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  • Letter
  • Open Access

Possibility of mixed helical p-wave pairings in Sr2RuO4

Wen Huang1,* and Zhiqiang Wang2,†

  • 1Shenzhen Institute for Quantum Science and Engineering & Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China
  • 2James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA

  • *huangw3@sustech.edu.cn
  • †zqwang@uchicago.edu
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Phys. Rev. Research 3, L042002 – Published 7 October, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.L042002

Abstract

The exact nature of the unconventional superconductivity in Sr2RuO4 remains a mystery. At the phenomenological level, no superconducting order parameter proposed thus far seems able to coherently account for all essential experimental signatures. Among the latter is the prominent polar Kerr effect, which implies a nonzero ac anomalous Hall conductivity. Assuming the Kerr effect is intrinsic, it can be accounted for by a bulk chiral Cooper pairing with nonzero orbital angular momentum, such as p+ip or d+id, which, however, has difficulties in being reconciled with other experimental results. Given the situation, in this paper we propose alternative possibilities with complex mixtures of distinct helical p-wave order parameters, namely A1u+iA2u and B1u+iB2u in group theory nomenclature. These states essentially consist of two copies of chiral p-wave pairings with opposite chirality and different pairing amplitudes, and therefore they support intrinsic Hall and Kerr effects. We further show that these states exhibit salient features that may explain several other key observations in this material, including the absence of spontaneous edge current, a substantial Knight shift drop, and possibly signatures in uniaxial strain and ultrasound measurements.

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References (82)

