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Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress


Strontium ruthenate (Sr2RuO4) continues to present an important test of our understanding of unconventional superconductivity, because while its normal-state electronic structure is known with precision, its superconductivity remains unexplained. There is evidence that its order parameter is chiral, but reconciling this with recent observations of the spin part of the pairing requires an order parameter that is either finely tuned or implies a new form of pairing. Therefore, a definitive resolution of whether the superconductivity of Sr2RuO4 is chiral is important for the study of superconductivity. Here we report the measurement of zero-field muon spin relaxation—a probe sensitive to weak magnetism—on samples under uniaxial stresses. We observe stress-induced splitting between the onset temperatures of superconductivity and time-reversal symmetry breaking—consistent with the qualitative expectations for a chiral order parameter—and argue that this observation cannot be explained by conventional magnetism. In addition, we report the appearance of bulk magnetic order under higher uniaxial stress, above the critical pressure at which a Lifshitz transition occurs in Sr2RuO4.

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Fig. 1: Hypothesis and setup.
Fig. 2: Results on unstressed Sr2RuO4.
Fig. 3: Stress-induced splitting between Tc and TTRSB.
Fig. 4: Magnetic order.
Fig. 5: Experimental phase diagram.

Data availability

The data shown in Figs. 25 are available as the source data. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.


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This work has been financially supported by the Deutsche Forschungsgemeinschaft (GR 4667/1, GRK 1621 and SFB 1143 projects C02 and C09) and the Max Planck Society. Y.M., T.M. and J.S.B. acknowledge the financial support of JSPS Kakenhi (JP15H5852, JP15K21717 and JP17H06136) and the JSPS Core-to-Core Program. N.K. acknowledges the financial support from JSPS Kakenhi (no. JP18K04715) and JST-Mirai Program (no. JPMJMI18A3). A.N. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 701647. This work was partially performed at the Swiss Muon Source (SμS), PSI, Villigen. We acknowledge fruitful discussions with A. Amato, B. Andersen, E. Babaev, S. Blundell, A. Charnukha, D. Efremov, I. Eremin, C. Kallin, A. Ramires, B. Ramshaw, A. Rømer, T. Scaffidi, M. Sigrist, C. Timm and S. Yonezawa. We also acknowledge H.-S. Xu for his contribution in the crystal growth as well as T. Shiroka and C. Wang for technical support. We thank A. Gilman, P. P. Orth and R. M. Fernandes for results from the Landau theory of a two-component order parameter.

Author information

Authors and Affiliations



V.G., S.G., R.S., J.-C.O., A.N., D.D., Z.G., H.L. and H.-H.K. performed the μSR measurements. V.G. performed the heat capacity measurements. V.G., S.G. and H.L. analysed the raw μSR data. S.G., A.N., M.E., F.B. and C.W.H. built and characterized the stress apparatus. M.E.B. and J.P. characterized the samples. N.K., D.A.S., J.S.B. and T.M. grew the samples. V.G., Y.M., A.P.M., H.L., C.W.H. and H.-H.K. supervised the research. V.G., C.W.H. and H.-H.K. wrote the paper. C.W.H. designed the stress apparatus. H.-H.K. initiated this study. All the authors discussed the results and implications, and they commented on the manuscript.

Corresponding authors

Correspondence to Vadim Grinenko, Clifford W. Hicks or Hans-Henning Klauss.

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The authors declare no competing interests.

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Peer review informationNature Physics thanks Morten Eskildsen, Jeff Sonier and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 The sample setup.

a, Photographs of Sample F, showing the locations of the ac susceptibility coil, the hematite masks, and the force-sensing strain gauges. b, A view from the muon direction. The beam diameter is 1 cm. (c) A schematic of the sample holder, in which the moving portion is coloured blue. This holder slots into a force generator, described in Ref. 59. d, Force-displacement data for Samples D-F. For Sample D, a chip broke from one of the two pieces of Sr2RuO4 at a load of 500 N. For Samples E and F, the force-displacement relationships are straight lines, showing that the sample did not fracture under the load. Note that in Sample F the data fall on two distinct lines. This may have originated in a slip in the apparatus that altered the calibration of the displacement sensor; it is not of the form expected for damage to the sample.

Extended Data Fig. 2 TF data.

