to apply various shear strains to Sr2RuO4 samples, we used a shear piezoelectric device (Thorlabs, PL5FB) consisting of a single piezoelectric ceramic layer of PbZrxTi1−xO3 (PZT) with gold-plated electrodes on the top and bottom surfaces. The device provides a free-stroke displacement of 1.3 μm with the application of ±200V at room temperature. We cut the commercial 5 × 5 × 0.5 mm3 shear devices into four pieces of about 2 × 2 × 0.5 mm3 with a diamond-disc cutter, and polished the cut planes using a polishing machine (Musashino Denshi, MA-200) with a diamond slurry of 3 μm. We left a suitable surface roughness to secure strong adhesion to the sample crystal and to provide a clear optical contrast for measuring actual strain. For the control experiment, we used a tensile strain device (Thorlabs, PA3CE, dimension 2 × 2 × 2 mm3) consisting of piezoelectric layers stacked in series. The device provides a free-stroke expansion displacement of 2 μm with the application of +100V at room temperature.

The SrRuO4 single crystals used in this study were grown by a floating-zone method using an image furnace (Canon Machinery, SC-E15HD)34. The crystal orientation was determined by X-ray Laue patterns (Rigaku, RASCO BL-II with a CCD camera). We selected SrRuO4 crystals with a sharp superconducting transition at Tc = 1.5 K, ensuring the absence of metallic-Ru inclusions that would induce broad enhancement of Tc up to 3.5 K35. The SrRuO4 crystals were cut into a typical size of 0.4 × 1 mm2 and polished down to a thickness of 30 ± 5 μm with a diamond slurry of 3 μm. Each thin crystalline sample was glued to the active side surface of a piezo device with epoxy (EPO-TEK, 353ND) cured at 110–120°C for 20 min. The lap shear of EPO-TEK 353ND (13.5 MPa) is much larger than the typical lap shear of other low-temperature epoxy, such as Stycast 2850FT (3 MPa, with catalyst 24LV), ensuring good adhesion between the SrRuO4 and the piezo element.

High-voltage electrodes for the piezo elements were provided by copper wires (diameter 0.14 mm) attached using silver paste (EPO-TEK, H20E) cured at 120°C for 2 h. For SC measurements, each piezo-sample assembly was mounted on a bamboo holder on which ten insulated copper wires (diameter 0.1 mm) were used for thermal anchoring, glued with varnish (GE, 8530) as shown in Fig. 2a.

Image correlation to optically measure the strains
We employed an optical-imaging technique to quantify the shear and tensile strain applied by the piezoelectric devices to SrRuO4 samples. Digital photographs were taken with different Vpiezo using an optical microscope (MicroSupport, SS306) with 10–40× zoom. Digital image correlation (DIC) with DICe software was employed to compute in-plane displacements in the x and y directions23,24. The average strain over the sample and piezo surfaces was calculated line-by-line averaging the displacement data, while the error was estimated from the standard deviation of the data.

Calibration of strain
The conversion factor from Vpiezo to shear εxy and uniaxial εxx was measured at different temperatures between 30 and 295 K in an optical cryostat (Thermal block, custom design). Since the piezo displacement per unit volt depends on temperature, the capacitance of the piezo was used to bridge between the measurements at 30 K, lowest temperature accessible by our optical cryostat, and the measurements of Tc, which were performed at and below 2 K. The strain transfer ratio discussed in Fig. 1e was estimated on different samples and found to be rather constant among the several samples due to their comparable thicknesses.

Determination of Tc
Evaluation of Tc with exceptionally high precision and accuracy was crucial in this study. We adopted alternating-current (AC) susceptibility measurements using a mutual-inductance coil set36 and a lock-in amplifier (Stanford Research Systems, SR830) to probe both the imaginary χ′′ and real χ0 parts of the AC susceptibility defined as χ = ∂M/∂H = −iχ′′ + χ0. By driving an alternating current of the form I AC(t) = I0 cos(2πft) in the excitation coil, the resulting voltage in the pickup coil is V AC(t) = (1/α)Vsample I0 f((−χ′′ cos(2πft) + χ0 sin(2πft))), with α a constant related to the geometry of the coils, Vsample the sample volume, I0 the amplitude of the excitation current, and f the frequency37. The coil set consisted of a pair of in-series pick-up/reference coils (400 turns each, internal diameter 5 mm, length 5.5 mm, copper wires diameter 50 μm) wound in opposite directions, and an excitation coil (2400 turns, internal diameter 6.0 mm, length 36 mm, copper wires diameter 100 μm), producing a magnetic field of 84 μT mA−1. An alternating current (frequency 17.7 kHz, root mean squared amplitude of 1 mA) was sourced to the excitation coil using the voltage output of the lock-in amplifier and a series resistance of 10 kΩ. In-phase (Vx) and out-of-phase (Vy) lock-in components were calibrated at 2 K by considering the phase delay of the circuit involving only the excitation coil. Several values of current were tested to ensure that the effect of heating was below 1 mK near Tc. The sample temperature was measured with a thermometer (Lake Shore, Cernox CX-1030-SD-HT) mounted on the copper plate to which each piezo-sample assembly and mutual-inductance coil set were thermally anchored.

