# Fileset [10.1103-PhysRevB.110.064514.pdf](https://mdr.nims.go.jp/filesets/f17a5264-1174-43e3-bc9e-675fa799aa36/download) ## Creator [Fabian Jerzembeck](https://orcid.org/0000-0003-0003-8140), You-Sheng Li, [Grgur Palle](https://orcid.org/0000-0001-8361-4822), Zhenhai Hu, [Mehdi Biderang](https://orcid.org/0000-0002-6666-1659), [Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), Dmitry A. Sokolov, [Sayak Ghosh](https://orcid.org/0000-0003-4168-7198), [Brad J. Ramshaw](https://orcid.org/0000-0002-3222-5007), [Thomas Scaffidi](https://orcid.org/0000-0002-3143-0797), [Michael Nicklas](https://orcid.org/0000-0001-6272-2162), [Jörg Schmalian](https://orcid.org/0000-0003-4142-2448), Andrew P. Mackenzie, Clifford W. Hicks ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [

T_{c}

and the elastocaloric effect of

{Sr}_{2} {RuO}_{4}

under

〈 110 〉

uniaxial stress: No indications of transition splitting](https://mdr.nims.go.jp/datasets/5ffa110f-72fb-4d05-839e-a8e6cb1fddca) ## Fulltext $T_c$ and the elastocaloric effect of ${\rm Sr}_2{\rm RuO}_4$ under $\langle 110 \rangle $ uniaxial stress: No indications of transition splittingPHYSICAL REVIEW B 110, 064514 (2024)Editors’ SuggestionTc and the elastocaloric effect of Sr2RuO4 under 〈110〉 uniaxial stress: No indicationsof transition splittingFabian Jerzembeck ,1,* You-Sheng Li,1,2 Grgur Palle ,3 Zhenhai Hu,1 Mehdi Biderang ,4 Naoki Kikugawa ,5Dmitry A. Sokolov,1 Sayak Ghosh ,6 Brad J. Ramshaw ,6,7 Thomas Scaffidi ,8,4 Michael Nicklas ,1 Jörg Schmalian ,3,9Andrew P. Mackenzie,1,10,† and Clifford W. Hicks1,11,‡1Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany2Department of Physics, National Taiwan University, Taipei 10617, Taiwan, Republic of China3Institute for Theory of Condensed Matter, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany4Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 1A7, Canada5National Institute for Materials Science, Tsukuba 305-0003, Japan6Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA7Canadian Institute for Advanced Research, Toronto, Ontario M5G 1M1, Canada8Department of Physics and Astronomy, University of California, Irvine, California 92697, USA9Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany10Scottish Universities Physics Alliance (SUPA), School of Physics and Astronomy,University of St. Andrews, St. Andrews KY16 9SS, United Kingdom11School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom(Received 7 June 2024; accepted 7 August 2024; published 26 August 2024)There is considerable evidence that the superconductivity of Sr2RuO4 has two components. Among thisevidence is a jump in the shear elastic modulus c66 at the critical temperature Tc, observed in ultrasoundmeasurements. Such a jump is forbidden for homogeneous single-component order parameters, and it impliesthat Tc should develop as a cusp under the application of shear strain with 〈110〉 principal axes. This shearstrain should split the onset temperatures of the two components, if they coexist, or select one component ifthey do not. Here, we report measurements of Tc and the elastocaloric effect of Sr2RuO4 under uniaxial stressapplied along the [110] lattice direction. Within experimental resolution, we resolve neither a cusp in the stressdependence of Tc, nor any second transition in the elastocaloric effect data. We show that reconciling these nullresults with the observed jumps in c66 requires extraordinarily fine tuning to a triple point of the Ginzburg-Landauparameter space. In addition, our results are inconsistent with homogeneous time-reversal symmetry breaking ata temperature T2 � Tc as identified in muon spin relaxation experiments.DOI: 10.1103/PhysRevB.110.064514I. INTRODUCTIONAlthough it has a critical temperature Tc of only 1.5 K,Sr2RuO4 has become one of the most studied unconventionalsuperconductors. This is in part because even though thenormal state of Sr2RuO4 is extraordinarily well-characterized,the pairing mechanism and superconducting order parameterremain unclear [1–6]. Given the extremely high purity ofthe crystals available for experimental investigation [7], thisshould be a soluble problem, and it has become a benchmarkfor the progress of the broader field of unconventional super-conductivity.*Contact author: fabian.jerzembeck@cpfs.mpg.de†Contact author: andy.mackenzie@cpfs.mpg.de‡Contact author: c.hicks.1@bham.ac.ukPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI. Openaccess publication funded by Max Planck Society.As often happens when a large number of experiments areperformed on a single material, the results and/or interpreta-tions of some experiments disagree. While it is appropriate fortheory to attempt to reconcile apparently contradictory results,the possibility of experimental error must also be kept in mind.It can be subtle. In the history of Sr2RuO4, a conflict existedfor nearly two decades between two different probes of thecompetition between superconducting condensation energyand magnetic polarization energy. Pauli critical field limit-ing [8,9] was consistent with even-parity spin-singlet order,but the magnetic polarizability of the superconducting statemeasured by the NMR Knight shift [10] contradicted that con-clusion, leading to extensive discussion of spin-triplet orderparameters. The issue was resolved only after a systematicerror in the original NMR measurements was uncovered [11].Although some researchers continue to explore the possibil-ity of spin-triplet pairing in Sr2RuO4 [12–15], the weightof recent evidence is now strongly in favor of spin-singlet,even-parity order [11,16–20].This experience provides strong motivation to checkother apparently settled experimental facts about thesuperconductivity of Sr2RuO4. A major question is whether2469-9950/2024/110(6)/064514(16) 064514-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-0003-8140https://orcid.org/0000-0001-8361-4822https://orcid.org/0000-0002-6666-1659https://orcid.org/0000-0003-3975-4478https://orcid.org/0000-0003-4168-7198https://orcid.org/0000-0002-3222-5007https://orcid.org/0000-0002-3143-0797https://orcid.org/0000-0001-6272-2162https://orcid.org/0000-0003-4142-2448https://ror.org/01c997669https://ror.org/05bqach95https://ror.org/04t3en479https://ror.org/03dbr7087https://ror.org/026v1ze26https://ror.org/05bnh6r87https://ror.org/01sdtdd95https://ror.org/04gyf1771https://ror.org/04t3en479https://ror.org/02wn5qz54https://ror.org/03angcq70https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.110.064514&domain=pdf&date_stamp=2024-08-26https://doi.org/10.1103/PhysRevB.110.064514https://creativecommons.org/licenses/by/4.0/FABIAN JERZEMBECK et al. PHYSICAL REVIEW B 110, 064514 (2024)the superconducting order parameter breaks time-reversalsymmetry. It has long been widely accepted as fact that itdoes, on the basis of muon spin rotation (μSR), Kerr rota-tion, and Josephson junction data [21–26]. However, someexpected experimental signatures have not been observed[27,28].Recently, it has become possible to test for two expectedconsequences of time-reversal symmetry breaking (TRSB)in Sr2RuO4 under uniaxial pressure. The uniaxial pressureshould break the degeneracy of the order parameter com-ponents required to produce a TRSB state, yielding, first, acusp in the stress dependence of Tc centered on zero pres-sure [29,30], and second, a splitting of the transition undernonzero pressure that should be observable in thermodynamicdata. Under uniaxial pressure along the [100] lattice direc-tion, neither effect has been observed, in spite of severalsearches [31–33]. However, transition splitting was observedin μSR measurements, a nonthermodynamic probe, under[100] uniaxial stress [23]. One possible interpretation ofthis discrepancy is that the thermodynamic measurements ofRefs. [31–33] were not sensitive enough to detect the secondtransition.There is therefore a premium on extending thermodynamicstudies to a situation in which there is more guidance onexpected thermodynamic quantities. Recent observations ofa jump in the elastic modulus c66 at Tc, determined via ul-trasound measurements [34,35], provide such guidance foruniaxial stress applied along the [110] lattice direction. Thisstress axis has largely been neglected because the couplingof the electronic structure of Sr2RuO4 to stress applied alongthe [110] direction is weak [31,36]. However, the observedmagnitudes of the jumps in c66 imply, through Ehrenfestrelations that we derive below, that the cusp and splittingshould, surprisingly, be easily observable under stress alongthis direction.We report results of high-resolution studies of both themagnetic susceptibility and elastocaloric effect under [110]uniaxial pressure. Within tight limits, we