Advertisement

Journal of Superconductivity and Novel Magnetism

, Volume 32, Issue 11, pp 3473–3481 | Cite as

Non-adiabatic Dynamics in d + id-Wave Fermionic Superfluids

  • Ammar A. KirmaniEmail author
  • Maxim Dzero
Original Paper

Abstract

We consider a problem of non-adiabatic dynamics of a 2D fermionic system with d + id-wave symmetry of pairing amplitude. Under the mean-field approximation, we determine the asymptotic behavior of the pairing amplitude following a sudden change of coupling strength. We also study an extended d + id pairing system for which the long-time asymptotic states of the pairing amplitude in the collisionless regime can be determined exactly. By using numerical methods, we have identified three non-equilibrium steady states described by different long-time asymptotes of the pairing amplitude for both the non-integrable and the integrable versions of d + id-wave models. We found that despite of its lack of integrability, long-time dynamics resulting from pairing quenches in the non-integrable d + id model are essentially similar to the ones found for its exactly integrable extended d + id model. We also obtain the long-time phase diagram of the extended d + id model through the Lax construction that exploits underlying integrability showing that the dynamic phases obtained by numerics are consistent with the dynamics of the exactly integrable approach. Both models describe a topological fermionic system with a topologically non-trivial BCS phase appearing at weak coupling strength. We show that the presence of oscillating order parameter region in the chiral d + id pairing dynamics differs from the d-wave (\(d_{x^{2}-y^{2}}\)), which may be used to probe pairing symmetries of chiral superconductors.

Keywords

Dynamics in d + id pairing Transient and long time dynamics of chiral-d + id superfluids and superconductors 

Notes

Acknowledgments

The authors are grateful to Emil Yuzbashyan for his comments on the manuscript and numerous stimulating discussions.

Funding Information

This study was financially supported by the National Science Foundation grant NSF-DMR-1506547. The work of one of us (M.D.) was financially supported in part by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0016481.