Data availability
The data that support the findings of this study is available from the corresponding author upon request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Mattoni et al. presents a noteworthy study to measure the response of Tc to shear strains in Sr2RuO4 (SRO). The coupling of Tc to shear strains in SRO has received a lot of attention in the last several years due to its implications on the superconducting pairing of SRO. In particular, as the authors note, a linear coupling between Tc and shear strain indicates a two-component order parameter in this highly studied unconventional SC. Multiple studies involving various techniques have provided insights regarding this without clear resolution, and any additional evidence to show the presence/absence of this coupling is important for the field. The experiments have been performed carefully and the data analyzed through comparison to existing literature via thermodynamic relations. However, I have some questions for the authors before I can recommend publication in Nature Communications.\n1. How accurate is the strain estimation on the sample based on the optical method the authors use? Did they perform any simulations to validate the strain on sample as a function of temperature, given it’s a complicated system of piezo stack attached to sample with glue and there are thermal stresses involved when cooled down. Is there any estimation of strain relaxation from where the sample is attached to piezo to top surface? The exact strain state of the sample is critical for interpreting the results.\n2. It is surprising to me that no change in Tc is observed under a B1g shear strain (Fig 2(f)). This strain is supposed to push one of the Fermi surfaces towards the van Hove singularity and strongly enhance Tc (see Barber et al., PRL 120, 076602, 2018 and Barber et al., PRB 100, 245139, 2019). The fact that such response is not seen calls into question the observations under B2g shear strain, which is the main result of this paper. Can the authors please comment on this?\n3. As the authors mention, there are pre-strains of up to 0.3% in the sample due to cooldown. Is this seen in their optical imaging technique? If yes, are the shear strains on top of that? Also, the shear strains are of much smaller magnitude compared to 0.3%, so their effect may be tiny or lost in the presence of much larger pre-strains. In comparison, the ultrasound measurements are under zero-strain conditions and strain cell measurements can traverse zero strain at low temperatures. I am curious to know if zero strain can be traversed in the experiment described here.", "reviewer_2": "Journal: Nature Communications\nSubmission number: NCOMMS-25-50704\nTitle: Direct evidence for the absence of coupling between shear strain and superconductivity in Sr2RuO4\nAuthors: Giordano Mattoni et al.\nSummary:\nThis manuscript reports experimental results regarding the superconducting state of strontium ruthenate (SRO). SRO is sometimes referred to as an archetype of unconventional superconductivity because it can be synthesized extremely purely and those provide exceptionally clean data. However, after more than 30 years of research, something as fundamental as the symmetry of the order parameter of SRO remains an open question. The reason is that although every thermodynamics (apart from ultrasounds) measurement suggest that the superconducting order parameter be one-component, ultrasounds experiments, among others, suggest that it is two-component. These discrepancies are very challenging to resolve, as they are fairly fundamental.\nIn this work, the others explore a different way to measure a similar quantity as what is done in ultrasounds, but instead of interacting with the solid and in particular its superconducting state via ultrasounds to generate the shear modes that suggest two-component, they directly apply the appropriate deformation mechanically and measure the change in critical temperature. They find the opposite conclusion then ultrasounds, namely that the superconducting order parameter be one-component.\nThis very important work offers a new perspective of a mystery that someone we were not able to solve yet. Although it agrees with many other scenarios, it opens a new question as to how this method differs from ultrasounds. Thus, I believe this work deserves to be published in a prestigious journal such as Nature Communications.\nReview:\nThe Results section starts by explaining how strain is applied statically by using a shear piezo. The amplitude of the strain deformation relies heavily on digital image correlation, a method I’m not familiar with. My main question is how reliable is it? There doesn’t seem to be any reference about it in the text or Methods section. In particular, looking at Fig. 1c, one can clearly see an asymmetry between positive and negative V_piezo. And why does the curve have the form of a hysteresis? I would expect those deformations to be elastic and reversible.\nOnce this is established, the authors present their measurements of the critical temperature as a function of shear strain. Overall, they show that Tc does not change sufficiently to be significant, with either of the three shear strains applied. To extract Tc, they look at the peak in the imaginary part of the AC susceptibility and the drop in its real part. They conclude that the drop is more reliable. Here are some comments on this section:\nIn Fig. 2b, c and d, the curves and points are attributed a color, which I have no idea what it is. Are those different samples? The colors don’t seem to make any sense as they are almost attributed in random order. Could the authors please specify?\nThe fact that the drop in the real part is more reliable is surprising to me. Indeed, the peak is much sharper than the drop, so it appears more intuitive to me to take the peak. Moreover, the peak is defined as the point where the first derivative is zero, while the drop, I suppose, is where the second derivative is zero. Typically, the higher the derivative, the more noisy, yet somehow it appears to be the contrary here, which is strange to me.\nI don’t quite understand how the AC susceptibility is obtained here. There is no reference or equation in the main text, and when inspecting the Methods section, the susceptibility is the derivative of the magnetization with respect to the applied field. I don’t understand how this leads to a complex number. I would appreciate a bit more formal mathematical description of what’s being plotted. I also don’t understand from the plot why the y-label is V_x and V_y.