resolve neither acusp nor transition splitting. We show that our results cannotbe plausibly reconciled with the observed jumps in c66 un-der assumption of a homogeneous superconducting state—thelevel of tuning implied is implausibly fine.Combining our results with those from previous work on[001] uniaxial pressure allows a prediction for the dependenceof Tc on hydrostatic pressure. We find good agreement withmeasurements of Tc under hydrostatic pressure [37], whichshows that our data are thermodynamically consistent withprevious results. However, our data are not consistent withrecently reported μSR results, in which a transition splittingunder [110] stress was reported [38].While presenting negative results is infrequently done, tworeasons motivated our efforts for doing so. In the context ofSr2RuO4, we believe that our findings make a bulk, ther-modynamic superconducting state that breaks time-reversalsymmetry exceedingly unlikely. They also call into ques-tion the existence of any two-component order parameter inSr2RuO4. The robustness of our conclusion stems from thespecial place held by thermodynamics in understanding thephysics of many-body systems. Our results, therefore, narrowdown the search for the symmetry of the pairing state of aFIG. 1. Schematic dependence of the phase transition tem-peratures on strain ε110. Orange line: for single-component andsome two-component order parameters, the onset temperature ofsuperconductivity derives only from the A1g-symmetric strain com-ponents, εd and ε3, and so it varies smoothly across ε110 = 0. Darkblue: For (dxz, dyz ) pairing and accidentally degenerate (s, dxy ) and(dx2−y2 , gxy(x2−y2 ) ) pairing a cusp, i.e., a sudden change of slope inTc(ε110) occurs, due to coupling to the shear strain component ε6.Light blue: in some cases (see Fig. 4) there is a second transition atT2 below Tc, also with a cusp. The inset shows the components of theapplied strain when uniaxial stress is applied along a 〈110〉 direction.material that has been emblematic of the field of unconven-tional superconductivity.II. STRAIN COMPONENTSTo frame the discussion in the paper, we introduce a no-tation for strains. We will use the symbols ε110 and σ110 todenote the strain and stress along the [110] lattice direction,under conditions of uniaxial stress. When these symbols areused, it is assumed that there are also transverse strains due tothe Poisson effect. Based on the elastic moduli at 4 K reportedin Ref. [34], σ110 = (187 GPa) × ε110.The strain can be resolved into components. We choosehere to resolve it into shear strain ε6, c-axis strain ε3, andin-plane dilatation εd ≡ ε1 + ε2, where ε1 through ε6 are thestrain components expressed in the standard Voigt notation.While εd and ε3 transform under the trivial representation A1gof the point group, ε6 transforms under B2g. These three straincomponents are illustrated in the inset of Fig. 1. The 4 Kelastic moduli from Ref. [34] yieldεd = αd σ110, ε3 = α3 σ110, ε6 = α6 σ110, (1)with αd = 0.003 07 GPa−1, α3 = −0.001 02 GPa−1, andα6 = 0.007 65 GPa−1.For all possible order parameters, nonzero εd and ε3 resultin a smooth variation of Tc which is linear in strain to leadingorder. We label this line as Tc0 in Fig. 1. A leading-order cou-pling to ε6 is permitted only for certain two-component orderparameters, and it results in a cusp in the strain dependence ofTc, that is, a discontinuity in slope of magnitude 2|dTc/dε6|:�Tc(εd , ε3, ε6) = dTcdεdεd + dTcdε3ε3 +∣∣∣∣dTcdε6∣∣∣∣|ε6| + · · · , (2)where the ellipsis denotes higher-order terms, and it will besuppressed from now on. Below, we show that for even-parity064514-2Tc AND THE ELASTOCALORIC EFFECT OF … PHYSICAL REVIEW B 110, 064514 (2024)pairing states, such a cusp occurs for symmetry-protected two-component order parameters that combine (dxz, dyz ) Cooperpairs, and for accidentally degenerate two-component orderparameters that combine (s, dxy) or (dx2−y2 , gxy(x2−y2 ) ) Cooperpairs. From Eq. (1) it follows that Tc as a function of σ110, ourexperimental control parameter, obeys�Tc(σ110) =⎛⎝∑i=d,3dTcdεiαi + sgn(σ110)∣∣∣∣dTcdε6∣∣∣∣α6⎞⎠σ110. (3)This behavior is illustrated in Fig. 1. For a single-componentsuperconducting order parameter and for two-componentstates other than the ones listed above, |dTc/dε6| = 0. It isthis distinct behavior with respect to ε6 that allows for the keyconclusions of this paper.In some cases, one expects a second transition at a temper-ature T2 < Tc where a composite of the two order parametercomponents breaks an additional symmetry. If this happens,one expects behavior similar to Eq. (3), but with the crucialdistinction that the sign in front of the cusp is negative:�T2(σ110) =⎛⎝∑i=d,3dT2dεiαi − sgn(σ110)∣∣∣∣dT2dε6∣∣∣∣α6⎞⎠σ110, (4)as sketched in Fig. 1. From the slopes |dTc/dε6| and (if asecond transition occurs) |dT2/dε6|, an upper bound on thejump in elastic constant c66 may be obtained; this relation ispresented below.III. RESULTS: MEASUREMENT OF Tc(σ110)To probe the dependence of Tc on σ110, we studied themagnetic susceptibility of single crystals of Sr2RuO4. Stresswas applied using piezoelectric-based apparatus that incorpo-rated both force and displacement sensors [39]. Samples weresculpted into dumbbell shapes using a Xe plasma focused ionbeam, a step that allows higher stresses to be reached [40].To measure magnetic susceptibility, concentric coils of a fewturns each were wound around the central neck portion of thesamples, and their mutual inductance was measured. For eachsample, the zero-stress point was identified by deliberatelybreaking the sample under tension, then measuring Tc withthe two parts separated [41].Three samples were measured. Samples 1 and 2 were takenfrom the same original crystal, in which the growth directionwas almost exactly along [110], while sample 3 was takenfrom a crystal where the growth direction was about 15◦ awayfrom [110]. In all cases, the samples were cut from the originalcrystal such that the pressure was applied along [110] within� 3◦. Samples 1 and 2 both withstood tensile stresses of upto σ110 ≈ +0.2 GPa, while sample 3 broke under very lowtensile stress. Samples 1 and 3 were compressed to σ110 <−2 GPa.For sample 1, there was some hysteresis in Tc(σ110). Themost likely origin was a hysteretic component of the appliedstress that had 〈100〉 principal axes: Tc of Sr2RuO4 respondsmuch more sensitively to 〈100〉 than 〈110〉 shear stress [31].After measurement of sample 1, the apparatus was modifiedto attenuate transmission of stress components other than thedesired [110] uniaxial stress. This step drastically reduced the-2.0 -1.5 -1.0 -0.5 0.00 00.020.040.060.080.100.12σ110 (GPa)σ110 σ110dTc/dσ 110 (K/GPa)dTc/dσ 110dTc/dσ 1101.251.301.351.401.45T c (K)sample 1sample 2quadratic fitssample 3last data pointfirst data point(d)(e)(f) (g)σ110 (GPa):-0.2800.32-1.28-0.95-0.64-1.27-0.91-1.571840616212M (nH)0.22-0.01-0.53-1.05-1.61-2.09sample 1sample 2sample 3piecewise fitsglobal fits1.2 1.4 1.6T (K)1.2 1.4 1.6T (K)1.2 1.4 1.6T (K)-0.430(a) sample 1 (b) sample 2 (c) sample 3FIG. 2. (a)–(c) Temperature dependence of the mutual induc-tance M of the susceptibility coils wrapped around the sample, forsamples 1, 2, and 3, respectively. The numbers indicate the appliedstress, σ110, in GPa, and σ110 < 0 denotes compression. (d) Tc, de-termined as the points where M crossed the thresholds indicated inpanels (a)–(c), against stress. To illustrate the level of drift, the pointsare colored by the order in which they were measured. For sample1, due to hysteresis only data from decreasing-σ ramps are shown.The black lines are quadratic fits to the data. (e) Points: dTc/dσ110determined from piecewise linear fits. Straight lines: dTc/dσ110 fromthe quadratic fits in panel (d). (f) Expected form of dTc/dσ110, if thereis a sharp cusp on top of a quadratic background. (g) Expected formof dTc/dσ110 if the cusp is broadened by, for example, internal straininhomogeneity.hysteresis for samples 2 and 3. The modification is describedin Appendix A.Raw data—the mutual inductance M of the susceptibilitycoils—for all three samples are shown in Figs. 2(a)–2(c). Forall, the transition width was about 50 mK and it did not064514-3FABIAN JERZEMBECK et al. PHYSICAL REVIEW B 110, 064514 (2024)increase much as stress was applied. This transition widthcould be a consequence of an inhomogeneous defect density,and/or an internal field of 〈100〉 shear strain due to defects.It is not a consequence of whatever internal field of 〈110〉shear strain may be present. It can be seen in Fig. 2(d) thatσ110 ≈ −1 GPa is required to suppress Tc by 50 mK, which isan unrealistically large internal stress except in the immediatevicinity of defects [42].Figure 