References

  1. 1.
    Gaudin, M.: La fonction d’onde de Bethe. Masson, Paris (1983)zbMATHGoogle Scholar
  2. 2.
    Sklyanin, E. K.: . J. Sov. Math. 47, 2473 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Sklyanin, E. K.: . Progr. Theoret. Phys. Suppl. 118, 35 (1995)ADSCrossRefGoogle Scholar
  4. 4.
    Yuzbashyan, E. A., Altshuler, B. L., Kuznetsov, V. B., Enolskii, V. Z.: . J. Phys. A Math. Gen. 38, 7831 (2005a)Google Scholar
  5. 5.
    Yuzbashyan, E. A., Altshuler, B. L., Kuznetsov, V. B., Enolskii, V. Z.: . Phys. Rev. B 72, 220503 (2005b)Google Scholar
  6. 6.
    Yuzbashyan, E. A., Tsyplyatyev, O., Altshuler, B. L.: . Phys. Rev. Lett. 96, 097005 (2006)ADSCrossRefGoogle Scholar
  7. 7.
    Yuzbashyan, E. A., Dzero, M., Gurarie, V., Foster, M. S.: . Phys. Rev. A 91, 033628 (2015). https://link.aps.org/doi/10.1103/PhysRevA.91.033628 ADSCrossRefGoogle Scholar
  8. 8.
    Foster, M. S., Dzero, M., Gurarie, V., Yuzbashyan, E. A.: . Phys. Rev. B 88, 104511 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    Foster, M. S., Gurarie, V., Dzero, M., Yuzbashyan, E. A.: . Phys. Rev. Lett. 113, 076403 (2014). http://link.aps.org/doi/10.1103/PhysRevLett.113.076403 ADSCrossRefGoogle Scholar
  10. 10.
    Peronaci, F., Schiró, M., Capone, M.: . Phys. Rev. Lett. 115, 257001 (2015). https://link.aps.org/doi/10.1103/PhysRevLett.115.257001 ADSCrossRefGoogle Scholar
  11. 11.
    Barankov, R. A., Levitov, L. S: pre-print arXiv:0704.1292 (2007)
  12. 12.
    Dong, Y., Dong, L., Gong, M., Pu, H.: arXiv:1406.3821. 1406.3821v1 (2014)
  13. 13.
    Dzero, M., Kirmani, A. A., Yuzbashyan, E. A.: . Phys. Rev. A 92, 053626 (2015). https://link.aps.org/doi/10.1103/PhysRevA.92.053626 ADSCrossRefGoogle Scholar
  14. 14.
    Fausti, D., Tobey, R. I., Dean, N., Kaiser, S., Dienst, A., Hoffmann, M. C., Pyon, S., Takayama, T., Takagi, H., Cavalleri, A.: . Science 331, 189 (2011). ISSN 0036-8075, http://science.sciencemag.org/content/331/6014/189.full.pdf, http://science.sciencemag.org/content/331/6014/189 ADSCrossRefGoogle Scholar
  15. 15.
    Matsunaga, R., Shimano, R.: . Phys. Rev. Lett. 109, 187002 (2012). http://link.aps.org/doi/10.1103/PhysRevLett.109.187002 ADSCrossRefGoogle Scholar
  16. 16.
    Matsunaga, R., Hamada, Y. I., Makise, K., Uzawa, Y., Terai, H., Wang, Z., Shimano, R.: . Phys. Rev. Lett. 111, 057002 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    Hu, W., Kaiser, S., Nicoletti, D., Hunt, C. R., Gierz, I., Hoffmann, M. C., Le Tacon, M., Loew, T., Keimer, B., Cavalleri, A.: . Nat. Mater. 13, 705 EP (2014).  https://doi.org/10.1038/nmat3963 ADSCrossRefGoogle Scholar
  18. 18.
    Matsunaga, R., Tsuji, N., Fujita, H., Sugioka, A., Makise, K., Uzawa, Y., Terai, H., Wang, Z., Aoki, H., Shimano, R.: . Science 345, 1145 (2014). ISSN 0036-8075, http://science.sciencemag.org/content/345/6201/1145.full.pdf. http://science.sciencemag.org/content/345/6201/1145 ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Mankowsky, R., Först, M., Loew, T., Porras, J., Keimer, B., Cavalleri, A.: . Phys. Rev. B 91, 094308 (2015). https://link.aps.org/doi/10.1103/PhysRevB.91.094308 ADSCrossRefGoogle Scholar
  20. 20.
    Graf, J., Jozwiak, C., Smallwood, C. L., Eisaki, H., Kaindl, R. A., Lee, D.-H., Lanzara, A.: . Nat. Phys. 7, 805 EP (2011).  https://doi.org/10.1038/nphys2027 CrossRefGoogle Scholar
  21. 21.
    Smallwood, C. L., Hinton, J. P., Jozwiak, C., Zhang, W., Koralek, J. D., Eisaki, H., Lee, D.-H., Orenstein, J., Lanzara, A.: . Science 336, 1137 (2012). ISSN 0036-8075, http://science.sciencemag.org/content/336/6085/1137.full.pdf, http://science.sciencemag.org/content/336/6085/1137 ADSCrossRefGoogle Scholar
  22. 22.