  1. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg, Superconductivity in a layered perovskite without copper, Nature (London) 372, 532 (1994).
  2. Y. Maeno, T. M. Rice, and M. Sigrist, The intriguing superconductivity of strontium ruthenate, Phys. Today 54(1), 42 (2001).
  3. A. P. Mackenzie and Y. Maeno, The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing, Rev. Mod. Phys. 75, 657 (2003).
  4. C. Kallin and A. J. Berlinsky, Is Sr2RuO4 a chiral p-wave superconductor?, J. Phys. Condens. Matter 21, 164210 (2009).
  5. C. Kallin, Chiral p-wave order in Sr2RuO4, Rep. Prog. Phys. 75, 042501 (2012).
  6. Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K. Ishida, Evaluation of spin-triplet superconductivity in Sr2RuO4, J. Phys. Soc. Jpn. 81, 011009 (2012).
  7. Y. Liu and Z. Q. Mao, Unconventional superconductivity in Sr2RuO4, Physica C 514, 339 (2015).
  8. C. Kallin and A. J. Berlinsky, Chiral superconductors, Rep. Prog. Phys. 79, 054502 (2016).
  9. A. P. Mackenzie, T. Scaffidi, C. W. Hicks, and Y. Maeno, Even odder after twenty-three years: The superconducting order parameter puzzle of Sr2RuO4, npj Quantum Mater. 2, 40 (2017).
  10. A. J. Leggett and Y. Liu, Symmetry properties of superconducting order parameter in Sr2RuO4, J. Supercond. Nov. Magn. 34, 1647 (2021).
  11. G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, Time-reversal Symmetry-breaking Superconductivity in Sr2RuO4, Nature (London) 394, 558 (1998).
  12. V. Grinenko, S. Ghosh, R. Sarkar, J. Orain, A. Nikitin, M. Elender, D. Das, Z. Guguchia, F. Brückner, M. E. Barber, J. Park, N. Kikugawa, D. A. Sokolov, J. S. Bobowski, T. Miyoshi, Y. Maeno, A. P. Mackenzie, H. Luetkens, C. W. Hicks, and H. Klauss, Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress, Nat. Phys. 17, 748 (2021).
  13. J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, High Resolution Polar Kerr Effect Measurements of Sr2RuO4: Evidence for Broken Time-Reversal Symmetry in the Superconducting State, Phys. Rev. Lett. 97, 167002 (2006).
  14. F. Kidwingira, J. D. Strand, D. J. van Harlingen, and Y. Maeno, Dynamical superconducting order parameter domains in Sr2RuO4, Science 314, 1267 (2006).
  15. S. Ghosh, A. Shekhter, F. Jerzembeck, N. Kikugawa, D. A. Sokolov, M. Brando, A. P. Mackenzie, C. W. Hicks, and B. J. Ramshaw, Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4, Nat. Phys. 17, 199 (2021).
  16. S. Benhabib, C. Lupien, I. Paul, L. Berges, M. Dion, M. Nardone, A. Zitouni, Z. Q. Mao, Y. Maeno, A. Georges, L. Taillefer, and C. Proust, Ultrasound evidence for a two-component superconducting order parameter in Sr2RuO4, Nat. Phys. 17, 194 (2021).
  17. J-L. Zhang, Y. Li, W. Huang, and F-C. Zhang, Hidden anomalous Hall effect in Sr2RuO4 with chiral superconductivity dominated by the Ru dxy orbital, Phys. Rev. B 102, 180509(R) (2020).
  18. A. Pustogow, Y. Luo, A. Chronister, Y. Su, D. A. Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks, N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown, Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance, Nature (London) 574, 72 (2019).
  19. K. Ishida, M. Manago, and Y. Maeno, Reduction of the O17 knight shift in the superconducting state and the heat-up effect by NMR pulses on Sr2RuO4, J. Phys. Soc. Jpn. 89, 034712 (2020).
  20. J. R. Kirtley, C. Kallin, C. W. Hicks, E.-A. Kim, Y. Liu, K. A. Moler, Y. Maeno, and K. D. Nelson, Upper limit on spontaneous supercurrents in Sr2RuO4, Phys. Rev. B 76, 014526 (2007).
