The document summarizes research into the puzzling pairing in the non-centrosymmetric superconductor LaNiC2. MuSR experiments showed the superconducting state breaks time-reversal symmetry through spontaneous magnetic fields. Theoretically, this implies non-unitary triplet pairing with weak spin-orbit coupling. The transition may split into two stages due to the influence of spin-orbit coupling on relativistic and non-relativistic instabilities. While experimental evidence points to time-reversal symmetry breaking, the specific pairing symmetry is still unknown, as is why the pairing is non-unitary. The research highlights that noncentrosymmetric superconductors cannot be fully understood through Rashba coupling alone and unconventional pairing extends

1. Puzzling pairing in thenon-centrosymmetric superconductor LaNiC2 Jorge Quintanilla SEPnet, University of Kent Hubbard Theory Centre, Rutherford Appleton Laboratory Adrian Hillier (RAL) Bob Cywinski (Huddersfield) James F. Annett (Bristol) Bayan Mazidian (Bristol and RAL) Collaborators: STFC, SEPnet Funding: CMMP’10, University of Warwick, 15 December 2010

2. LaNiC2 – a weakly-correlated, paramagnetic superconductor? W. H. Lee et al., Physica C 266, 138 (1996) V. K. Pecharsky, L. L. Miller, and Zy, Physical Review B 58, 497 (1998) specific heat susceptibility ΔC/TC=1.26 (BCS: 1.43) Tc=2.7 K

3. ISIS muSR

4. Relaxation due to electronic moments _  sample e  e forward detector backward detector (longitudinal) Moment size ~ 0.1G (~ 0.01μB) Timescale: > 10-4s ~ Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

5. Relaxation due to electronic moments _  sample e  e forward detector backward detector (longitudinal) Moment size ~ 0.1G (~ 0.01μB) Timescale: > 10-4s ~ Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

6. Relaxation due to electronic moments _  sample e  e forward detector backward detector (longitudinal) Moment size ~ 0.1G (~ 0.01μB) Timescale: > 10-4s ~ Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

7. Relaxation due to electronic moments _  sample e  e forward detector backward detector (longitudinal) Moment size ~ 0.1G (~ 0.01μB) Timescale: > 10-4s ~ Spontaneous, quasi-static fields appearing at Tc ⇒ superconducting state breaks time-reversal symmetry [ c.f. Sr2RuO4 - Luke et al., Nature (1998) ] Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

8. Symmetry of the gap function See J.F. Annett Adv. Phys. 1990.

9. Neutrondiffraction Data from D1B @ ILL Orthorhombic Amm2 C2v a=3.96 Å b=4.58 Å c=6.20 Å Note no inversion centre. C.f. CePt3Si (1), Li2Pt3B & Li2Pd3B (2), ... (1) Bauer et al. PRL’04 (2) Yuan et al. PRL’06

11. Singlet, triplet, or both? singlet [ 0(k) even ] triplet [ d(k) odd ]

12. Singlet, triplet, or both? Neglect (for now!) spin-orbit coupling: Singlet and triplet representations of SO(3): Γns = - (Γns)T, Γnt = + (Γnt)T Impose Pauli’s exclusion principle:  either singlet or triplet

13. Character table 180o Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

14. Character table These must be combined with the singlet and triplet representations of SO(3). Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

15. Possible order parameters Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

16. Possible order parameters Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

17. Possible order parameters Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

18. Non-unitary d x d* ≠ 0 Possible order parameters Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

19. breaks only SO(3) x U(1) x T Non-unitary d x d* ≠ 0 Possible order parameters * * C.f. Li2Pd3B & Li2Pt3B, H. Q. Yuan et al. PRL’06 Hillier, Quintanilla & Cywinski, PRL 102 117007 (2009)