    Böker, J., Volkov, P. A., Efetov, K. B., Eremin, I.: . Phys. Rev. B 96, 014517 (2017). https://link.aps.org/doi/10.1103/PhysRevB.96.014517 ADSCrossRefGoogle Scholar
  23. 23.
    Laughlin, R. B.: . Phys. Rev. Lett. 80, 5188 (1998). https://link.aps.org/doi/10.1103/PhysRevLett.80.5188 ADSCrossRefGoogle Scholar
  24. 24.
    Horovitz, B., Golub, A.: . Phys. Rev. B 68, 214503 (2003). https://link.aps.org/doi/10.1103/PhysRevB.68.214503 ADSCrossRefGoogle Scholar
  25. 25.
    Liu, F., Liu, C.-C., Wu, K., Yang, F., Yao, Y.: . Phys. Rev. Lett. 111, 066804 (2013). https://link.aps.org/doi/10.1103/PhysRevLett.111.066804 ADSCrossRefGoogle Scholar
  26. 26.
    Carmi, E., Koren, G., Averbach, A.: Nature (London) 404. https://www.nature.com/articles/35009062(2000)
  27. 27.
    Dagan, Y., Deutscher, G.: . Phys. Rev. Lett. 87, 177004 (2001). https://link.aps.org/doi/10.1103/PhysRevLett.87.177004 ADSCrossRefGoogle Scholar
  28. 28.
    Krishana, K., Ong, N. P., Li, Q., Gu, G. D., Koshizuka, N.: . Science 277, 83 (1997). http://science.sciencemag.org/content/277/5322/83 CrossRefGoogle Scholar
  29. 29.
    Nandkishore, R., Levitov, L. S., Chubukov, A. V.: Nat. Phys. 8.  https://doi.org/10.1038/nphys2208 (2012)ADSCrossRefGoogle Scholar
  30. 30.
    Richardson, R.: . Phys. Lett. 3, 277 (1963). ISSN 0031-9163ADSCrossRefGoogle Scholar
  31. 31.
    Richardson, R., Sherman, N.: . Nucl. Phys. 52, 221 (1964). ISSN 0029-5582CrossRefGoogle Scholar
  32. 32.
    Gaudin, M.: Travaux de Michel Gaudin: Les Modeles Exactement Resolus. Les Editions de Physiques, Paris (1995)Google Scholar
  33. 33.
    Dukelsky, J., Esebbag, C., Schuck, P.: . Phys. Rev. Lett. 87, 066403 (2001). https://link.aps.org/doi/10.1103/PhysRevLett.87.066403 ADSCrossRefGoogle Scholar
  34. 34.
    Ortiz, G., Somma, R., Dukelsky, J., Rombouts, S.: . Nucl. Phys. B 707, 421 (2005). ISSN 0550-3213. http://www.sciencedirect.com/science/article/pii/S0550321304008879 ADSCrossRefGoogle Scholar
  35. 35.
    Dunning, C., Ibañez, M., Links, J., Sierra, G., Zhao, S.-Y.: . J. Stat. Mech: Theory Exp. 2010, P08025 (2010). http://stacks.iop.org/1742-5468/2010/i=08/a=P08025 CrossRefGoogle Scholar
  36. 36.
    Rombouts, S. M. A., Dukelsky, J., Ortiz, G.: . Phys. Rev. B 82, 224510 (2010). https://link.aps.org/doi/10.1103/PhysRevB.82.224510 ADSCrossRefGoogle Scholar
  37. 37.
    Marquette, I., Links, J.: . Nucl. Phys. B 866, 378 (2013). ISSN 0550-3213. http://www.sciencedirect.com/science/article/pii/S0550321312005056 ADSCrossRefGoogle Scholar
  38. 38.
    Read, N., Green, D.: . Phys. Rev. B 61, 10267 (2000). https://link.aps.org/doi/10.1103/PhysRevB.61.10267 ADSCrossRefGoogle Scholar
  39. 39.
    Sato, M., Takahashi, Y., Fujimoto, S.: . Phys. Rev. B 82, 134521 (2010). http://link.aps.org/doi/10.1103/PhysRevB.82.134521 ADSCrossRefGoogle Scholar
  40. 40.
    Anderson, P. W.: . Phys. Rev. 112, 1900 (1958)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Barankov, R. A., Levitov, L. S., Spivak, B. Z.: . Phys. Rev. Lett. 93, 160401 (2004)ADSCrossRefGoogle Scholar
  42. 42.
    Dzero, M., Yuzbashyan, E. A., Altshuler, B. L.: . Europhys. Lett. 85, 20004 (2008)ADSCrossRefGoogle Scholar
  43. 43.
    Scaramazza, J. A., Smacchia, P., Yuzbashyan, E. A.: . Phys. Rev. B 99, 054520 (2019). https://link.aps.org/doi/10.1103/PhysRevB.99.054520 ADSCrossRefGoogle Scholar
  44. 44.
    Cao, Y., Fatemi, V., Fang, S., Watanabe, K., Taniguchi, T., Kaxiras, E., Jarillo-Herrero, P.: . Nature 556, 43 EP (2018).  https://doi.org/10.1038/nature26160 ADSCrossRefGoogle Scholar
  45. 45.
    Kennes, D. M., Lischner, J., Karrasch, C.: . Phys. Rev. B 98, 241407 (2018). https://link.aps.org/doi/10.1103/PhysRevB.98.241407 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsKent State UniversityKentUSA

Personalised recommendations