  21. C. W. Hicks, J. R. Kirtley, T. M. Lippman, N. C. Koshnick, M. E. Huber, Y. Maeno, W. M. Yuhasz, M. B. Maple, and K. A. Moler, Limits on superconductivity-related magnetization in Sr2RuO4 and PrOs4Sb12 from Scanning SQUID Microscopy, Phys. Rev. B 81, 214501 (2010).
  22. D. Vollhardt and P. Wölfle, The Superfluid Phases of Helium 3 (Dover, New York, 1990).
  23. K. D. Nelson, Z. Q. Mao, Y. Maeno, and Y. Liu, Odd-Parity Superconductivity in Sr2RuO4, Science 306, 1151 (2004).
  24. J. F. Annett, G. Litak, B. L. Györffy, and K. I. Wysokiński, Spin-orbit coupling and symmetry of the order parameter in strontium ruthenate, Phys. Rev. B 73, 134501 (2006).
  25. T. Scaffidi, J. C. Romers, and S. H. Simon, Pairing symmetry and dominant B band in Sr2RuO4, Phys. Rev. B 89, 220510(R) (2014).
  26. L. D. Zhang, W. Huang, F. Yang, and H. Yao, Superconducting pairing in Sr2RuO4 from weak to intermediate coupling, Phys. Rev. B 97, 060510(R) (2018).
  27. A. T. Rømer, D. D. Scherer, I. M. Eremin, P. J. Hirschfeld, and B. M. Andersen, Knight Shift and Leading Superconducting Instability from Spin Fluctuations in Sr2RuO4, Phys. Rev. Lett. 123, 247001 (2019).
  28. A. T. Rømer, D. D. Scherer, I. M. Eremin, P. J. Hirschfeld, and B. M. Andersen, Fluctuation-driven Superconductivity in Sr2RuO4 from weak repulsive interactions, Mod. Phys. Lett. B 34, 2040052 (2020).
  29. H. S. Roising, T. Scaffidi, F. Flicker, G. F. Lange, and S. H. Simon, Superconducting order of Sr2RuO4 from a three-dimensional microscopic model, Phys. Rev. Research 1, 033108 (2019).
  30. Z. Wang, X. Wang, and C. Kallin, Spin-orbit coupling and spin-triplet pairing symmetry in Sr2RuO4, Phys. Rev. B 101, 064507 (2020).
  31. N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum hall effect, Phys. Rev. B 61, 10267 (2000).
  32. E. Taylor and C. Kallin, Intrinsic Hall Effect in a Multiband Chiral Superconductor in the Absence of an External Magnetic Field, Phys. Rev. Lett. 108, 157001 (2012).
  33. K. I. Wysokiński, J. F. Annett, and B. L. Györffy, Intrinsic Optical Dichroism in the Chiral Superconducting State of Sr2RuO4, Phys. Rev. Lett. 108, 077004 (2012).
  34. Z. Wang, J. Berlinsky, G. Zwicknagl, and C. Kallin, Intrinsic Ac anomalous hall effect of nonsymmorphic chiral superconductors with an application to UPt3, Phys. Rev. B 96, 174511 (2017).
  35. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.3.L042002 for details of the calculation of σH and θK, and further explanations of the absence of cross-gradient terms in the GL free energy that are relevant to spontaneous current.
  36. M. Matsumoto and M. Sigrist, Quasiparticle states near the surface and the domain wall in a px±ipy-wave superconductor, J. Phys. Soc. Jpn. 68, 994 (1999).
  37. A. Furusaki, M. Matsumoto, and M. Sigrist, Spontaneous hall effect in a chiral p-wave superconductor, Phys. Rev. B 64, 054514 (2001).
  38. P. E. C. Ashby and C. Kallin, Suppression of spontaneous supercurrents in a chiral P-wave superconductor, Phys. Rev. B 79, 224509 (2009).
  39. Y. Imai, K. Wakabayashi, and M. Sigrist, Properties of edge states in a spin-triplet two-band superconductor, Phys. Rev. B 85, 174532 (2012).
  40. Y. Imai, K. Wakabayashi, and M. Sigrist, Topological and edge state properties of a three-band model for Sr2RuO4, Phys. Rev. B 88, 144503 (2013).
  41. S. Lederer, W. Huang, E. Taylor, S. Raghu, and C. Kallin, Suppression of spontaneous currents in Sr2RuO4 by surface disorder, Phys. Rev. B 90, 134521 (2014).
  42. A. Bouhon and M. Sigrist, Current inversion at the edges of a chiral P-wave superconductor, Phys. Rev. B 90, 220511(R) (2014).