20. Non-unitary pairing Spin-up superfluid coexisting with spin-down Fermi liquid.

21. Non-unitary pairing Spin-up superfluid coexisting with spin-down Fermi liquid. C.f.

22. Non-unitary pairing Spin-up superfluid coexisting with spin-down Fermi liquid. C.f. The A1 phase of liquid 3He.

23. Non-unitary pairing Spin-up superfluid coexisting with spin-down Fermi liquid. C.f. The A1 phase of liquid 3He. Ferromagnetic superconductors. F. Hardy et al., Physica B359-61, 1111-13 (2005) [ See A. de Visser in Encyclopedia of Materials: Science and Technology (Eds. K. H. J. Buschow et al.), Elsevier, 2010]

24. Ferromagnetic superconductors But LaNiC2 is a paramagnet ! A. de Visser in Encyclopedia of Materials: Science and Technology (Eds. K. H. J. Buschow et al.), Elsevier, 2010

25. Isn’t there a more simple explanation?

27. The role of spin-orbit coupling (SOC) G = [SO(3)×Gc]×U(1)×T

28. The role of spin-orbit coupling (SOC) G = [SO(3)×Gc]×U(1)×T

29. The role of spin-orbit coupling (SOC) G = [SO(3)×Gc]×U(1)×T

30. The role of spin-orbit coupling (SOC) G = [SO(3)×Gc]×U(1)×T

31. G = Gc,J×U(1)×T The role of spin-orbit coupling (SOC)

32. G = Gc,J×U(1)×T The role of spin-orbit coupling (SOC)

33. G = Gc,J×U(1)×T The role of spin-orbit coupling (SOC)

34. The role of spin-orbit coupling (SOC) z E.g. reflection through a vertical plane perpendicular to the y axis: x y Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

35. The role of spin-orbit coupling (SOC) z E.g. reflection through a vertical plane perpendicular to the y axis: x y Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

36. The role of spin-orbit coupling (SOC) z E.g. reflection through a vertical plane perpendicular to the y axis: x y Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

37. The role of spin-orbit coupling (SOC) z E.g. reflection through a vertical plane perpendicular to the y axis: x y Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

38. The role of spin-orbit coupling (SOC) z E.g. reflection through a vertical plane perpendicular to the y axis: This affects d(k) (a vector under spin rotations). x y Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

39. The role of spin-orbit coupling (SOC) z E.g. reflection through a vertical plane perpendicular to the y axis: This affects d(k) (a vector under spin rotations). It does not affect 0(k) (a scalar). x y Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

40. The role of spin-orbit coupling (SOC) Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

41. The role of spin-orbit coupling (SOC) None of these break time-reversal symmetry! Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

42. How could this happen? Gap matrices evolve smoothly as SOC is turned on. E.g. ( 1A1 ) ( A1 ) for B = C = D = 0 Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

43. How could this happen? Some instabilities split in two under the influence of SOC: ( 3A1(b) ) E.g. ( B2 ) ( B1 ) Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

44. Relativistic and non-relativistic instabilities: a complex relationship 1A1 A1 1A2 A2 B2 3A1(b) 1B1 B1 B1 B1 1B2 B2 3A2(b) B2 A2 3A1(a) A2 3B1(b) A1 A1 3A2(a) A1 B2 3B1(a) 3B2(b) A2 B1 3B2(a) spin-orbit coupling spin-orbit coupling Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

45. Relativistic and non-relativistic instabilities: a complex relationship A1 B1 A2 B2 singlet Pairing instabilities 1A1 1A2 1B1 1B2 non-unitary triplet pairing instabilities 3B1(b) 3B2(b) 3A1(b) 3A2(b) unitary triplet pairing instabilities 3A1(a) 3A2(a) 3B1(a) 3B2(a)

46. The role of spin-orbit coupling (SOC) The second (lower-Tc) instability can be symmetry-breaking because it is no longer an instability of the normal state: The experiments show a transition straight into the broken TRS phase ⇒ SOC must be small in LaNiC2 B2 (kz,0,0) 3A1 (b) (kz,ikz,0) B1 i(0,kz,0) N.B. singlet component must be very small too. SOC Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

47. Recap

48. Recap What have we learned about LaNiC2?

54. Why non-unitary?Take this home: