Journal of Low Temperature Physics

, Volume 73, Issue 1–2, pp 137–160 | Cite as

Collective modes and a theory of response ofd-wave superconductors

  • Dai S. Hirashima
  • Hiroshi Namaizawa


We present a theory of the response ofd-wave superconductors to weak applied fields, by taking account of the Coulomb interaction and all the collective degrees of freedom as well as crystal symmetry. We choose two representative phases: the dγ phase, which has point nodes in the energy gap, and theY2−1 phase, which has line as well as point nodes. The former is a self-consistent solution for cubic as well as spherical symmetries and the latter is one for spherical, cubic, and hexagonal symmetries. We obtain obviously gauge-invariant expressions for the order-parameter fluctuations and the currents, having forms common not only to thed-wave states, but also to thep-wave states studied earlier. We also investigate the collective excitations; in the long-wavelength limit for spherically symmetric systems, there are, on the frequency-temperature plane, seven branches for eachd-wave phase considered, in addition to the common plasma mode and orbital Goldstone modes resulting from the spontaneous breakdown of the rotational invariance. In theY2−1-phase two eigenmodes are found to become gapless at a finite temperature, below which they are purely imaginary. This implies instability of the phase. The effect of crystal anisotropy on the collective spectra is also studied.


Point Node Collective Mode Crystal Symmetry Rotational Invariance Collective Excitation 
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  1. 1.
    G. R. Stewart,Rev. Mod. Phys. 56, 755 (1984).Google Scholar
  2. 2.
    H. R. Ott, H. Rudigier, T. M. Rice, K. Ueda, Z. Fisk, and J. L. Smith,Phys. Rev. Lett. 52, 1915 (1984).Google Scholar
  3. 3.
    D. J. Bishop, C. M. Varma, B. Batlogg, F. Bucher, Z. Fish, and J. L. Smith,Phys. Rev. Lett. 53, 1009 (1984).Google Scholar
  4. 4.
    V. Müller, D. Maurer, E. W. Scheidt, C. Roth, K. Lüders, E. Bucher, and H. E. Bömmel,Solid State Commun. 57, 319 (1986).Google Scholar
  5. 5.
    B. S. Shivaram, Y. H. Jeong, T. F. Rosenbaum, and D. G. Hinks,Phys. Rev. Lett. 56, 1078 (1986).Google Scholar
  6. 6.
    B. Golding, D. J. Bishop, B. Batlogg, W. H. Haemmerle, Z. Fisk, J. L. Smith, and H. R. Ott,Phys. Rev. Lett. 55, 2479 (1985).Google Scholar
  7. 7.
    D. E. MacLaughlin, Cheng Tien, W. G. Clark, M. D. Lan, Z. Fisk, J. L. Smith, and H. R. Ott,Phys. Rev. Lett. 53, 1833 (1984).Google Scholar
  8. 8.
    D. Einzel, P. J. Hirschfeld, F. Gross, B. S. Chandrasekher, A. Andres, H. R. Ott, J. Beuers, Z. Fisk, and J. L. Smith,Phys. Rev. Lett. 56, 2513 (1986).Google Scholar
  9. 9.
    B. Batlogg, D. J. Bishop, B. Golding, E. Bucher, J. Hufnagl, Z. Fisk, J. L. Smith, and H. R. Ott,Phys. Rev. B 33, 5906 (1986).Google Scholar
  10. 10.
    C. M. Varma,Bull. Am. Phys. Soc. 29, 404 (1984); P. W. Anderson,Phys. Rev. B 30, 1549 (1984).Google Scholar
  11. 11.
    K. Miyake, T. Matsuura, H. Jichu, and Y. Nagaoka,Prog. Theor. Phys. 72, 1063 (1984); F. J. Ohkawa and H. Fukuyama,J. Phys. Soc. Jpn. 53, 4344 (1984).Google Scholar
  12. 12.
    J. E. Hirsch,Phys. Rev. Lett. 54, 1317 (1985); A. Auerbach and K. Levin,Phys. Rev. Lett. 57, 877 (1986); M. Cyrot,Solid State Commun. 60, 253 (1986); K. Miyake, S. Schmitt-Rink, and C. M. Varma,Phys. Rev. B 34, 6554 (1986); M. Lavagna, A. J. Millis, and P. A. Lee,Phys. Rev. Lett. 58, 266 (1987); D. J. Scalapino, E. Loh, Jr., and J. E. Hirsch,Phys. Rev. B 35, 6694 (1987).Google Scholar
  13. 13.
    P. W. Anderson,Phys. Rev. B 30, 4000 (1984); G. E. Volovik and L. P. Gorkov,JETP Lett. 39, 674 (1984);Sov. Phys. JETP 61, 843 (1985); K. Ueda and T. M. Rice,Phys. Rev. B 31, 7114 (1985); E. I. Blount,Phys. Rev. B 32, 2935 (1985).Google Scholar
  14. 14.
    S. Schmitt-Rink, K. Miyake, and C. M. Varma,Phys. Rev. Lett. 57, 2575 (1986).Google Scholar
  15. 15.
    P. Hirschfeld, D. Vollhardt, and P. Wölfle,Solid State Commun. 59, 111 (1986); P. J. Hirschfeld, P. Wölfle, and D. Einzel,Phys. Rev. B 37, 83 (1988).Google Scholar
  16. 16.
    H. Monien, K. Scharnberg, and D. Walder,Solid State Commun. 63, 263 (1987).Google Scholar
  17. 17.
    D. S. Hirashima and H. Namaizawa,Prog. Theor. Phys. 74, 400 (1985); Addenda,77, 1004 (1987).Google Scholar
  18. 18.
    D. S. Hirashima and H. Namaizawa,Prog. Theor. Phys. 77, 563 (1987).Google Scholar
  19. 19.
    D. S. Hirashima and H. Namaizawa,Prog. Theor. Phys. 77, 585 (1987).Google Scholar
  20. 20.
    H. Monien, K. Scharnberg, L. Tewordt, and N. Schopohl,Phys. Rev. B 34, 3478 (1986);J. Low Temp. Phys. 65, 13 (1986).Google Scholar
  21. 21.
    H. Monien, K. Scharnberg, L. Tewordt, and D. Walker,Solid State Commun. 61, 581 (1987).Google Scholar
  22. 22.
    T. Wölkhausen, Diplomarbeit, University of Hamburg (1987), unpublished; K. Scharnberg, private communication.Google Scholar
  23. 23.
    P. W. Anderson,Phys. Rev. 112, 1900 (1958).Google Scholar
  24. 24.
    P. Wölfle,Phys. Lett. A 119, 40 (1986).Google Scholar
  25. 25.
    H. Monien, L. Tewordt, and K. Scharnberg,Solid State Commun. 63, 1027 (1987).Google Scholar
  26. 26.
    K. Miyake and C. M. Varma,Phys. Rev Lett. 57, 1627 (1987).Google Scholar
  27. 27.
    N. D. Mermin,Phys. Rev. A 9, 868 (1974).Google Scholar
  28. 28.
    P. W. Anderson and P. Morel,Phys. Rev. 123, 1911 (1961).Google Scholar
  29. 29.
    P. Kumar and P. Wölfle,Phys. Rev. Lett. 59, 1954 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Dai S. Hirashima
    • 1
  • Hiroshi Namaizawa
    • 1
  1. 1.Institute of Physics, College of Arts and SciencesUniversity of TokyoTokyo

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