  43. W. Huang, S. Lederer, E. Taylor, and C. Kallin, Nontopological nature of the edge current in a chiral P-wave superconductor, Phys. Rev. B 91, 094507 (2015).
  44. T. Scaffidi and S. H. Simon, Large Chern Number and Edge Currents in Sr2RuO4, Phys. Rev. Lett. 115, 087003 (2015).
  45. M. Sigrist and K. Ueda, Phenomenological theory of unconventional superconductivity, Rev. Mod. Phys. 63, 239 (1991).
  46. X. Wang, Z. Wang, and C. Kallin, Spontaneous edge current in higher chirality superconductors, Phys. Rev. B 98, 094501 (2018).
  47. S. A. Kivelson, A. C. Yuan, B. J. Ramshaw, and R. Thomale, A proposal for reconciling diverse experiments on the superconducting state in Sr2RuO4, npj Quantum Mater. 5, 43 (2020).
  48. A. N. Petsch, M. Zhu, M. Enderle, Z. Q. Mao, Y. Maeno, I. I. Mazin, and S. M. Hayden, Reduction of the Spin Susceptibility in the Superconducting State of Sr2RuO4 Observed by Polarized Neutron Scattering, Phys. Rev. Lett. 125, 217004 (2020).
  49. A. Chronister, A. Pustogow, N. Kikugawa, D. A. Sokolov, F. Jerzembeck, C. W. Hicks, A. P. Mackenzie, E. D. Bauer, and S. E. Brown, Evidence for even parity unconventional superconductivity in Sr2RuO4, Proc. Natl. Acad. Sci. (USA) 118, e2025313118 (2021).
  50. W. Huang, Y. Zhou, and H. Yao, Exotic Cooper pairing in Multiorbital Models of Sr2RuO4, Phys. Rev. B 100, 134506 (2019).
  51. A. Ramires and M. Sigrist, Superconducting order parameter of Sr2RuO4: A microscopic perspective, Phys. Rev. B 100, 104501 (2019).
  52. S. O. Kaba and D. Sénéchal, Group-theoretical classification of superconducting states of strontium ruthenate, Phys. Rev. B 100, 214507 (2019).
  53. C. M. Puetter and H.-Y. Kee, Identifying spin-triplet pairing in spin-orbit coupled multi-band superconductors, Europhys. Lett. 98, 27010 (2012).
  54. A. J. Leggett, Spin Susceptibility of a Superfluid Fermi Liquid, Phys. Rev. Lett. 14, 536 (1965).
  55. P. Steffens, Y. Sidis, J. Kulda, Z. Q. Mao, Y. Maeno, I. I. Mazin, and M. Braden, Spin Fluctuations in Sr2RuO4 from Polarized Neutron Scattering: Implications for Superconductivity, Phys. Rev. Lett. 122, 047004 (2019).
  56. E. Pavarini and I. I. Mazin, First-principles study of spin-orbit effects and NMR in Sr2RuO4, Phys. Rev. B 74, 035115 (2006).
  57. C. W. Hicks, D. O. Brodsky, E. A. Yelland, et al., Strong increase of Tc of Sr2RuO4 under both tensile and compressive strain, Science 344, 283 (2014).
  58. A. Steppke, L. Zhao, M. E. Barber, T. Scaffidi, et al., Strong peak in Tc of Sr2RuO4 under uniaxial pressure, Science 355, eaaf9398 (2017).
  59. C. A. Watson, A. S. Gibbs, A. P. Mackenzie, C. W. Hicks, and K. A. Moler, Micron-scale measurements of low anisotropic strain response of local Tc in Sr2RuO4, Phys. Rev. B 98, 094521 (2018).
  60. S. Yonezawa, T. Kajikawa, and Y. Maeno, Specific-heat evidence of the first-order superconducting transition in Sr2RuO4, J. Phys. Soc. Jpn. 83, 083706 (2014).
  61. Y.-S. Li, N. Kikugawa, D. A. Sokolov, F. Jerzembeck, A. S. Gibbs, Y. Maeno, C. W. Hicks, M. Nicklas, and A. P. Mackenzie, High-sensitivity Heat-capacity Measurements on Sr2RuO4 under Uniaxial Pressure, Proc. Natl. Acad. Sci. (USA) 118, e2020492118 (2021).
  62. T. Scaffidi, Degeneracy between even- and odd-parity superconductivity in the quasi-1D hubbard model and implications for Sr2RuO4, arXiv:2007.13769.
  63. R. Gupta, T. Saunderson, S. Shallcross, M. Gradhand, J. Quintanilla, and J. Annett, Superconducting subphase and substantial knight shift in Sr2RuO4, Phys. Rev. B 102, 235203 (2020).