2(d) shows Tc(σ110) of the three samples, where Tcis taken as the point where M crosses the thresholds indicatedin panels (a)–(c). To convey the level of drift, the data pointsare colored by the order in which they were measured. Dueto the hysteresis, for sample 1 only data points taken underdecreasing σ110 are shown.Linear fits over the range −0.2 < σ110 < 0.2 GPa givedTc/dσ110 = 68.3, 65.8, and 58.5 mK/GPa for samples 1, 2,and 3, respectively. Including uncertainty on the force sensorcalibration, we take dTc/dσ110|σ110=0 = 64 ± 7 mK/GPa.For none of the samples is there an obvious cusp at σ110 =0. To look more closely, we determine the slopes dTc/dσ110from piecewise linear fits, choosing windows small enoughthat for samples 1 and 2 there are windows at σ110 > 0. Re-sults from this fitting are shown in Fig. 2(e). If there werea sharp cusp in Tc(σ110) at σ110 = 0, the points at σ110 > 0would be above the trend from σ110 < 0. They are not. Alsoshown in Fig. 2(e) is dTc/dσ110 determined from quadraticfits to Tc(σ110) over the entire measured stress range. Therms difference between the piecewise-fitted and these globallyfitted slopes is 10 mK/GPa.A cusp could be broadened by internal strain. In Fig. 2(f)we illustrate the expected form of dTc/dσ110 if there is asharp cusp, and in Fig. 2(g) if it is rounded. With rounding,dTc/dσ110 deviates from the background over a range of stressaround σ110, such that even in sample 3, where essentially notensile stress could be applied, the effects of a cusp could havebecome visible. No such deviation is visible in the data.We take a conservative upper limit on any change inslope dTc/dσ110 across a cusp, �(dTc/dσ110), of 20 mK/GPa,whether the cusp is sharp or rounded. This upper limit implies∣∣∣∣dTcdε6∣∣∣∣ = 12α6�(dTcdσ110)< 1.3 K, (5)which is our first key experimental result. The implication ofthis tight upper bound on |dTc/dε6| will be discussed in detaillater in the paper.IV. ELASTOCALORIC EFFECT EXPERIMENTSIt is clear from the Tc(σ110) data that the superconductivitycouples much more weakly to shear strain with 〈110〉 than〈100〉 principal axes; data under 〈100〉 uniaxial stress arepublished in, for example, Refs. [31,33]. It might not thenseem that the elastocaloric effect (the change in sample tem-perature induced by applied stress) under [110] stress wouldbe a particularly effective probe of the physics. However, thatis not necessarily the case. Under adiabatic conditions, theelastocaloric coefficient η is given byη ≡ dTdε= −TC∂S∂ε, (6)where C is the heat capacity, S is the entropy, and the direc-tion of strain ε is fixed by the conditions of the experiment.Suppose, for a moment, single-component superconductivity.The Ginzburg-Landau free energy of the single-componentsuperconducting state is then given byF = Fn + a(T, ε)2|ψ |2 + u4|ψ |4. (7)Here, a(T, ε) = a0(T − Tc0(ε)), and Fn is the free energy inthe normal state. The entropy below Tc0 is given byS(T, ε) = Sn + a0(T − Tc0(ε))2u, (8)where Sn = −∂Fn/∂T is the normal state entropy. It followsfrom Eq. (6) that the change in η at Tc, �η, equals�ηηn(Tc)= −(1 − 1ηn(Tc)dTcdε)×(1 + Cn(Tc)�C)−1, (9)where ηn is the normal-state elastocaloric coefficient, Cn is thenormal-state heat capacity, and �C is the heat-capacity jumpat Tc. In the case of two-component order and near zero stress,Eq. (9) applies for strains ε that transform under the trivialrepresentation: εd or ε3. For two-component order and undernonzero stress that splits the transitions, Eq. (9) applies forstrains εd , ε3, and ε6.Equation (9) allows for rather general insights into thestrain dependence of Tc. Suppose the term dTc/dε is negligi-ble compared to ηn. Since 1 + Cn/�C > 1, the magnitude ofη would fall at Tc, but it would not change sign. This behavioris in contrast to the behavior under [100] uniaxial stress, wheredTc/dε is much larger and η is observed to change sign at Tc[43]. Importantly, Eq. (9) shows that �η can be substantialeven if dTc/dε = 0.It has been demonstrated that for Sr2RuO4 under uniaxialstress at low temperatures, η can be measured with a highersignal-to-noise ratio than heat capacity [33,43]. Therefore, inthe case in which the superconducting transition splits intotransitions at Tc and T2, the elastocaloric effect is an idealprobe to search for thermodynamic signatures of the transitionat T2.Elastocaloric effect data from two samples under [110]uniaxial pressure are shown in Fig. 3. In the normal state,η is negative over the range of pressures and temperaturesstudied, meaning that S in the normal state increases whensamples are tensioned. Inspection of the data shows that, atall strains measured, only one transition is observed. There isno visible sign of uniaxial pressure-dependent splitting of themain transition—any structure in the transition that is seenat zero pressure (likely due to slight inhomogeneity of thestrain field and/or defect density) remains the same at nonzeropressure. We may therefore proceed with analysis under atentative hypothesis of single-component order. The observedbehavior across Tc is as expected for small |dTc/dε110|: η issmaller below Tc, but its sign is unchanged.The small size of the signal makes data analysis morechallenging than for [100] uniaxial stress [43]. To analyzethe data, we assume that the normal-state heat capacity ofSr2RuO4 at low temperatures is given byCn = (γ0 + γ1ε110)T + βT 3, (10)064514-4Tc AND THE ELASTOCALORIC EFFECT OF … PHYSICAL REVIEW B 110, 064514 (2024)-35-30-25-20-15-10-5η(K)-0.00GPa-0.07GPa-0.14GPa-0.21GPa-0.31GPa-0.42GPa-0.52GPa1.0 1.2 1.4 1.6 1.8 2.0T (K)-30-25-20-15-10-5η(K)0.07 GPa-0.04GPa-0.14GPa-0.24GPa-0.34GPa-0.44GPasample A(a)(b)sample BFIG. 3. Elastocaloric coefficient η vs temperature for two sam-ples. For clarity, all curves apart from the black ones are shiftedvertically with respect to each other. The method of analysis isdiscussed in the text and Appendix B.which yieldsηn = −γ1Tγ0 + γ1ε110 + βT 2. (11)The elastocaloric effect is a recently introduced technique[44], and experimental uncertainties remain, especially inquantifying the actual strain oscillation amplitude, δε, and thedegree of adiabaticity, A. (A = 1 denotes perfect adiabaticity,and A = 0 denotes complete dissipation of temperature os-cillations into the stress cell.) However, the key quantity inEq. 9, �η/ηn(Tc), can be obtained directly from the jumpin measured thermocouple voltage at Tc, without knowledgeof A or δε. That quantity can then be used to solve for γ1,which is the only quantity in Eqs. (10) and (11) that is notfixed by experimental data. We set γ0, β, and �C/Cn toliterature values of 37.5 mJ/mol K2, 0.197 mJ/mol K4, and0.65, respectively [34,45]. In Eq. (9), we set ε to be ε110, cor-responding to our measurement conditions, and we set dTc/dεto 64 mK/GPa × 187 GPa = 12.0 K. Averaging results fromthe two samples, we obtain γ1 = 0.43 J/mol K2.All the terms in Eq. (11) are now known, and we may thenapply Eq. (11) to extract A(T ) × δε(T ). In effect, where theraw data deviate from the form given in Eq. (11), we make anassumption that it is more likely to be due to T dependence ofAδε than that there are terms in Cn beyond those in Eq. (10)that are important at low T and ε110. We then extrapolateA(T )δε(T ) to T < Tc to obtain a best estimate for the ECEin the superconducting state. This is what is shown in Fig. 3.Further details are given in Appendix B.V. DISCUSSIONThe qualitative finding from these 〈110〉 uniaxial pressureexperiments on Sr2RuO4 is that we do not observe eitherof two key predicted features of two-component supercon-ducting states: neither a cusp in Tc, nor splitting of thesuperconducting transition into two transitions at Tc and T2.In this discussion, we review the consistency of these findingswith those of other experiments, and the implications fortheories of the superconducting order parameter of Sr2RuO4.In both sections, we frame the discussion with the quantitativebounds that we have placed on the putative existence of a cuspor of transition splitting, rather than on categorical statementsthat neither exists.A. Comparison with hydrostatic pressure dependence of TcIn this work, we have determined that dTc/dσ110 = 64 ±7 mK/GPa. As shown in Appendix D, combining this re-sult with previous measurement under pressure applied alongthe c-axis that dTc/dσ001 = 76 ± 5 mK/GPa [40] enables aprediction for the dependence of Tc on hydrostatic pressureσhyd: dTc/dσhyd = 204 ± 12 mK/GPa. This allows a usefulcross-check on the accuracy of our uniaxial pressure data,because the dependence of Tc on hydrostatic pressure has beenmeasured in four independent experiments [24,37,46,47].However, since the Tc of most studied samples were substan-tially below 1.5 K, which is pointing to strong disorder, wefocus on the hydrostatic pressure dependence of Ref. [37],optimal-Tc experiment, which found σhyd: dTc/σhyd = 220 ±20 mK/GPa. The agreement with the derivation from uniaxialstress measurements is reassuring.B. Comparison with μSR under [110] uniaxial pressureRecently, a μSR study under 〈110〉 uniaxial pressure in-ferred transition splitting, with a TRSB transition at T2 = Tc −(0.7 ± 0.2) K/GPa [38], which corresponds to dT2/dε110 ≈131 K. The transition at T2 would be within our measuredtemperature range for each of the stresses shown for sample Bin Fig. 3, and it would have been visible as long as |�η2| werelarger than ∼0.2 K, or 2% of the jump in ECE at Tc, |�ηc|.As with previous comparisons of μSR and heat-capacity dataunder [100] uniaxial pressure [23], it is hard to imagine a tran-sition to a second homogeneous thermodynamic state yieldingsuch a small anomaly. We note the low statistical signifi-cance of the splitting reported in Ref. [38]; a simple repeatof that measurement would be useful. However, we believethat the direction of travel is toward fundamental reevalu-ation of the interpretation of μSR data in unconventionalsuperconductors.C. Relationship with jumps in c66 observedin ultrasound experimentsOne of the motivations for the current experiments was toperform a careful comparison with the results of ultrasoundexperiments, which have resolved jumps in the elastic con-stant c66 at the superconducting transition. The observation ofa jump in this elastic constant is particularly significant be-cause it implies the existence of some kind of two-component064514-5FABIAN JERZEMBECK et al. PHYSICAL REVIEW B 110, 064514 (2024)superconducting order parameter; a single-component orderparameter gives a jump in other elastic constants, but not inc66. Jumps of 0.03 and 0.15 MPa were reported on the basisof separate pulse-echo measurements at 169 and 201 MHz,respectively [35], and of 1.05 MPa on the basis of a resonantultrasound experiment performed at much lower frequenciesof approximately 2 MHz [34]. The difference between thetwo pulse-echo results was attributed to possible mode mixingin the 201 MHz experiment. It has been suggested that thedifference between the pulse-echo and resonant ultrasoundresults is a consequence of the very different measurementfrequencies, with the higher frequencies thought to suppressthe jump from its intrinsic thermodynamic value [35]. For thepurposes of the analysis that follows, we will take the quotednumbers at face value and examine the extent to which theyare consistent with our results, under an assumption that allthe experiments are giving information on bulk, homogeneousthermodynamic phases.To frame our thermodynamic analysis, we first give someimportant results about Ehrenfest relations, giving the fullderivation in Appendix E. Broadly speaking, there are twopossibilities for two-component order parameters that givea jump in c66 in Sr2RuO4. One is that degeneracy of thecomponents is symmetry-protected, i.e., the components areequivalent on a tetragonal lattice. Given the strong evidencefor even-parity, spin-singlet superconductivity in Sr2RuO4[11,19,43,48], these two components would have dxz anddyz symmetry. The other possibility is accidental degeneracybetween certain non-symmetry-related components, namelythe pair (s, dxy) or the pair (dx2−y2 , gxy(x2−y2 ) ). For both pairs,multiplying the components yields a composite order thatcouples linearly to shear strain with 〈110〉 principal axes, thatis, ε6. Single-component order parameters, in contrast, do notyield a jump in c66 at Tc. Neither do two-component orderparameters which do not couple to ε6 shear strain: the pairs(s, dx2−y2 ), (s, gxy(x2−y2 ) ), (dxy, dx2−y2 ), and (dxy, gxy(x2−y2 ) ). Atable of these possibilities is shown in Fig. 4.Consider first the symmetry-protected possibility,(dxz, dyz ). If, at temperatures well below Tc, the componentscombine to break time-reversal symmetry, then under shearstrain with 〈110〉 principal axes the transition would splitinto a transition at Tc into dxz ± dyz order, followed by atransition at T2 into (dxz ± dyz ) ± i(dxz ∓ dyz ) order. Thispossibility is illustrated in Fig. 4(a). If the componentscombine to form B2g-nematic order (that is, dxz ± dyz order),there would be no second transition at lower temperature.There would be a first-order transition along the ε6 = 0line, where the favored orientation of the nematicity flips.This possibility is illustrated in Fig. 4(b). Finally, if thecomponents form B1g-nematic order, i.e., dxz or dyz withoutcoexistence of the two components, then 〈110〉 shear strain isagain expected to yield a split transition. The order parameterjust below Tc would be dxz ± dyz, and at T2 there would be asymmetry-breaking transition at which the principal axes ofthe nematicity rotate away from the 〈110〉 axes and towardsthe 〈100〉 axes.The accidentally degenerate pairs (s, dxy) and(dx2−y2 , gxy(x2−y2 ) ) could also combine to yield TRSB orB2g-nematic orders—for example, s ± idxy or s ± dxy.The T -ε6 phase diagrams would be qualitatively theε6 ε6 ε6non-TRSBTRSB(dxz, dyz)(s, dxy),(s, dx²−y²),(s, g),(dxy, dx²−y²),(dxy, g),(dx²−y², g)TcT2TcT2(b) B2g-nematic (c) B1g-nematic(a)ε6 ε6ε6ε6(e) B2g-nematic (f) no coexistence(d)(g)single-componentFIG. 4. T -ε6 phase diagrams for various possible order parame-ters. g denotes gxy(x2−y2 ). The first row illustrates (dxz, dyz ) order andthe ways in which these components can combine at temperatureswell below Tc. In panel (a), they form TRSB order, dxz ± idyz. Inpanel (b), they form B2g-nematic order, dxz ± dyz. In panel (c), theyform B1g-nematic order: condensation of either component alone,without coexistence. The next row illustrates the equivalent possibil-ities for accidentally degenerate two-component orders that couplelinearly to ε6. The bottom row illustrated the expected strain depen-dence of order parameters, which do not couple linearly to ε6. Inall panels, single black lines indicate second-order transitions, dou-ble lines indicate first-order transitions, and color gradients indicatecrossovers. Further explanation is provided in the text.same as for the (dxz, dyz ) pair; see Figs. 4(d) and 4(e).The remaining possibility—absence of coexistence—isqualitatively different from the (dxz, dyz ) case, because forthese accidentally degenerate pairs there is no diagonalreflection symmetry x ↔ y which protects the B2g nematicstate. Just below Tc and with ε6 = 0, the two componentswould combine to yield B2g-nematic order, and a cuspeddependence of Tc on ε6. As T is further reduced, one of thecomponents would come to dominate, but this would be asmooth crossover. This possibility is illustrated in Fig. 4(f).Analysis of each situation yields the following Ehrenfestrelations (Appendix E 3). For B2g-nematic order constructedfrom (dxz, dyz ) components,�c66 = �C0Tc0∣∣∣∣dTcdε6∣∣∣∣2, (12)where �C0 is the jump in heat capacity at the superconductingtransition in the unstressed system, and Tc0 is Tc in the un-stressed system. On the other hand, for both dxz ± idyz TRSBand (dxz, dyz ) B1g-nematic order,�c66 = �C0Tc0∣∣∣∣dTcdε6∣∣∣∣∣∣∣∣dT2dε6∣∣∣∣. (13)064514-6Tc AND THE ELASTOCALORIC EFFECT OF … PHYSICAL REVIEW B 110, 064514 (2024)This equation is obtained from Eq. (E41) in Appendix E,with α in that equation set to zero. Here, we label as �C0the heat-capacity jump without splitting—anywhere along thetransition line when splitting does not occur, and at ε6 = 0when it does. Tc0 is the critical temperature in the absence ofsplitting.We begin with the simpler of the above Ehrenfest rela-tions, Eq. (12), which applies to dxz ± dyz B2g-nematic order.