\nAgain in Fig. 2, I would suggest in e, f and g to write the irreducible representations associated to each shear mode in bigger font. I consider that this is the important information there and it should be more visible.\nAfter presenting data suggesting that the critical temperature doesn’t change with applied shear strains, they apply a typical tensile strain as a control, to demonstrate that their experiment works. In particular, they recover the effect of uniaxial strain on the critical temperature. This is a good idea, as we can clearly see the evolution of the critical temperature as a function of strain. However, I don’t really understand why zero strain is not at the minimum of the parabola. This is explained in the text, but then it is quite confusing to me that the authors don’t write the effective e_xy instead of setting it to zero. This is confusing with the rest of the literature.\nThen, we go into the Discussion section, where the authors compare the present data with that reported in the ultrasounds papers. Although I believe the main point of this section is to say that the two methods present dramatically different results, I thought it was very difficult to read. Here are some reasons why:\nThe section start by defining the Ginzburg-Landau free energy for an Eg order parameter of the form d_xz + (i) d_yz. This is also detailed in the Supplementary Material. I do not understand why the authors focus on this order parameter, as it is one choice among many possibilities and I do not believe that it is the leading candidate for the two-component order parameter. Indeed, I seems pretty clear to me that the leading component would be d_xy (actually discussed by some of my own works Refs [1, 2, 3]. The second component is often suggested to be a g-wave, so the two-component order parameter would be B_1g + A_2g rather than E_g. It wasn’t clear at all for me whether the discussion was also valid for that scenario. In fact, I don’t think taking any specific scenario is necessary, and it seems like invoking E_g without discussing the other ones is not very fair to the current discussion on the proposed symmetries of the order parameter.\nIn Fig. 4, the authors show what the other results should look like in comparison to the results from this paper. Similarly in Table 1, the results are presented. Something that this work clearly suggests is that there is a misunderstanding in interpreting ultrasounds experiments, although they don’t clearly state it. I don’t think they should avoid the question and if they cannot explain why their results differ so much, then they should still mention it clearly and raise it as a new question waiting to be answered. This point is not present in either the abstract or the conclusion and I would appreciate understanding why.\nMore on that last point, they authors kind of hint, in the Section 1A of the Supplementary Material, that the difference between ultrasounds and their work could come from the frequency-dependence of the ultrasounds, while their work is static. In particular, I’m talking about Eq. S9. From my own works Refs [2, 3], we propose an odd-frequency A_2g as the second component, along with B_1g as the first. Odd-frequency would substantially change the expression S9. In particular, it should have a zero value at zero frequency, and non-zero elsewhere. Could this scenario help explain the different interpretations between the current work and the ultrasounds?\nOverall, I think the results presented in this work are extremely important for the field and bring an old open question to a different direction, which could eventually lead to its resolution. Therefore, I support this work being published in Nature Communications. However, I think it could benefit from having its presentation slightly revised, based at least on the comments provided in this review.", "reviewer_3": "Mattoni et al report direct shear strain measurements on the unconventional superconductor Sr2RuO4, with the aim of resolving the puzzle of its order parameter symmetry. This comes against the backdrop of mutually inconsistent results from ultrasound Ref. [20,21] and uniaxial strain [22] experiments, which strongly motivate the present study. The results of the study broadly agree with the findings of Ref. [22], providing strong experimental evidence that there is a fundamental open question about the interpretation of the ultrasound experiments. I believe these results to be significant and valuable to the community. What is more, the experimental methodology introduced here to directly apply and quantify shear strain is new, and I believe broadly valuable, beyond this specific application to Sr2RuO4.\nHowever, before I can recommend publication, I would like to resolve a few questions:\n1) I do not understand the statement that the diagonal shear strain is equivalent to e_xx - e_yy. Isn’t e_xx - e_yy precisely the uniaxial strain component that generates the large quadratic change of Tc with strain? How is that consistent with the <90mK%^-2 reported in Table 1?\n2) The measurements of Tc vs e_{110} were reported in Ref. [1], and e_{110} is dominantly e_{xy}. Ref [1] reports a change of Tc of 100mK/%. The authors should comment on the possible origin of discrepancy between that result and <6mK/% found here.\n3) I think it is a bit misleading to refer to the ‘pre-strain’ as biaxial in the discussion of the tensile control experiment, as it is its B1g component that leads to the offset of 0 (if I understood that correctly)\n4) To better clarify the experimental configuration, a coordinate system (x, y, z) should be noted on Fig. 1(a,b, including the inset)\n5) While I agree with the authors that the hysteresis typical for these piezos, the systematic errors it introduces in strain calibration should be estimated."}, "author_response": "We thank the reviewer for considering our study to be “noteworthy” and for describing our experiments as “carefully performed.” We agree that providing additional evidence on the presence or absence of coupling to shear strain is important for the field, and we respond to all the reviewer’s comments in detail below.\n1.1: How accurate is the strain estimation on the sample based on the optical method the authors use? Did they perform any simulations to validate the strain on sample as a function of temperature, given it’s a complicated system of piezo stack attached to sample with glue and there are thermal stresses involved when cooled down. Is there any estimation of strain relaxation from where the sample is attached to piezo to top surface? The exact strain state of the sample is critical for interpreting the results.