Advertisement

Canonical pair condensation in a flat-band BCS superconductor

  • Jacques TempereEmail author
  • Dolf Huybrechts
Regular Article
  • 25 Downloads

Abstract

The standard approach of the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity is to introduce a self-consistent mean-field approximation, and a variational ansatz for the many-body ground state. The resulting mean-field Hamiltonian no longer commutes with the total number operator, and the variational search takes place in Fock space rather than in a Hilbert space of states with fixed number of particles. This is a disadvantage when studying small systems where the canonical ensemble predictions differ from the corresponding grand-canonical results. To remedy this, alternative approaches such as Richardson’s method have been put forward. Here, we derive the exact many-body ground state of a model Hamiltonian corresponding to the deep-BCS or flat-band regime, without having to resort to Richardson’s set of coupled nonlinear equations. This allows to write the exact many-body ground state in a way that makes the difference with the BCS variational wave function particularly clear. We show that the exact wave function consists of a superposition of many-pair states in such a way that the mean-field averaging corresponds to a summation over these many-pair states. This explains why many expectation values calculated with the BCS variational wave function give the same result as when calculated with the exact wave function, even though these wave functions are different. In the canonical (fixed-number) approach, pairing is investigated using the second-order reduced density matrix and calculating its largest eigenvalue. When interpreted as the order parameter of the superconducting state, this can be compared directly to the behavior of the mean-field gap. Finally, we show that a clear difference between the canonical approach and the BCS grand canonical estimates appears when evaluating pair condensate fluctuations as well as the pair entanglement entropy.