�C0 in Sr2RuO4 is 38 mJ/mol K [49,50]. Combining thiswith the largest and smallest literature values for �c66, 1.05and 0.03 MPa, yields values for |dTc/dε6| of 48 and 8 K,respectively. As stated above, we resolve no cusp, with ourexperiment placing an upper limit on |dTc/dε6| of 1.3 K. Ifboth our experiments and the ultrasound studies are probingbulk, homogeneous thermodynamic states, there is thereforea discrepancy of between a factor of 37 and 6 between ourupper limit on |dTc/dε6| and predictions from Eq. (12) usingmeasured values of �c66. We can therefore rule out bulkB2g-nematic order, namely dxz ± dyz order, as the origin of theobserved jumps in c66.In the case of dxz ± idyz or B1g-nematic dxz or dyz order,where it is Eq. (13) that applies, the very small upper limit thatour measurements place on |dTc/dε6| could in principle becompensated by a very large value of |dT2/dε6|. However, thenumbers are stark. For �c66 = 0.03 MPa, |dT2/dε6| > 51 Kwould be required to be compatible with our finding that|dTc/dε6| < 1.3 K. For the largest measured value of �c66,1.05 MPa, |dT2/dε6| > 1785 K would be required.A difference of this magnitude between |dT2/dε6| and|dTc/dε6| would require a very high degree of tuning. Toquantify this fine-tuning, let us parametrize the free energyof (dxz, dyz ) order in the following way [Eq. (E13)]:F = a2(|�1|2 + |�2|2) + u4(|�1|2 + |�2|2)2 + γ u|�1|2|�2|2+ γ ′ u2(�∗1�∗1�2�2 + �∗2�∗2�1�1). (14)�1 and �2 are, respectively, the amplitudes of the dxz anddyz components. γ ′ > 0 favors combining them with a π/2phase shift, yielding dxz ± idyz TRSB order, while γ ′ < 0 fa-vors dxz ± dyz B2g-nematic order. γ > 0 disfavors coexistenceof the two components; B1g-nematic order is obtained whenγ > |γ ′|. For a generic microscopic theory, the three quarticcoefficients are expected to be comparable in magnitude, giv-ing γ , γ ′ on the order of 1. The γ -γ ′ phase diagram is shownin Fig. 5.As discussed above, our data are not consistent withB2g-nematic order, but TRSB and B1g-nematic orders are inprinciple possible. By exploiting the Ehrenfest relation (13),one may bound the ratior ≡∣∣∣∣dTcdε6∣∣∣∣∣∣∣∣dT2dε6∣∣∣∣−1(15)from above to obtain r < (1.3 K)/(51 K) = 0.026. Here wehave used the smallest reported �c66 > 0.03 MPa to obtain aconservative estimate. One may show that in the B1g-nematicregion r = γ + γ ′, and that in the TRSB region r = 2γ ′/(1 +γ − γ ′) [Eq. (E46)]. Therefore, the regions of the phase dia-gram in Fig. 5 colored blue would be consistent with our data:the interaction between the two components must be tunedB1g-nematicunstableB2g-nematic TRSB-1-0.5-0.5 0.5000.5γ'γr < 0.026TRSBandr < 0.026andr' < 0.05FIG. 5. Phase diagram of the Ginzburg-Landau parameter spacefor (dxz, dyz ) superconductivity, based on the free energy written inEq. (14). γ and γ ′ are parameters of this free energy that definewhether the ground state is B1g-nematic, B2g-nematic, or TRSB. Theblue region is the region consistent with the condition |dTc/dε6| <1.3 K and �c66 > 0.03 MPa. Within the TRSB region, the red patchnear the γ = γ ′ = 0 triple point is additionally consistent with theheat-capacity data of Ref. [33] under 〈100〉 uniaxial stress. The upperbounds on the ratios r, r′ are explained in the main text.right to the cusp of B2g nematicity, but without the order beingB2g-nematic.We can add to this diagram constraints imposed by mea-surements under 〈100〉 uniaxial stress. The analogous ratior′ ≡∣∣∣∣ dTcd (ε1 − ε2)∣∣∣∣∣∣∣∣ dT2d (ε1 − ε2)∣∣∣∣−1(16)is inversely related to the ratio of the heat-capacity jumpsat the upper and lower transitions [Eq. (E62)]. In Ref. [33],it was shown that the heat-capacity anomaly at any secondtransition is at most 5% of that at Tc, corresponding (again, fordxz ± idyz order) to the condition r′ < 0.05. Within the TRSBregion of the phase diagram, r′ = −(γ − γ ′)/(1 + γ − γ ′).Therefore, the region within the TRSB phase consistent withboth sets of experiments is only the red region in Fig. 5. Thatis, if the order parameter of Sr2RuO4 is a TRSB order param-eter constructed from components with symmetry-protecteddegeneracy, the interaction between these components wouldneed to be doubly fine-tuned so as to lie extremely close to thetriple point in the phase diagram.So far, there are no data ruling out B1g nematicity at thesame level that B2g nematicity is excluded here. The upperlimit on |dTc/d (ε1 − ε2)| is 11 K [32], not as tight as thelimit set here on |dTc/dε6|. Therefore, within the B1g-nematicregion we require only the single level of fine-tuning describedabove, γ + γ ′ < 0.026, but we emphasize that it is a stringentcondition.In the case of TRSB order constructed from acciden-tally degenerate components, the Ehrenfest relation Eq. (13)064514-7FABIAN JERZEMBECK et al. PHYSICAL REVIEW B 110, 064514 (2024)becomes an inequality:�c66 � �C0Tc0∣∣∣∣dTcdε6∣∣∣∣∣∣∣∣dT2dε6∣∣∣∣, (17)and we still obtain an upper bound on �c66 derivable fromthe strain variations of the transition temperatures. This equa-tion is Eq. (E41) from Appendix E 3, with α = 0. As under thedxz ± idyz hypothesis, reconciling our data with the observed�c66 would still require fine tuning, with |dT2/dε6| vastlylarger than |dTc/dε6|.In the absence of coexistence—the case illustrated inFig. 4(f)—no similarly strong conclusions can be drawn.Equation (12) would apply directly at the superconductingtransition, as it must, but under the hypothesis of accidentaldegeneracy there would be a crossover just below Tc fromstrain-induced B2g nematicity to single-component order. Inexperimental data, it is likely that the changes in c66 from thetwo features would merge into a single resolution-limited step,so that Eq. (12) would not be applicable in practice. Neverthe-less, it seems almost certain that reconciling observed valuesof �c66 with our upper limit on |dTc/dε6| would still requirea high degree of tuning.D. Implications of the lack of transition splittingin elastocaloric measurementsThe lower limit |dT2/dε6| > 51 K established above fordxz ± idyz and B1g-nematic (dxz, dyz ) order is made even morestringent by analysis of our elastocaloric effect data. For smallsplitting, i.e., Tc − T2 Tc0, we require�CcTc+ �C2T2= �C0Tc0, (18)where �Cc and �C2 are, respectively, the jumps in heat ca-pacity at Tc and T2. The condition that degeneracy of the twocomponents is symmetry-protected imposes the additionalcondition that�CcTcT2�C2=∣∣∣∣dT2dε6∣∣∣∣∣∣∣∣dTcdε6∣∣∣∣−1. (19)Making use of these conditions, it is straightforward toderive the ECE. The derivation is shown in Appendix C. InFig. 6, we show the range of |dTc/dε6| and |dT2/dε6| overwhich a second transition at T2 would have been observablein our elastocaloric effect data. To obtain this plot, we set aconservative observability threshold that the jump in η at T2be at least 0.2 K. Although the noise level for sample B issmaller than this, inhomogeneity broadening might be largerthan for the transition at Tc, due to its potentially steeperslope. This observability threshold yields the curved line thatbounds the observable region on the left. Added to this plotis the relation between |dTc/dε6| and |dT2/dε6| fixed by theconstraint �c66 = 0.03 MPa. It crosses out of the observableregion at |dT2/dε6| = 144 K. That is, to account for nonob-servation of a second transition in the ECE data under anassumption of (dxz, dyz ) order and taking �c66 = 0.03 MPa,we require |dTc/dε6| < 0.47 K and |dT2/dε6| > 144 K,which is a substantial tightening of the condition |dTc/dε6| <1.3 K obtained from direct measurement of Tc(ε110).0 0.504002001446001.0 1.5dTcdε6(K)dT2dε6(K)excluded byECE dataΔc66=0.03 MPaFIG. 6. Region of observability of a second transition in elas-tocaloric effect data. In the red region, a jump in the elastocaloriceffect would have been resolvable at T2 in our data, under an as-sumption of dxz ± idyz or B1g-nematic, (dxz, dyz ) superconductivity.The thick line is the set of points defined by the condition �c66 =0.03 MPa.The upper and lower boundaries of the observable regioncorrespond to a requirement:42 K <∣∣∣∣ dT2dε110∣∣∣∣ < 750 K.The lower limit corresponds to a change of 0.1 K at the largeststrain applied to sample B, −0.45 GPa/187 GPa = −2.4 ×10−3. At this strain, T2 would be ≈0.07 K less than Tc, whichwe estimate as the minimum separation at which a secondtransition would become distinct from the main transition.The upper bound corresponds to a requirement that T2 notdrop below our lowest measurement temperature, 1 K, at thesmallest applied strain, ≈0.10 GPa/187 GPa = 5.3 × 10−4.The |dT2/dε6| − |dTc/dε6| curve that would yield �c66 =1.05 MPa lies entirely above the observable region in Fig. 6,and therefore, if we take �c66 = 1.05 MPa, our ECE data donot further tighten the constraint |dTc/dε6| < 1.3 K obtainedfrom direct measurement.This analysis does not strictly apply to fine-tuned acci-dental degeneracies, but the situation for those is expected tobe similar. There might be alternative fine-tuning routes thatyield small �C2 with modest |dT2/dε6|, but our data never-theless imply at least two levels of fine-tuning with accidentaldegeneracy: the accidental degeneracy itself, and the tuningrequired to obtain the observed �c66 in a way consistent withthe null results in this report.E. Implications for understanding ultrasound dataThe above analysis, combined with the fact that the lit-erature values of �c66 have a large variation, causes us tospeculate that all the reported ultrasound measurements of�c66 substantially exceed the value (which may well be zero)that can be attributed to a homogeneous, two-component orderparameter alone. We propose as subjects for future study theeffects on the ultrasound measurements of quenched disor-der, particularly extended defects, in