\nFirst of all, we would like to clarify that our shear strain experiment does not use a piezoelectric stack, but rather a single-piece piezoelectric actuator (Methods, bottom-left of Page 7). This setup is mechanically simpler and more predictable than multilayer stacks, which helps reduce uncertainties in how strain is applied to the sample. The sample is glued directly to the piezo, which allows the strain to be transmitted through the glue layer into the sample.\nSecond, we note that in our measurements we image the top surface of the sample using an optical method. Because any strain applied by the piezo must pass through the glue and then across the sample thickness, it is possible that some strain is lost along the way. However, since we are directly measuring the top surface, this gives us a lower bound on the strain that actually reaches the sample because the strain is certainly larger close to the piezo surface.\nTo better understand how much strain is transferred through the sample, we have now performed finite element simulations of shear strain for different sample thicknesses. These simulations show that for a sample thickness of 30 µm, which matches the thickness of the samples used in our experiments, about 80% of the shear strain applied by the piezo reaches the top surface. We note that this number is in good agreement with the experimentally measured strain transfer of 75% shown in Fig. 1e of the main text. This supports the accuracy of our strain estimation and helps confirm that the strain that we measure is a good estimate of the sample’s actual strain.\nFor the reviewer's convenience, we report the newly added Supplementary Figure S2 below.\n1.2: It is surprising to me that no change in Tc is observed under a B1g shear strain (Fig 2(f)). This strain is supposed to push one of the Fermi surfaces towards the van Hove singularity and strongly enhance Tc (see Barber et al., PRL 120, 076602, 2018 and Barber et al., PRB 100, 245139, 2019). The fact that such response is not seen calls into question the observations under B2g shear strain, which is the main result of this paper. Can the authors please comment on this?\nWe thank the reviewer for this important and thoughtful comment. We agree that there is a discrepancy between our observation of a negligible change in Tc under B1g shear strain and prior uniaxial stress experiments along [100]. Uniaxial stress experiments observe that Tc increases with uniaxial stress and they associate this Tc increase with the B1g symmetry channel. This association was justified because uniaxial strain is mainly in the B1g channel, considering the contributions from the Poisson effect. The additionally induced A1g component (essentially corresponding to hydrostatic pressure and c-axis pressure which are known to suppress Tc) is then considered to be smaller. On this background, we cannot explain why we do not observe any comparable Tc increase with B1g strain. This is an experimental fact for which we do not have a conclusive explanation. Our setup is different because it applies a pure shear strain B1g with no accompanying variation of the non-shear component A1g. Note also that the deformations that our device induces are smaller than those by the uniaxial strain setup, such that the Fermi surface remains still very far from the VHS.\nWe agree that understanding the full response of Tc to various strain symmetries remains an open and subtle problem. However, the absence of a Tc response under B1g strain in our geometry does not, in our view, undermine the robustness of the distinct and reproducible response under B2g shear, which remains the central result of this work.\nWe now added the following discussion in the top-left of page 6 of the main text:\nInterestingly, we observe no measurable change in Tc under exy_110, which corresponds to pure B1g shear strain (see Fig. 2(f)). This seems in contrast to previous uniaxial stress studies along [100], where an increase in Tc was reported and attributed to the B1g symmetry channel [Barber et al., Phys. Rev. B 100, 245139 (2019)]. In those works, the strain state includes a combination of symmetry components due to the Poisson effect, including a significant A1g contribution. By contrast, our setup applies a pure B1g shear, with negligible variation of the A1g component. Furthermore, the strain magnitude in our experiment is smaller, such that the Fermi surface remains far from the van Hove singularity. At present, the absence of a Tc response under pure B1g strain remains an open question, but it highlights the complexity of how different strain symmetries couple to the superconducting state.\n1.3: As the authors mention, there are pre-strains of up to 0.3% in the sample due to cooldown. Is this seen in their optical imaging technique? If yes, are the shear strains on top of that? Also, the shear strains are of much smaller magnitude compared to 0.3%, so their effect may be tiny or lost in the presence of much larger pre-strains. In comparison, the ultrasound measurements are under zero-strain conditions and strain cell measurements can traverse zero strain at low temperatures. I am curious to know if zero strain can be traversed in the experiment described here.\nWe agree with the reviewer that it would be very informative to directly observe the thermal contraction-induced pre-strain using our optical imaging technique. However, this measurement is extremely challenging because the thermal contraction of our optical setup along the z axis causes the sample to drift out of focus during cooldown, limiting our ability to track it reliably. The pre-strains of up to 0.3% are thus an upper bound based solely on the estimated thermal contraction mismatches derived from material properties from literature. The actual pre-strains in the sample are likely smaller.\nAs we discuss in Supplementary Fig. S4, a key point is that the thermally induced pre-strain and the piezo-driven strain lie in different symmetry channels. For example, in the configuration where the piezo voltage applies strain in the B2g channel (i.e., [100] shear sigma_xy), the pre-strain from thermal contraction lies in the B1g channel, or vice versa. This separation of symmetry channels is crucial, as it allows us to study the effect of the applied shear strain around the zero value in one channel, even when it is small in magnitude compared to the thermal pre-strain in the other channel. Importantly, we are able to sweep the piezo voltage to traverse zero strain in the channel of interest.