Graphical abstract

Keywords

Solid State and Materials 

References

  1. 1.
    J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 106, 162 (1957) ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957) ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    G.M. Eliashberg, Sov. Phys. JETP 11, 696 (1960) [Zh. Eksp. Teor. Fiz. 38, 966 (1960)] Google Scholar
  4. 4.
    D.A. Kirzhnits, E.G. Maksimov, D.I. Khomskii, J. Low Temp. Phys. 10, 79 (1973) ADSCrossRefGoogle Scholar
  5. 5.
    J. Nagamatsu et al., Nature 410, 63 (2001) ADSCrossRefGoogle Scholar
  6. 6.
    Y. Kamihara et al., J. Am. Chem. Soc. 130, 3296 (2008) CrossRefGoogle Scholar
  7. 7.
    C.A. Regal, M. Greiner, D. Jin, Phys. Rev. Lett. 92, 040403 (2004) ADSCrossRefGoogle Scholar
  8. 8.
    M. Zwierlein et al., Phys. Rev. Lett. 92, 120403 (2004) ADSCrossRefGoogle Scholar
  9. 9.
    V.A. Khodel, V.R. Shaginyan, JETP Lett. 51, 553 (1990) ADSGoogle Scholar
  10. 10.
    R.W. Richardson, Phys. Lett. 3, 277 (1963) ADSCrossRefGoogle Scholar
  11. 11.
    M. Gaudin, J. Phys. 37, 1087 (1976) MathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Dukelsky, S. Pittel, G. Sierra, Rev. Mod. Phys. 76, 643 (2004) ADSCrossRefGoogle Scholar
  13. 13.
    A.K. Kerman, Ann. Phys. 12, 300 (1961) ADSCrossRefGoogle Scholar
  14. 14.
    M. Barranco, S. Hernandez, R.J. Lombard, Z. Phys. D 22, 659 (1992) ADSCrossRefGoogle Scholar
  15. 15.
    V.A. Khodel, V.R. Shaginyan, V.V. Khodel, Phys. Rep. 249, 1 (1994) ADSCrossRefGoogle Scholar
  16. 16.
    N.B. Kopnin, T.T. Heikkilä, G.E. Volovik, Phys. Rev. B 83, 220503 (2011) ADSCrossRefGoogle Scholar
  17. 17.
    M. Tovmasyan, S. Peotta, P. Törmä, S.D. Huber, Phys. Rev. B 94, 245149 (2016) ADSCrossRefGoogle Scholar
  18. 18.
    S. Peotta, P. Törmä, Nat. Commun. 6, 8944 (2015) CrossRefGoogle Scholar
  19. 19.
    R. Nandkishore, L. Levitov, A. Chubukov, Nat. Phys. 8, 158 (2012) CrossRefGoogle Scholar
  20. 20.
    B. Uchoa, Y. Barlas, Phys. Rev. Lett. 111, 046604 (2013) ADSCrossRefGoogle Scholar
  21. 21.
    E. Tang, L. Fu, Nat. Phys. 10, 964 (2014) CrossRefGoogle Scholar
  22. 22.
    D. Yudin, D. Hirschmeier, H. Hafermann, O. Eriksson, A.I. Lichtenstein, M.I. Katsnelson, Phys. Rev. Lett. 112, 070403 (2014) ADSCrossRefGoogle Scholar
  23. 23.
    V.A. Khodel, J.W. Clark, K.G. Popov, V.R. Shaginyan, JETP Lett. 101, 413 (2015) ADSCrossRefGoogle Scholar
  24. 24.
    V.J. Kauppila, F. Aikebaier, T.T. Heikkilä, Phys. Rev. B 93, 214505 (2016) ADSCrossRefGoogle Scholar
  25. 25.
    N.B. Kopnin, JETP Lett. 94, 81 (2011) ADSCrossRefGoogle Scholar
  26. 26.
    N.B. Kopnin, M. Ijäs, A. Harju, T.T. Heikkilä, Phys. Rev. B 87, 140503(R) (2013) ADSCrossRefGoogle Scholar
  27. 27.
    L. Liang, T.I. Vanhala, S. Peotta, T. Siro, A. Harju, P. Torma, Phys. Rev. B 95, 024515 (2017) ADSCrossRefGoogle Scholar
  28. 28.
    R. Ojajärvi, T. Hyart, M.A. Silaev, T.T. Heikkilä, Phys. Rev. B 98, 054515 (2018) ADSCrossRefGoogle Scholar
  29. 29.
    T. Löthman, A.M. Black-Schaffer, Phys. Rev. B 96, 064505 (2017) ADSCrossRefGoogle Scholar
  30. 30.
    T.T. Heikkila, N.B. Kopnin, G.E. Volovik, JETP Lett. 94, 233 (2011) ADSCrossRefGoogle Scholar
  31. 31.
    Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, P. Jarillo-Herrero, Nature 556, 43 (2018) ADSCrossRefGoogle Scholar
  32. 32.
    Y. Cao, V. Fatemi, A. Demir, S. Fang, S.L. Tomarken, J.Y. Luo, J.D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R.C. Ashoori, P. Jarillo-Herrero, Nature 556, 80 (2018) ADSCrossRefGoogle Scholar
  33. 33.
    G.E. Volovik, JETP Lett. 53, 222 (1991) ADSGoogle Scholar
  34. 34.
    P. Nozières, J. Phys. I 2, 443 (1992) Google Scholar
  35. 35.
    T.T. Heikkilä, G.E. Volovik, Flat bands as a route to high-temperature superconductivity in graphite, in Basic Physics of Functionalized Graphite, edited by P.D. Esquinazi (Springer International Publishing, Switzerland, 2016), pp. 123–143 Google Scholar
  36. 36.
    Y.N. Ovchinnikov, V.Z. Kresin, Eur. Phys. J. B 45, 5 (2005) ADSCrossRefGoogle Scholar
  37. 37.
    D. Sels, M. Wouters, Phys. Rev. E 92, 022123 (2015) ADSCrossRefGoogle Scholar
  38. 38.
    O. Penrose, L. Onsager, Phys. Rev. 104, 576 (1956) ADSCrossRefGoogle Scholar
  39. 39.
    C.N. Yang, Rev. Mod. Phys. 34, 694 (1962) ADSCrossRefGoogle Scholar
  40. 40.
    A. Leggett, Quantum Liquids (Oxford University Press, Oxford, UK, 2006) Google Scholar
  41. 41.
    T. Sowinski, M. Gajda, K. Rzazewski, Europhys. Lett. 109, 26005 (2015) ADSCrossRefGoogle Scholar
  42. 42.
    C. Weiss, J. Tempere, Phys. Rev. E 94, 042124 (2016) ADSCrossRefGoogle Scholar
  43. 43.
    M. Sato, Y. Tanaka, K. Yada, T. Yokoyama, Phys. Rev. B 83, 224511 (2011) ADSCrossRefGoogle Scholar
  44. 44.
    A.P. Schnyder, S. Ryu, Phys. Rev. B 84, 060504 (2011) ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Theory of Quantum and Complex Systems, Universiteit AntwerpenAntwerpenBelgium

Personalised recommendations