the crystals. Possible064514-8Tc AND THE ELASTOCALORIC EFFECT OF … PHYSICAL REVIEW B 110, 064514 (2024)inhomogeneities of the superconducting state itself shouldalso be carefully considered. For example, there is evidencefor domain walls in the superconducting state [25,26,51], andtheir motion, which was not considered here, might affectultrasound measurements. We note in passing that attribut-ing the variation in the ultrasound results to differences inthe measurement frequency does not help to resolve thediscrepancies that our work has highlighted. The reverse istrue, because it is more likely that lower-frequency mea-surements would yield the thermodynamically correct result,but it is the lowest-frequency measurement that yielded thelargest �c66. Finally, we note that the limit |dTc/dε6| < 1.3 Kis independent of anything concluded from Eqs. (13)–(17).Within the standard Ginzburg-Landau theory described inAppendix E, this limit is satisfied automatically by any single-component order parameter, for which |dTc/dε6| = 0. Fortheories involving two-component order parameters it sets akey constraint, along with the others regarding the observ-ability of transition splitting in ECE measurements. Theseconstraints are stringent, and it remains to be seen whetherthey can be met by any plausible microscopic theory of atwo-component order parameter.VI. CONCLUSIONWe have measured Tc and the elastocaloric effect inSr2RuO4 under uniaxial stress applied along the crystalline[110] direction, and we found no sign of a two-componentsuperconducting state. Namely, we found neither a cusp in thestrain dependence of Tc around zero strain, nor a second tran-sition in elastocaloric effect data under nonzero applied strain.To reconcile our data with even the smallest reported jumpin c66 under a hypothesis of homogeneous two-componentsuperconductivity requires an extreme level of tuning forall proposed order parameters, while some are effectivelyruled out. The difficulty in obtaining clear thermodynamicevidence for two-component superconductivity, both in theresults reported here and in previous measurements under[100] uniaxial stress, means that the possibility that the super-conducting order parameter of Sr2RuO4 is single-component,without breaking time-reversal symmetry, must be seriouslyconsidered. The true experimental conditions and/or theinterpretations of the measurement results, both thermody-namic and nonthermodynamic, that have shown evidence fortime-reversal symmetry breaking should be reinvestigated.However, it is a fact that a large number of experimentalprobes have found unusual behavior in Sr2RuO4, such asthe jump in c66, increased ultrasound dissipation below Tc[51], nonzero Kerr rotation [21], and anomalous switchingbehavior in junctions [25,52], among others. Therefore, evenif the bulk order parameter turns out to be both spin-singletand single-component, it appears probable that there is nev-ertheless something unusual about the superconductivity ofSr2RuO4, and that the nature of this “unusualness” has per-haps not been identified even in approximate form by theresearch community.The data that support the findings of this study are openlyavailable from the Max Planck Digital Library [53].necksmounting tabflexuressample caps10 mmsample 3:FIG. 7. Photograph of the sample carrier used to reduce hystere-sis for samples 2 and 3. A photograph of sample 3 is also shown.ACKNOWLEDGMENTSThis work was supported by the Max Planck Society.A.P.M. (Project No. A10) and G.P., J.S. (Project No. B01) ac-knowledge the financial support of the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) CRCTRR 288-422213477 ElastoQMat. C.W.H. acknowledgessupport from the Engineering and Physical Sciences ResearchCouncil (U.K.) (EP/X012158/1). J.S. acknowledges supportby a Weston Visiting Professorship at the Weizmann Instituteof Science, where part of this work was performed. T.S. andM.B. acknowledge the support of NSERC, in particular theDiscovery Grant (RGPIN-2020-05842), the Accelerator Sup-plement (RGPAS-2020-00060), and the Discovery LaunchSupplement (DGECR-2020-00222). N.K. is supported byJSPS KAKENHI (No. JP18K04715, No. JP21H01033, No.JP22K19093, and No. 24K01461).APPENDIX A: SAMPLE CARRIER DESIGNFOR LOW HYSTERESISTo cancel differential thermal contraction between thepiezoelectric actuators and the body of the uniaxial stress cell,there are three actuators. The outer two are joined electri-cally, and ideally move identically. If they do not, a torqueis generated within the cell that deforms the cell body andcan yield a transverse displacement applied across the sample.That is a problem for measurement of Sr2RuO4 under 〈110〉uniaxial stress: a transverse displacement across the samplegenerates shear strain with 〈100〉 principal axes within thesample, and Tc of Sr2RuO4 responds very sensitively to 〈100〉shear strain. Hysteresis in this torque, from hysteresis in theactuator motion, resulted for sample 1 in hysteresis in Tc thatwas large compared with the signal we aimed to measure.To reduce this hysteresis, the sample carrier photographed inFig. 7 was used for samples 2 and 3. It incorporates necks thatattenuate the transmission of transverse displacements to thesample.APPENDIX B: CALIBRATION OF THE ECEThe elastocaloric effect η ≡ dT/dε is obtained by ap-plying a small ac strain to a sample and measuring the ac064514-9FABIAN JERZEMBECK et al. PHYSICAL REVIEW B 110, 064514 (2024)1.0(b)1.2 1.4 1.6 1.8 2.0T (K)0.00.51.01.52.02.5a(T)δε(T)(10-5)σ110 (GPa):sample Asample B-0.7 -0.5 0.1-0.1-0.31.0(a)-101.2 1.4 1.6 1.8 2.0T (K)thermocouplevoltage(nV)TcfitVn(Tc)Vs(Tc )FIG. 8. (a) Extraction of Tc from elastocaloric data. The data hereare from sample A at 0 GPa. (b) Derivation of A(T )δε(T ) for samplesA and B to obtain agreement with Eq. (11) in the normal state.temperature response. This ac strain can be superimposedonto much larger dc strains to measure the elastocaloriceffect at different applied pressures. Details can be found inRefs. [43,44]. The temperature response is measured with aAu/AuFe thermocouple, which is glued by epoxy (Dupont6838) to the center of the sample.An example of extraction of �η/ηn is shown in Fig. 8(a).The measured thermocouple voltage is extrapolated fromabove and below the transition into the transition region.The temperature at which the data pass through the medianline between these extrapolations is identified as Tc. The ex-trapolation from below to Tc is identified as Vs(Tc), and theextrapolation from above is identified as Vn(Tc), and �η/ηn isset to (Vs − Vn)/Vn. As discussed above, applying Eq. (9) withdTc/dε set to 12.0 K yields γ1 = 0.43 J/mol K2.The conversion from thermocouple voltage V to η is givenbyη(T ) = V (T )S(T )A(T )δε(T ),where S(T ) is the Seebeck coefficient of the thermocouple,0 < A < 1 is the degree of adiabaticity, and δε is the strain os-cillation amplitude. S(T ) was determined by a thermocouplecalibration in reference to a calibrated RuO2 thermometer inRefs. [54,55]. We then derive the form of A(T ) × δε(T ) thatyields agreement in the normal state with Eq. (11). A(T ) ×δε(T ) is obtained as a third-order polynomial, α0 + α2T 2 +α3T 3; the T -linear term is omitted because linear variation inthe T → 0 limit is not generally expected. Results are shownin Fig. 8(b). These derived curves are then extrapolated tobelow Tc, yielding results shown in Fig. 3.APPENDIX C: JUMP IN ECEIn Sec. IV of the main text we showed how the elastocaloriccoefficient for a single-component order parameter is derivedfrom a Ginzburg-Landau ansatz. In this Appendix, we derivethe elastocaloric coefficient of two-component order. Equa-tion (9) can be rewritten toη = ηn +(−ηn + dTc0dε)(1 + Tc0�C0CnT)−1. (C1)In the case of strain-split transitions, Eq. (C1) still appliesat the first transition, at Tc. We have only a relabelling ofquantities, following the above-described notation for splittransitions. Let η1 be the elastocaloric coefficient for thetemperature range T2 < T < Tc. We haveη1 = ηn +(−ηn + dTcdε)(1 + Tc�CcCnT)−1. (C2)Let η2 be the elastocaloric coefficient at T < T2. To obtainη2, we take Eq. (C2) and replace ηn with η1, �Cc with �C2,and Tc with T2. We assume that Tc and T2 are both in thestrain-linear regime, and we apply the condition Eq. (19).After simplification, the result isη2 = ηn +(−ηn + dTc0dε)(1 + Tc0�C0CnT)−1. (C3)If ε is ε6, then Eq. (C3) simplifies toη2 = ηn − ηn(1 + Tc0�C0CnT)−1. (C4)Equation (C3) is precisely the same as Eq. (C1), though witha slight difference in interpretation: in Eqs. (C3) and (C4),Tc0 is the transition temperature that would be obtained in theabsence of splitting.To evaluate the above expressions and obtain the left-hand bounding line in Fig. 6, we set γ0 = 0.038 J/mol K2,Tc0 = 1.5 K, γ1 = 0.43 J/mol K2, β = 0.000 197 J/mol K4,and ε110 = −5 × 10−4. Because the relevant data were takenalmost entirely under compressive stress, we set dTc/dε110 =12.0 K − α6 × dTc/dε6.APPENDIX D: COMPONENT ANALYSISAs a cross-check we can calculate from the stress de-pendence of Tc under [110] pressure, dTc/dσ110 = −64 ±7 mK/GPa, and the stress dependence of Tc under [001]pressure, dTc/dσ001 = −76 ± 5 mK/GPa [40], the stress de-pendence of Tc under hydrostatic pressure and compare thisto the experimental results. Based on the 4 K elastic modulireported in [34], we obtain the following relations betweenstrain and applied stress:dεddσ110= 0.003 07 GPa−1,dε3dσ110= −0.001 02 GPa−1,dεddσ001= −0.002 04 GPa−1,dε3dσ001= 0.004 57 GPa−1,dεddσhyd= 0.004 11 GPa−1,dε3dσhyd= 0.002 54 GPa−1.