\nWe have now improved the description of Supplementary Figure S4 to clarify this point as follows.\nWe thank the reviewer for the thoughtful assessment of our work. We appreciate the recognition that our approach, directly applying shear deformations, offers a new and complementary perspective to ultrasound-based techniques, and that it “offers a new perspective of a mystery that so far we were not able to solve.” We hope our responses to all reviewer's comments below will support the reviewer’s view that this work “deserves to be published in a prestigious journal such as Nature Communications.”\n2.1: The Results section starts by explaining how strain is applied statically by using a shear piezo. The amplitude of the strain deformation relies heavily on digital image correlation, a method I’m not familiar with. My main question is how reliable is it? There doesn’t seem to be any reference about it in the text or Methods section.\nDigital Image Correlation (DIC) is a well-established, non-contact optical technique that enables high-precision measurement of surface strain and displacement by tracking local contrast features across a sequence of digital images. The technique achieves sub-pixel accuracy and has been widely adopted in materials science, solid mechanics, and structural engineering for characterizing mechanical properties. The methods for DIC have been developed and refined over several decades, making DIC a reliable standard tool in both academic research and industrial applications. To clarify the reliability and broader usage of this technique, we have now added the following two representative references: [Schreier et al., Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications (2009)] and [McCormick & Lord, Materials Today 13, 52–54 (2010)] as Refs. [23] and [24].\n2.2: In particular, looking at Fig. 1c, one can clearly see an asymmetry between positive and negative V_piezo. And why does the curve have the form of a hysteresis? I would expect those deformations to be elastic and reversible.\nHysteresis is an intrinsic characteristic of piezoelectric actuators, caused by the nonlinear and history-dependent response of ferroelectric domains within the material (see for example [Damjanovic, The Science of Hysteresis, Vol. 3, 2006]). The hysteresis loop observed in Fig. 1c reflects this well-known behavior and is unrelated to the elastic shear deformations experienced by the Sr2RuO4 sample.\nThe asymmetry between positive and negative V_piezo arises from the initial polarization state of the actuator. In this case, the actuator state is not centered within the hysteresis loop but starts from one of its edges, resulting in an offset response.\nDue to the presence of hysteresis, there is an uncertainty in the conversion from applied voltage to shear strain, which is largest at V_piezo = 0. We estimate this maximum uncertainty from the vertical width of the hysteresis loop in Fig. 1c to be approximately δe_xy = 0.02%. This uncertainty becomes negligible at the edges of the hysteresis loop, thus not affecting our estimates for dTc/de_xy in Fig. 2. We have now used different symbols (open/closed) in Fig.2 to indicate positive/negative voltage sweep directions, and added this clarification with the corresponding new Ref. [25] at the bottom of Page 2 in the main text as:\nThe shear–voltage characteristics in Fig. 1c are nearly linear, with a small hysteresis that is typical of piezoelectric actuators [25]. While this hysteresis is generally negligible for the purposes of this study, it introduces an uncertainty in the shear strain that is most significant around zero applied voltage, where the upper bound estimated from the vertical width of the hysteresis is δe_xy ≈ 0.02%. Importantly, this uncertainty is minimal at the largest applied voltages, where the strain response is well defined and reproducible.\n2.3: Once this is established, the authors present their measurements of the critical temperature as a function of shear strain. Overall, they show that Tc does not change sufficiently to be significant, with either of the three shear strains applied. To extract Tc, they look at the peak in the imaginary part of the AC susceptibility and the drop in its real part. They conclude that the drop is more reliable. Here are some comments on this section:\nIn Fig. 2b, c and d, the curves and points are attributed a color, which I have no idea what it is. Are those different samples? The colors don’t seem to make any sense as they are almost attributed in random order. Could the authors please specify?\nWe thank the reviewer for pointing out that the meaning of the color coding in Fig. 2b–d was missing. The colours represent the chronological order in which the measurements were taken, and are used to help visualize any possible time-dependent effects or drifts. We have now added a colour scale to panel d to clarify this, and updated the figure caption accordingly.\n2.4: The fact that the drop in the real part is more reliable is surprising to me. Indeed, the peak is much sharper than the drop, so it appears more intuitive to me to take the peak. Moreover, the peak is defined as the point where the first derivative is zero, while the drop, I suppose, is where the second derivative is zero. Typically, the higher the derivative, the more noisy, yet somehow it appears to be the contrary here, which is strange to me.\nWhile the peak in X’’ for sample S5 is sharply defined, it is important to note that this feature is only about 0.5 μV above the baseline in Fig.2b. In contrast, the drop observed in X’ spans approximately 6 μV in Fig. 2c, which is more than an order of magnitude larger. Because of this significantly larger signal, the drop in X’ is generally more reliable for determining Tc, especially in the case of small samples where signal levels are low. For samples different from S5, the peak in X’’ is less well-defined or obscured by noise, whereas the drop in X’ remains consistently robust and clearly visible. This is the reason why we chose the drop in X’ to identify Tc.\nWe now added the following sentence in the left side of Page 4 of the main text to clarify this point:\nSince the peak in X’’ is about an order of magnitude smaller than the drop in X’, the latter provides a more reliable criterion across different samples and will be used to determine Tc throughout this work.