(a-b) Transverse-field μSR time spectra above and below Tc, with a muon spin polarization 45 with respect to the sample c axis and Bc. (c-d) Fourier transforms of the time spectra shown in panels (a) and (b). The fit to determine the sample contribution is done in the time domain.

Extended Data Fig. 3 Testing different backgrounds.

a, A(t) resulting from the single histogram analysis of Sample D at − 0.43 GPa under different assumptions about the background: non-relaxing, and with 55% of the background muon spins taken to relax following the relaxation function of copper. b, Results from the single histogram analysis for the relaxation rate λ taking different portions of copper in the background. c, Data in panel (b) with temperature independent constants subtracted. Assuming different backgrounds is seen to have essentially no effect on temperature-dependent relative changes in λ extracted from analysis. d, Fourier transforms of the transverse-field μSR time spectra of Sample A2 above and below Tc. e, Comparison of the temperature dependence of the zero field muon spin relaxation rate (left scale) and the specific heat data (right scale) for Samples A and A2.

Extended Data Fig. 4 Control measurement.

a, Photograph of the ‘sample’ for these measurements: a hematite mask is put in place of Sr2RuO4. Therefore the signal originates from muons stopping in hematite and in the non-magnetic background (cryostat walls, sample holder frame, and etc.). b, Strongly damped oscillations of the muon spin polarisation can be seen in the first 50 ns. There is no notable change in their form between 1.6 and 0.4 K. c, Weak transverse-field measurements, showing the signal from muons that did not implant into the hematite. d, Zero-field measurements: the polarisation of the muons implanting outside the hematite decays on a μs time scale. Fits exclude the first 50 ns, shown in panel (b). e, The zero-field muon spin relaxation rate has no significant temperature dependence.

Extended Data Fig. 5 Further characterization of the samples.

Additional sample characterisation data for samples a-e. Penetration depths were determined through transverse-field μSR measurements. Heat capacity data for Samples c,d, and e were recorded from portions of the samples that were removed from the holder after the μSR measurement; for Sample a the measurement was performed on a portion of the sample before the μSR measurement. Note that for Sample c, we show heat capacity data from Sample e, which was extracted from Sample c. Susceptibility data were recorded in situ using the susceptibility coils shown in Extended Data Fig. 1. (g-h) Additional data on Samples E and F at, respectively, -0.70 and -0.79 GPa. (h-i) In situ ac susceptibility raw data at different uniaxial stresses for Samples e and f.

Extended Data Fig. 6 Testing different models to describe magnetic order.

Zero-field asymmetry A(t) in the magnetic state for Sample F at -1.05 GPa, fitted with a Bessel function describing incommensurate spin density wave order, and a damped cosine describing ferromagnetism or commensurate magnetic order. The damped cosine fit performs noticably worse at early times, and also fails to capture the third oscillation at t ≈ 3.5μs. Note that in the single histogram analysis a functional form for A(t) must be assumed to fit the dark count rate, which results in different experimental A(t) for the two forms, even though both are derived from a single data set.

Extended Data Fig. 7 Knight shift.

a, Temperature dependence of the muon Knight shift for Sample A2 in Bab = 8 T. The Knight shift is defined as Kab = [BSRO − BAg]/BAg, where BSRO and BAg are the magnetic fields at the muon stopping site in Sr2RuO4 and the Ag sample holder, respectively. The inset shows a Fourier transform of the high transversal field time spectra at T = 0.12 K. The frequencies for the Ag and Sr2RuO4 are well-resolved. b, Temperature dependence of the muon spin relaxation rate for Sample A2 and the Ag sample holder in Bab = 8 T.

Supplementary information

Supplementary Information

Further discussion of possible magnetic mechanisms and a Ginzburg–Landau analysis.

Source data

Source Data Fig. 2

μSR and specific heat data at zero stress shown in Fig. 2.

Source Data Fig. 3

μSR and a.c. susceptibility data at different uniaxial pressures shown in Fig. 3.

Source Data Fig. 4

μSR and a.c. susceptibility data at P = −1.05 GPa shown in Fig. 4.

Source Data Fig. 5

Superconducting Tc, TTRSB and magnetic transition TN temperatures at different values of the uniaxial pressure shown in Fig. 5.

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Grinenko, V., Ghosh, S., Sarkar, R. et al. Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress. Nat. Phys. 17, 748–754 (2021).

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