  64. K. Deguchi, M. A. Tanatar, Z. Q. Mao, T. Ishiguro, and Y. Maeno, Superconducting double transition and the upper critical field limit of Sr2RuO4 in parallel magnetic fields, J. Phys. Jpn. Soc. 71, 2839 (2002).
  65. S. Yonezawa, T. Kajikawa, and Y. Maeno, First-Order Superconducting Transition of Sr2RuO4, Phys. Rev. Lett. 110, 077003 (2013).
  66. K. Deguchi, Z. Q. Mao, H. Yaguchi, and Y. Maeno, Gap Structure of the Spin-Triplet Superconductor Sr2RuO4 Determined from the Field-Orientation Dependence of the Specific Heat, Phys. Rev. Lett. 92, 047002 (2004).
  67. S. Nishizaki, Y. Maeno, and Z. Q. Mao, Changes in the superconducting state of Sr2RuO4 under magnetic fields probed by specific heat, J. Phys. Soc. Jpn. 69, 572 (2000).
  68. I. A. Firmo, S. Lederer, C. Lupien, A. P. Mackenzie, J. C. Davis, and S. A. Kivelson, Evidence from tunneling spectroscopy for a quasi-one-dimensional origin of superconductivity in Sr2RuO4, Phys. Rev. B 88, 134521 (2013).
  69. E. Hassinger, P. Bourgeois-Hope, H. Taniguchi, S. Rene de Cotret, G. Grissonnanche, M. S. Anwar, Y. Maeno, N. Doiron-Leyraud, and Louis Taillefer, Vertical Line Nodes in the Superconducting Gap Structure of Sr2RuO4, Phys. Rev. X 7, 011032 (2017).
  70. J. F. Dodaro, Z. Wang, and C. Kallin, Effects of deep superconducting gap minima and disorder on residual thermal transport in Sr2RuO4, Phys. Rev. B 98, 214520 (2018).
  71. D. F. Agterberg, T. M. Rice, and M. Sigrist, Orbital Dependent Superconductivity in Sr2RuO4, Phys. Rev. Lett. 78, 3374 (1997).
  72. M. E. Zhitomirsky and T. M. Rice, Interband Proximity Effect and Nodes of Superconducting Gap in Sr2RuO4, Phys. Rev. Lett. 87, 057001 (2001).
  73. A. Ramires and M. Sigrist, Identifying detrimental effects for multiorbital superconductivity: Application to Sr2RuO4, Phys. Rev. B 94, 104501 (2016).
  74. W.-S. Wang, C.-C. Zhang, F.-C. Zhang, and Q.-H. Wang, Theory of Chiral P-Wave Superconductivity with Near Nodes for Sr2RuO4, Phys. Rev. Lett. 122, 027002 (2019).
  75. O. Gingras, R. Nourafkan, A. M. S. Tremblay, and M. Còté, Superconducting Symmetries of Sr2RuO4 from First-Principles Electronic Structure, Phys. Rev. Lett. 123, 217005 (2019).
  76. Y. Li and W. Huang, Possible ‘symmetry-imposed’ near-nodal two-dimensional p-wave pairing in Sr2RuO4, arXiv:1909.03141.
  77. H. G. Suh, H. Menke, P. M. R. Brydon, C. Timm, A. Ramires, and D. F. Agterberg, Stabilizing even-parity chiral superconductivity in Sr2RuO4, Phys. Rev. Research 2, 032023(R) (2020).
  78. W. Chen and J. An, interorbital p- and d-wave pairings between dxz/dyz and dxy orbitals in Sr2RuO4, Phys. Rev. B 102, 094501 (2020).
  79. A. W. Lindquist and H-Y. Kee, Distinct reduction of knight shift in superconducting state of Sr2RuO4 under uniaxial strain, Phys. Rev. Research 2, 032055(R) (2020).
  80. A. Sumiyama, D. Kawakatsu, J. Gouchi, A. Yamaguchi, G. Motoyama, Y. Hirose, R. Settai, and Y. Ōnuki, Search for spontaneous magnetization of superconductors with broken time-reversal symmetry, JPS Conf. Proc. 3, 015017 (2014).
  81. R. Willa, M. Hecker, R. M. Fernandes, and J. Schmalian, Inhomogeneous time-reversal symmetry breaking in Sr2RuO4, Phys. Rev. B 104, 024511 (2021).
  82. L-H. Hu, X. Wang, and T. Shang, Spontaneous magnetization in unitary superconductors with time reversal symmetry breaking, Phys. Rev. B 104, 054520 (2021).

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Phys. Rev. Research 3, L042002– Published 7 October, 2021
  • Received 24 February 2021
  • Revised 11 August 2021
  • Accepted 13 September 2021
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