\n2.5: I don’t quite understand how the AC susceptibility is obtained here. There is no reference or equation in the main text, and when inspecting the Methods section, the susceptibility is the derivative of the magnetization with respect to the applied field. I don’t understand how this leads to a complex number. I would appreciate a bit more formal mathematical description of what’s being plotted. I also don’t understand from the plot why the y-label is V_x and V_y.\nThe AC susceptibility is measured by driving an alternating current I_AC(t) = I_0 cos(2πft) with frequency f through an excitation coil and measuring the induced voltage in the pickup coil, V_AC(t). This voltage is measured using a lock-in amplifier, which separates the signal into two components: the in-phase response (Vx) and the out-of-phase response (Vy). Since these are the raw quantities we measure (in microvolts), we plot them directly on the y-axis in the panels of Fig. 2.\nAs described in [Nikolo M., Am. J. Phys. 63(1), 1995], the time-dependent signal measured by the lock-in amplifier can be written as\nV_AC(t) = (1/alpha) * V_sample * I_0 * f * (-χ'' cos(2π f t) + χ' sin(2π f t)),\nwhere alpha is a constant related to the geometry of the coils, V_sample is the sample volume, I_0 is the amplitude of the excitation current, and f is its frequency. This expression shows that the in-phase component (Vx) corresponds to the imaginary part of the susceptibility (χ''), while the 90-degree out-of-phase component (Vy) corresponds to the real part (χ').\nWe have now added this equation to the Methods section (left side of Page 8) and included a reference to the paper above as [37] for readers seeking a more detailed explanation.\n2.6: Again in Fig. 2, I would suggest in e, f and g to write the irreducible representations associated to each shear mode in bigger font. I consider that this is the important information there and it should be more visible.\nWe appreciate the reviewer’s suggestion and have increased the font size of the irreducible representations in panels e, f, and g to 8pt to improve their visibility. Please note, however, that the final font sizes may be adjusted during the production process to comply with Nature Communications' formatting standards.\n2.7: After presenting data suggesting that the critical temperature doesn’t change with applied shear strains, they apply a typical tensile strain as a control, to demonstrate that their experiment works. In particular, they recover the effect of uniaxial strain on the critical temperature. This is a good idea, as we can clearly see the evolution of the critical temperature as a function of strain. However, I don’t really understand why zero strain is not at the minimum of the parabola. This is explained in the text, but then it is quite confusing to me that the authors don’t write the effective e_xy instead of setting it to zero. This is confusing with the rest of the literature.\nWe thank the reviewer for recognizing the value of our control experiment using the tensile piezo device. In this case, the presence of thermally induced pre-strain, which is in the same symmetry channel as the voltage-induced uniaxial strain, means that zero piezo voltage does not correspond to zero effective uniaxial strain. We believe that the reviewer is questioning why we do not write the effective e_xx (and not e_xy, which was possibly a typo in their comment).\nTo clarify this point, we have now updated the x-axis label in Fig. 3 of the main text from e_xx to Δe_xx to explicitly indicate that the plotted strain is the additional strain applied on top of an unknown pre-strain. As discussed in our response to Reviewer 1.3, it is unfortunately difficult to precisely determine the magnitude of this pre-strain. However, based on a parabolic fit to the experimental data, we estimate that the minimum of the curve should occur at Δe_xx ≈ +0.1%, suggesting the presence of a pre-strain of similar magnitude.\nSince this is a control measurement and the effective value of uniaxial strain is not central to our main conclusions, we hope that this clarification and updated figure will satisfactorily address the reviewer’s concern.\n2.8: Then, we go into the Discussion section, where the authors compare the present data with that reported in the ultrasounds papers. Although I believe the main point of this section is to say that the two methods present dramatically different results, I thought it was very difficult to read. Here are some reasons why:\nThe section start by defining the Ginzburg-Landau free energy for an Eg order parameter of the form d_xz + (i) d_yz. This is also detailed in the Supplementary Material. I do not understand why the authors focus on this order parameter, as it is one choice among many possibilities and I do not believe that it is the leading candidate for the two-component order parameter. Indeed, I seems pretty clear to me that the leading component would be d_xy (actually discussed by some of my own works Refs [1, 2, 3]. The second component is often suggested to be a g-wave, so the two-component order parameter would be B_1g + A_2g rather than E_g. It wasn’t clear at all for me whether the discussion was also valid for that scenario. In fact, I don’t think taking any specific scenario is necessary, and it seems like invoking E_g without discussing the other ones is not very fair to the current discussion on the proposed symmetries of the order parameter.\nFirst, we would like to note that we are unfortunately unable to see the references [1, 2, 3] cited by the reviewer, which may have been removed during the editorial or review process.\nWe focus on the Eg order parameter as an illustrative example that allows us to discuss the coupling between a two-component order parameter and the strain in different deformation channels. Eg has the advantage that it couples to all relevant representations of the strain. This, in turn, also provides the basic expressions for the Ehrenfest relations for different ultrasound polarisations which we list up to prefactors of the order one in Table I. Treating the case of d+ig would lead to analogous results with the exception that some deformation channels do not couple and therefore would not give rise to Ehrenfest relations. It is not the aim of this publication to present an exhaustive list of possible cases of multi-component order parameters. We added the following comment on the case of d+ig in the Supplementary Information, bottom of Page 15 of the blue-marked document. In any case, our results give no evidence for a two-component order parameter of any kind.