(D1)In these expressions, εd = ε1 + ε2. So we have0.064 ± 0.007 K = 0.003 07dTcdεd− 0.001 02dTcdε3,0.076 ± 0.005 K = −0.002 04dTcdεd+ 0.004 57dTcdε3.Solving these equations yieldsdTcdεd= 31.0 ± 2.6 K,dTcdε3= 30.4 ± 1.8 K.064514-10Tc AND THE ELASTOCALORIC EFFECT OF … PHYSICAL REVIEW B 110, 064514 (2024)TABLE I. Irreducible representations (irreps) of D4h under whichthe bilinear forms φμ transform. The bilinears are constructedfrom two-component order parameters � = (�1, �2) according toEq. (E1). � belongs to the 2D irreps Eg,u on the left, whereas onthe right �1,2 belong to two 1D irreps �1,2, respectively. The ±irrep superscripts describe the behavior under time reversal. Onlyaccidentally degenerate pairs whose �1 ⊗ �2 = B2g are analyzed inthis Appendix.� ∈ Eg,u �1 ∈ �1, �2 ∈ �2Bilinear Irrep Bilinear Irrepφ0 A+1g φ0 A+1gφx B+2g φx (�1 ⊗ �2)+φy A−2g φy (�1 ⊗ �2)−φz B+1g φz A+1gUsing the third line of Eq. (D1) yields dTc/dσhyd = 0.204 ±0.012 K/GPa.APPENDIX E: GINZBURG-LANDAU ANALYSISHere we analyze the response of a two-component or-der parameter � = (�1,�2)ᵀ to σ6 shear stress within theGinzburg-Landau framework, under the assumptions of ho-mogeneous strain and superconductivity. While the analysisof a symmetry-protected two-component order parameter hadalready been done for the D4h point group [34,35], the caseof accidental degeneracy has not been analyzed in the liter-ature to the degree of detail required for our analysis. Thesymmetry-protected case corresponds to the two-dimensionalirreducible representations Eg and Eu whose wave functionswe may write as (dxz, dyz) and (px, py), respectively. Acci-dental degeneracy could, in principle, be between any pairof one-dimensional irreducible representations (irreps). Weconsider only those degenerate pairs that couple linearly to σ6,which are A1g ⊕ B2g (s, dxy) and B1g ⊕ A2g (dx2−y2 , gxy(x2−y2 )).Odd-parity 1D irrep pairs, such as A1u ⊕ B2u and B1u ⊕A2u, are also possible in principle, but not deemed likely dueto Pauli limiting [8,9] and NMR Knight shift experiments[11,17–20]. Quadratic coupling to σ6 does not induce a jumpin the shear elastic modulus c66 nor does it split the transition.Before we proceed with the Ginzburg-Landau analysis, letus briefly discuss how we obtained the phase diagrams shownin Fig. 4. Introduce the bilinear formsφμ = �†τμ�, (E1)where τ0 is the 2 × 2 unit matrix and τx,y,z are Pauli matri-ces. The transformation properties of φμ are summarized inTable I. A sufficient condition for a cusp in Tc(σ6) is that thereexists a φμ that transforms like the shear strain σ6 ∈ B+2g. Inour case, this is only possible for φx. If φx acquires a nonzeroexpectation value below Tc at ε6 = 0, then strain acts like aconjugate field that lifts the degeneracy between ±〈φx〉, andonly one transition takes place since the symmetry associatedwith φx is already broken. If, on the other hand, φx is not thebilinear that acquires a finite expectation value below Tc atε6 = 0, an additional symmetry can still break, resulting in asecond transition.The Ginzburg-Landau expansion of the free energy in theabsence of stress is given byF = Fn + a2φ0 + u4φ20 +∑μ=x,y,zvμ4φ2μ+ ã2φz + ṽ4φ0φz. (E2)From Table I, it is easy to see that this is the most general formof an invariant (A+t1g) function that is quadratic in φμ, and thatis quartic in �. Due to the Fierz identityφ20 =∑μ=x,y,zφ2μ, (E3)there is a redundancy between the vμφ2μ terms that we elimi-nate by settingvz = 0. (E4)In the case of a symmetry-protected degeneracy, φz trans-forms under B1g and therefore ã = ṽ = 0. Below the transitiontemperature Tc0, the quadratic coefficient changes sign. Toleading order in temperature, a(T ) is thus linear in T witha positive slope ȧ > 0:a = (T − Tc0) × ȧ, (E5)whereas the quartic coefficients are T -independent.When �1,2 belong to two 1D irreps, φz transforms triviallyand both ã and ṽ are allowed to be finite. However, since�1 and �2 are unrelated by symmetry, we may rescale them(�1,�2) �→ (s�1, s−1�2) by a factor s = ( u+ṽu−ṽ)1/8so thatafter the rescalingṽ = 0, (E6)which we assume henceforth. Regarding ã, in the expansionF = ȧ1(T − Tc0,1)|�1|2 + ȧ2(T − Tc0,2)|�2|2 + · · · the fine-tuning of the two transition temperatures corresponds to therequirement that Tc0,1 = Tc0,2 ≡ Tc0. Hence a(T ) is given byEq. (E5) with ȧ = ȧ1 + ȧ2, whileã(T ) = α × a(T ) (E7)for a T -independent coefficient α = ȧ1−ȧ2ȧ1+ȧ2. α can take anyvalue in between −1 and 1 and reflects the absence of asymmetry transformation connecting �1 and �2. Thus in thesymmetry-protected case the only formal difference is thatα = 0, given that ṽ = 0 in both cases.Let us now include elasticity. When strains εi are presentin the system, they couple to the superconductivity viaFc =6∑i=12∑a,b=1λiabεi�∗a�b, (E8)where λiab are the coupling constants. As it turn out, when theelastic free energy is quadratic in εi,Fε = 126∑i, j=1ci j,0εiε j, (E9)one may decouple the elastic and superconducting parts of thefree energy, greatly simplifying the free-energy minimization064514-11FABIAN JERZEMBECK et al. PHYSICAL REVIEW B 110, 064514 (2024)problem. Here ci j,0 is the elastic tensor in the absence of super-conductivity. This decoupling is accomplished by introducingthe “external” strainεi,0 ≡ εi +6∑j=12∑a,b=1c−1i j,0λiab�∗a�b, (E10)which is decoupled from � and directly related to the ex-ternal stress: εi,0 = ∑6j=1 c−1i j,0σ j . It is the strain that wouldbe obtained under a given set of stresses in the absence ofsuperconductivity.In practice, the difference between εi,0 and the total strain εiis negligible for Sr2RuO4: the larger of the two reported valuesof �c66 is ∼10−5c66,0 [34,35], and the experimental upperlimit on any spontaneous nematic strain is on the order of 10−8[Eq. (E48)], far smaller than the scale of the strains appliedin this work. For this reason, in the main text we make nodistinction between εi,0 and εi. Here, we retain this distinctionto be able to calculate the jump in the shear modulus �c66.In the presence of σ6 external shear stress, the total freeenergy after decoupling therefore equalsF = Fn + Fε0 + F�0, (E11)where the elastic part isFε0 = 12 c66,0ε26,0 − σ6ε6,0, (E12)and the superconducting part isF�0 = a2(|�1|2 + |�2|2) + αa2(|�1|2 − |�2|2)+ u4(|�1|2 + |�2|2)2 + γ u|�1|2|�2|2+ γ ′ u2(�∗1�∗1�2�2 + �∗2�∗2�1�1)+ σ6c−166,0λ6(�∗1�2 + �∗2�1). (E13)Here we have explicitly written out F�0 in terms of � insteadof φμ. The form of the coupling to σ6 ∈ B+2g follows fromTable I; for the accidentally degenerate case, we assumed�1 ⊗ �2 = B2g. By enacting (�1,�2) �→ (�1,−�2), we canalways make λ6 > 0, which we henceforth assume. For laterconvenience, we parametrized the quartic coefficients in termsof γ , γ ′ which are related to the vx,y of Eq. (E2) via vx =(γ + γ ′)u and vy = (γ − γ ′)u. In shifting from εi to εi,0, thequartic coefficients u, γ , γ ′ have been renormalized.The free energy is bounded from below when u > 0 andγ − |γ ′| > −1; these constraints define the physical part ofthe parameter space. To find the minimum of F�0, we use thespherical parametrization(�1�2)= �0(cos θsin θ eiφ)(E14)in terms of whichF�0 = A(θ, φ)a2�20 + U (θ, φ)u4�40, (E15)whereA(θ, φ) = 1 + α cos(2θ ) + β sin(2θ ) cos(φ), (E16)U (θ, φ) = 1 + �(φ) sin2(2θ ), (E17)�(φ) = γ + γ ′ cos(2φ). (E18)Here we have introduced the shorthandβ ≡ 2λ6ε6,0a. (E19)The saddle point equations for the nontrivial solution whose�20 = −aA/(uU ) > 0 are0 = sin(φ) sin(2θ )[γ ′ cos(φ) sin(2θ ) − βU2A], (E20)0 = � sin(2θ ) cos(2θ ) + UA[α sin(2θ ) − β cos(φ) cos(2θ )].(E21)1. Solutions for σ6 = 0In the absence of applied stress (β = 0), these saddle pointequations are easily solved. They give three classes of solu-tions:(i) �1 or �2 only: θ = 0 or π/2, � ∼ (1, 0) or (0,1).(ii) B2g-nematic: θ = 12 arccos(−αx+), φ = 0 or π ,� ∼ (1,±1).