\nWe restrict our discussion here to the chiral Eg superconducting phase to illustrate the coupling of a two-component order parameter to strain; Eg is the only two-dimensional even-parity irreducible representation of the point group D4h. Other two-component order parameters discussed in the literature are composites of two different one-dimensional representations, such as the d_x2-y2 + i g_xy(x2 - y2) state [Kivelson et al., 2020; Ghosh et al., 2021; Maeno et al., 2024], combining B1g for d_x2-y2 and A2g for g_xy(x2 - y2), assumed to have nearly the same transition temperature.\nThe advantage of considering the Eg order parameter is that all examined strain channels can couple and allow us to derive the corresponding Ehrenfest relations as listed in Table 1 (up to some prefactors of order one). This is not the case for any of the composite two-component order parameters. With this choice, we do not intend to promote any interpretation of the realised order parameter for this superconductor, as our experimental results do not specifically support a two-component order parameter.\n2.9: In Fig. 4, the authors show what the other results should look like in comparison to the results from this paper. Similarly in Table 1, the results are presented. Something that this work clearly suggests is that there is a misunderstanding in interpreting ultrasounds experiments, although they don’t clearly state it. I don’t think they should avoid the question and if they cannot explain why their results differ so much, then they should still mention it clearly and raise it as a new question waiting to be answered. This point is not present in either the abstract or the conclusion and I would appreciate understanding why.\nWe now discuss the discrepancy in clearer terms in the manuscript and added the following in the conclusions, left of Page 7:\nWe note, however, that there remains an unresolved discrepancy between the results of ultrasound and static deformation (stress/strain) experiments, which requires further investigation.\n2.10: More on that last point, they authors kind of hint, in the Section 1A of the Supplementary Material, that the difference between ultrasounds and their work could come from the frequency-dependence of the ultrasounds, while their work is static. In particular, I’m talking about Eq. S9. From my own works Refs [2, 3], we propose an odd-frequency A_2g as the second component, along with B_1g as the first. Odd-frequency would substantially change the expression S9. In particular, it should have a zero value at zero frequency, and non-zero elsewhere. Could this scenario help explain the different interpretations between the current work and the ultrasounds?\nIn the Supplementary Information, we discussed the inconsistencies among the ultrasound experiments, which were attributed to the frequency dependence of the measured renormalization of the elastic constants at the phase transition. This does not affect the interpretation of our static experiment. Moreover, the expression in Eq. S9 describes the non-resonant coupling of ultrasound to the superconducting order parameter and corresponds to overdamped dynamics. This description does not specifically rely on microscopic details apart from the magnitude of phenomenological parameters. Thus, odd-frequency pairing would not be directly reflected in Eq. S9.\nThe inconsistencies between our static and the ultrasound experiments are connected with the Ehrenfest relations, which are seemingly violated. At present, we have no solution to this discrepancy, but our results add new “data points” to the ongoing discussion.\nOverall, I think the results presented in this work are extremely important for the field and bring an old open question to a different direction, which could eventually lead to its resolution. Therefore, I support this work being published in Nature Communications. However, I think it could benefit from having its presentation slightly revised, based at least on the comments provided in this review.\nWe thank the reviewer again for the positive and encouraging comments. We believe that by addressing the points raised we significantly improved our work, and we hope that it is now suitable for publication.\nWe thank the reviewer for recognizing the relevance of our results to the ongoing discussion on the order parameter symmetry of Sr₂RuO₄. We also appreciate the comments on the broader applicability of the shear strain methodology introduced in this work. Such feedback encourages us and strengthens our confidence in the impact of this study on the community.\n3.1: I do not understand the statement that the diagonal shear strain is equivalent to e_xx - e_yy. Isn’t e_xx - e_yy precisely the uniaxial strain component that generates the large quadratic change of Tc with strain? How is that consistent with the <90mK%^-2 reported in Table 1?\nWe agree that there is a discrepancy between our observation of a negligible change in Tc under B1g shear strain and prior uniaxial stress experiments along [100]. Uniaxial stress experiments observe that Tc increases with uniaxial stress and they associate this Tc increase with the B1g symmetry channel. This association was justified because uniaxial strain is mainly in the B1g channel, considering the contributions from the Poisson effect. The additionally induced A1g component (essentially corresponding to hydrostatic pressure and c-axis pressure, which are known to suppress Tc) is then considered to be smaller. On this background, we cannot explain why we do not observe any comparable Tc increase with B1g strain. This is an experimental fact for which we do not have a conclusive explanation. Our setup is different because it applies a pure shear strain B1g with no accompanying variation of the non-shear component A1g. Note also that the deformations that our device induces are smaller than those by the uniaxial strain setup, such that the Fermi surface remains still very far from the VHS.\nWe now added the following discussion in the top-left of page 6 of the main text:\nInterestingly, we observe no measurable change in Tc under exy_110, which corresponds to pure B1g shear strain (see Fig. 2(f)). This seems in contrast to previous uniaxial stress studies along [100], where an increase in Tc was reported and attributed to the B1g symmetry channel [Barber et al., Phys. Rev. B 100, 245139 (2019)]. In those works, the strain state includes a combination of symmetry components due to the Poisson effect, including a significant A1g contribution. By contrast, our setup applies a pure B1g shear, with negligible variation of the A1g component. Furthermore, the strain magnitude in our experiment is smaller, such that the Fermi surface remains far from the van Hove singularity. At present, the absence of a Tc response under pure B1g strain remains an open question, but it highlights the complexity of how different strain symmetries couple to the superconducting state.