(iii) TRSB: θ = 12 arccos(−αx−), φ = ±π2 , � ∼ (1,±i).In these expressions,x± ≡ 1 + γ ± γ ′γ ± γ ′ . (E22)In the case of symmetry-protected degeneracy (α = 0), the�1-only and �2-only ground states are degenerate and maybe identified with B1g-nematic order.The free-energy values for these solutions areF�0 = − a24u×⎧⎪⎪⎪⎨⎪⎪⎪⎩(1 + α)2, �1 only,(1 − α)2, �2 only,1−α2x+1+γ+γ ′ , B1g − nem,1−α2x−1+γ−γ ′ , TRSB.(E23)The preferred global minimum isγ − |γ ′| > − |α|1 + |α| , α > 0 : �1 only,γ − |γ ′| > − |α|1 + |α| , α < 0 : �2 only,−1 < γ + γ ′ < − |α|1 + |α| , γ ′ < 0 : B2g-nem,−1 < γ − γ ′ < − |α|1 + |α| , γ ′ > 0 : TRSB. (E24)For the α = 0 case, the corresponding phase diagram is shownin Fig. 5 of the main text. For finite α, the B1g-nematic regionof Fig. 5 becomes �1 or �2 only, and its lower edge γ −|γ ′| = 0 is shifted downward by |α|1+|α| .2. Solutions for σ6 �= 0First consider T > Tc0. In this case, F�0 < 0 is only ob-tained when A(θ, φ) < 0. By minimizing Eq. (E16), we seethat the minimum of A(θ, φ) is 1 −√α2 + β2 and has φ = 0or π with θ = 0, which corresponds to B2g-nematic order.064514-12Tc AND THE ELASTOCALORIC EFFECT OF … PHYSICAL REVIEW B 110, 064514 (2024)HenceTc = Tc0 + λ6|ε6,0|ȧ2√1 − α2(E25)and the symmetry is B2g-nematic.Now consider reducing T below Tc. As illustrated in Fig. 4,a second transition takes place when the ground state breakstime-reversal symmetry, whether the degeneracy is symmetry-protected or not, and when the ground state is B1g-nematic. Inthe latter case, the degeneracy must be symmetry-protectedbecause only then is the (�1,�2) �→ (�2,�1) symmetrypresent, which forbids a smooth crossover between B1g- andB2g-nematic states.To determine the lower transition temperature T2, we needto solve the saddle point equations (E20) and (E21) and fig-ure out which solution yields the smallest free energy. Forthe nematic case, φ = 0 or π , and θ is determined by thetranscendental equation[cos(2θ ) + αx+] sin(2θ ) = βγ + γ ′ cos(2θ ). (E26)When the σ6 = β = 0 ground state is B2g-nematic, θ of theglobal minimum changes smoothly with σ6 and there is nosecond transition. The same happens when �1 or �2 are theground states and α = 0: we have a smooth crossover, as onecan show by analyzing the bifurcation of the solutions.When the ground state is B1g-nematic and α = 0, B1g-nematic solutions overtake the B2g-nematic solutions below|β| = γ + γ ′, yieldingT2 = Tc0 − λ6|ε6,0|ȧ2γ + γ ′ . (E27)When the ground state is TRSB with symmetry-protecteddegeneracy,T2 = Tc0 − λ6|ε6,0|ȧ1 + γ − γ ′γ ′ . (E28)Along the line γ = γ ′ that is the boundary between the B1gand TRSB regions of the α = 0 parameter space [Eq. (E24)],these two expressions for T2 agree. When the ground state isTRSB with accidental degeneracy,T2 = Tc0 − λ6|ε6,0|ȧ1 + γ − γ ′γ ′√1 − α2x2−. (E29)In the TRSB case, one may solve the saddle-point equations inclosed form:θ = 12 arccos(−αx−), (E30)φ = ± arccos⎛⎜⎝λε6,0a1 + γ − γ ′γ ′√1 − α2x2−⎞⎟⎠, (E31)F�0 = − a24u1 − α2x−1 + γ − γ ′ − λ26ε26,02uγ ′ . (E32)3. Ehrenfest relationsThe jump in the heat capacity across the superconductingtransition is given by�C0Tc0= − ∂2F�0∂T 2∣∣∣∣T =Tc0,σ6=0. (E33)From the free-energy expressions of Eq. (E23),�C0Tc0= ȧ22u×⎧⎪⎪⎨⎪⎪⎩(1 + |α|)2, �1 or �2 only,1−α2x+1+γ+γ ′ , B2g-nem,1−α2x−1+γ−γ ′ , TRSB.(E34)The shear elastic modulus c66 below Tc is given by1c66= 1c66,0− ∂2F�0∂σ 26∣∣∣∣T,σ6=0. (E35)The jump �c66 = c66,0 − c66|T =Tc0 is the difference betweenc66 just above Tc0 and that just below it.When the ground state is �1 or �2 only,�c66 = 2λ26u1 + |α|(γ + γ ′)(1 + |α|x+). (E36)This is derived by solving Eq. (E26) for small β. In the case ofsymmetry-enforced degeneracy (α = 0), that is, B1g-nematicground states, one obtains the following Ehrenfest relation:�c66 = �C0Tc0∣∣∣∣ dTcdε6,0∣∣∣∣∣∣∣∣ dT2dε6,0∣∣∣∣. (E37)In the general α = 0 case, we could try using Tc instead of T2above, but the corresponding dimensionless ratio can be anypositive real number, depending on the values of α and γ + γ ′which we do not know.When the ground state is B2g-nematic, by solving Eq. (E26)one finds that�c66 = 2λ26u1 − α2x3+(1 + γ + γ ′)(1 − α2x2+), (E38)and therefore�c66�C0Tc0∣∣ dTcdε6,0∣∣∣∣ dTcdε6,0∣∣ = (1 − α2)(1 − α2x3+)(1 − α2x+)(1 − α2x2+). (E39)When α = 0, this expression reduces to the standard Ehren-fest relation. The stability condition for B2g-nematic order[Eq. (E24)] corresponds to −1/|α| < x+ < 0 and the right-hand side can equal any number between (1 − α2) and +∞for α = 0 and x+ in this range.When the ground state is TRSB, the second derivative ofEq. (E32) with respect to ε6,0 yields�c66 = λ26uγ ′ . (E40)The corresponding Ehrenfest relation takes the form�c66�C0Tc0∣∣ dTcdε6,0∣∣∣∣ dT2dε6,0∣∣ =√1 − α2√1 − α2x2−1 − α2x−� 1. (E41)064514-13FABIAN JERZEMBECK et al. PHYSICAL REVIEW B 110, 064514 (2024)In the −1/|α| < x− < 0 region where TRSB is the groundstate [Eq. (E24)], the right-hand side takes values in between0 and 1, and for α = 0 it equals 1.4. Ratio relationsHere we show that the ratios of the jumps at the upper andlower transitions are related. These relations hold only for thesymmetry-protected case (α = 0). Denote �Cc and �c66,c thejumps at the upper transition (T = Tc), and �C2 and �c66,2the jumps at the lower transition (T = T2).The jumps at the upper transition are (α = 0, σ6 = 0)�CcTc= ȧ22u11 + γ + γ ′ , (E42)�c66,c = 2λ26u11 + γ + γ ′ . (E43)The jumps at the lower transition are (α = 0, σ6 = 0)�C2T2= ȧ22u×⎧⎨⎩γ+γ ′1+γ+γ ′ , B1g-nem,2γ ′(1+γ+γ ′ )(1+γ−γ ′ ) , TRSB,(E44)�c66,2 = 2λ26u×{ 1(γ+γ ′ )(1+γ+γ ′ ) , B1g-nem,1+γ−γ ′(1+γ+γ ′ )2γ ′ , TRSB.(E45)To find these expressions, we had to solve Eq. (E26) aroundthe β at which the solutions bifurcate. Note that �Cc/Tc +�C2/T2 and �c66,c + �c66,2 reproduce the previous �C0/Tc0and �c66 with α = 0. Combining, we obtain the ratio relations∣∣ dT2dε6,0∣∣∣∣ dTcdε6,0∣∣ =�CcTc�C2T2= �c66,2�c66,c={ 1γ+γ ′ , B1g-nem,1+γ−γ ′2γ ′ , TRSB.(E46)5. Nematic strainThe second term in Eq. (E10) defines the “internal” strain,which is the strain generated by the superconducting orderparameter:εnem6 = − λ6c66(�∗1�2 + �∗2�1)= − λ6c66�20 sin(2θ ) cos φ. (E47)Due to the proportionality to cos φ, when σ6 = 0 only theB2g-nematic states generate a nonzero ε6. Its value is boundedfrom above throughc66,0∣∣εnem6∣∣�C0Tc0∣∣ dTcdε6,0∣∣|T − Tc0|=√1 − α2√1 − α2x2+1 − α2x+� 1, (E48)where the right-hand side is in between 0 and 1 in therange −1/|α| < x+ < 0 where B2g-nematic order is preferred[Eq. (E24)], and for α = 0 equals 1.6. Case of B1g stressAs discussed in the main text, if one combines the measure-ments of the current paper with those performed under 〈100〉uniaxial stress [33], one can put tight constraints on whereprecisely Sr2RuO4 must be in the phase diagram of Fig. 5.To make contact with the measurements under 〈100〉 uni-axial stress, here we briefly summarize the results of theGinzburg-Landau analysis for B1g stress, σB1g = 12 (σ1 − σ2) =σ100/2. Superconductivity couples linearly to B1g stress onlyin the case of symmetry-protected degeneracy (α = 0), whichwe henceforth consider.The coupling to B1g stress takes the formF�0 = · · · + σB1gc−1B1gλB1g (|�1|2 − |�2|2), (E49)where cB1g = 12 (c11 − c12). By a rotation�̃ = 1√2(1 −11 1)� (E50)and reparametrizationũ = (1 + γ + γ ′)u, (E51)γ̃ = − 12γ − 32γ ′1 + γ + γ ′ , (E52)γ̃ ′ = − 12γ + 12γ ′1 + γ + γ ′ , (E53)one obtains a free energy identical in form to Eq. (E13). Henceall the previous formulas carry over if we replace u, γ , γ ′, λ6with ũ, γ̃ , γ̃ ′, λB1g , and exchange what one identifies as B1gwith B2g, and vice versa.The upper transition temperature is given byTc = Tc0 + 2λB1g∣∣εB1g,0∣∣ȧ. (E54)At finite B1g stress, the superconductivity is B1g-nematicslightly below Tc. When B1g-nematic pairing is the groundstate, there is no second transition. For the other two cases,T2 = Tc0 − 2λB1g∣∣εB1g,0∣∣ȧ×{−x+, B2g-nem,−x−, TRSB.(E55)The heat-capacity jumps are�CcTc= ȧ22u, (E56)�C2T2= ȧ22u×{−1/x+ for B2g-nem,−1/x− for TRSB.(E57)The jumps in the B1g elastic constants are�cB1g,c =2λ2B1gu, (E58)�cB1g,2 =2λ2B1gu×{−x+ for B2g-nem,−x− for TRSB.(E59)The total jumps are obtained by summing the jumps at theupper and lower transition, if it takes place.064514-14Tc AND THE ELASTOCALORIC EFFECT OF … PHYSICAL REVIEW B 110, 064514 (2024)The Ehrenfest relation for B1g-nematic states is�cB1g = �C0Tc0∣∣∣∣∣ dTcdεB1g,0∣∣∣∣∣∣∣∣∣∣ dTcdεB1g,0∣∣∣∣∣. (E60)The Ehrenfest relation when B2g-nematic or TRSB pairing ispreferred in the absence of stress is�cB1g = �C0Tc0∣∣∣∣∣ dTcdεB1g,0∣∣∣∣∣∣∣∣∣∣ dT2dεB1g,0∣∣∣∣∣. (E61)Ratio relations are∣∣ dT2dεB1g,0∣∣∣∣ dTcdεB1g,0∣∣ =�CcTc�C2T2= �cB1g,2�cB1g,c=⎧⎨⎩−x+, B2g-nem,−x−, TRSB.(E62)[1] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita,J. G. Bednorz, and F. Lichtenberg, Nature (London) 372, 532(1994).[2] A. P. 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