\n3.2: The measurements of Tc vs e_{110} were reported in Ref. [1], and e_{110} is dominantly e_{xy}. Ref [1] reports a change of Tc of 100mK/%. The authors should comment on the possible origin of discrepancy between that result and <6mK/% found here.\nA more recent paper (Ref. [22], [Jerzembeck F., et al., Phys. Rev. B 110(6), 064514 (2024)]) reports an upper bound of 13 mK/% for the change in Tc, which is reasonably close to our bound of 6 mK/% and may reflect differences in experimental resolution. Therefore, we believe that our results are rather consistent.\n3.3: I think it is a bit misleading to refer to the ‘pre-strain’ as biaxial in the discussion of the tensile control experiment, as it is its B1g component that leads to the offset of 0 (if I understood that correctly)\nWe understand the reviewer’s concern, and it is indeed possible that the offset in this experiment arises primarily from the B1g component of the pre-strain. However, the applied pre-strain has components along both the x and y directions, and therefore projects onto both the A1g and B1g symmetry channels. Since both symmetry channels involve linear combinations of e_xx and e_yy, they are both forms of biaxial strain in the general sense. This is the reason we referred to the pre-strain as “biaxial”. We have now clarified this point in the revised manuscript (right side of Page 4) to avoid any potential confusion.\n3.4: To better clarify the experimental configuration, a coordinate system (x, y, z) should be noted on Fig. 1(a,b, including the inset)\nIn response to the reviewer’s suggestion, we have now added a coordinate system (x, y, z) to Fig. 1a, 1b, as well as to Supplementary Figs. S2 and S4 to better clarify the experimental geometry. While the shear strain is applied in the x-y plane of the piezoelectric device’s coordinate system, Sr2RuO4 samples oriented along different crystallographic directions allow us to apply shear strains of the form e_xy, e_x'y', and e_zx in the sample’s coordinate system.\n3.5: While I agree with the authors that the hysteresis typical for these piezos, the systematic errors it introduces in strain calibration should be estimated.\nWe thank the reviewer for raising the point regarding systematic errors from hysteresis in strain calibration. As explained in our response to the second Reviewer's point 2.2, the hysteresis of the piezoelectric actuator introduces an uncertainty in strain conversion that is maximal near zero applied voltage, with an estimated upper bound of δe_xy ≈ 0.02%. This uncertainty decreases at larger voltages, where the strain response is well defined and reproducible, and therefore does not affect our key results or the determination of dTc/de_xy. To make this point visible to the reader, we have now differentiated the symbols of the points taken while sweeping the strain up or down by using closed/open circles in Figs. 2 e,f,g, respectively. Also, we have now included the following clarification in the main text (bottom of Page 2) as:\nThe shear–voltage characteristics in Fig. 1c are nearly linear, with a small hysteresis that is typical of piezoelectric actuators [25]. While this hysteresis is generally negligible for the purposes of this study, it introduces an uncertainty in the shear strain that is most significant around zero applied voltage, where the upper bound estimated from the vertical width of the hysteresis is δe_xy ≈ 0.02%. Importantly, this uncertainty is minimal at the largest applied voltages, where the strain response is well defined and reproducible."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have satisfactorily answered the questions I raised in my previous review and also updated the manuscript accordingly. There remains an open question regarding the lack of B1g response in the authors' measurements, however, that point has been addressed in the Discussion now. I am happy to recommend publication of the current manuscript.", "reviewer_2": "After reviewing the revised manuscript, I believe the authors successfully answered all my questions and comments, along with those of the other reviewers. I am comfortable with supporting the publication of this manuscript.", "reviewer_3": "The authors have satisfactorily responded to my comments, and I am happy to support the publication of this manuscript.\nA minor remaining comment: In response to my question 3.1 (which overlaps with 1.2 from Referee 1) the authors acknowledge a discrepancy with the previous literature, which remains to be understood, and point to possible origins, which is fine in principle. However, the text they added implies that the increase in Tc observed in other works may be due to A1g strain. I don't think that is what they mean to say, as the origin of the peak in Tc is quite well understood as arising from the proximity of the van Hove singularity along one of the high symmetry directions, but not the other. Perhaps they mean to say that *at low strain* the observed curvature could be affected by A1g strain?"}, "author_response": "We thank the reviewer for his/her comment. Indeed, that was \"not what we meant to say\" because the reported variations of Tc with 100 stress are mostly due to B1g strain due to the Poisson effect. What we meant to say is that in those works the A1g strain is also changed together with the B1g strain, possibly complicating the experimental interpretation, whereas our experiment applies a pure B1g strain with a constant A1g pre-strain. Nevertheless, the discrepancy between our results of B1g strain and previous results of uniaxial stress remains unresolved.\nWe understood that our previous sentence was rather ambiguous:\nIn those works, the strain state includes a combination of symmetry components due to the Poisson effect, including a significant A1g contribution. By contrast, our setup applies a pure B1g shear, with negligible variation of the A1g component.\nSo we replaced it with the following:\nIn those works, the strain state includes a combination of symmetry components due to the Poisson effect, with variation of both B1g and A1g strains. By contrast, our setup applies a pure B1g shear strain, with constant A1g strain."}]

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