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Two-Particle-Self-Consistent Approach for the Hubbard Model

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Strongly Correlated Systems

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 171))

Abstract

Even at weak to intermediate coupling, the Hubbard model poses a formidable challenge. In two dimensions in particular, standard methods such as the random phase approximation are no longer valid since they predict a finite temperature antiferromagnetic phase transition prohibited by the Mermin–Wagner theorem. The two-particle-self-consistent (TPSC) approach satisfies that theorem as well as particle conservation, the Pauli principle, the local moment and local-charge sum-rules. The self-energy formula does not assume a Migdal theorem. There is consistency between one- and two-particle quantities. The internal accuracy checks allow one to test the limits of validity of TPSC. Here I present a pedagogical review of TPSC along with a short summary of existing results and two case studies: (a) the opening of a pseudogap in two dimensions when the correlation length is larger than the thermal de Broglie wavelength and (b) the conditions for the appearance of d-wave superconductivity in the two-dimensional Hubbard model.

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Notes

  1. 1.

    See the contribution of Honerkamp in this volume.

  2. 2.

    Modifications have been proposed in zero dimension to use as impurity solver for DMFT [26].

  3. 3.

    In other references we often use iq n instead of iω n to denote the Matsubara frequency corresponding to the wave vector q.

  4. 4.

    The meaning of the superscripts differs from that in [7]. Superscripts \(\left (2\right )\left (1\right )\)here correspond, respectively, to \(\left (1\right )\left (0\right )\)in [7].

  5. 5.

    For the conductivity with vertex corrections [36], the f-sum rule with n kσobtained from G (2)is satisfied.

  6. 6.

    There is a misprint of a factor of 2 in [7]. It is corrected in [28].

  7. 7.

    In the Hubbard model the Fock term cancels with the same-spin Hartree term.

  8. 8.

    Appendix B of [7].

  9. 9.

    FLEX does not satisfy this consistency requirement. See Appendix E of [7]. In fact double-occupancy obtained from ΣGcan even become negative [55].

  10. 10.

    (See [7], Appendix E).

  11. 11.

    To remind ourselves of this, we may also adopt an additional vertical matrix notation convention and write (13.37) as \(\frac{\delta G} {\delta \phi } = {G}_{{}^{}}G + G\left [\frac{\frac{\delta \Sigma }{\delta G}} {\frac{\delta G}{\delta \phi }}\right ]G\).

  12. 12.

    See footnote (14) of [20] for a discussion of the choice of limit 1 + versus 1 − .

  13. 13.

    For n > 1, all particle occupation numbers must be replaced by hole occupation numbers.

  14. 14.

    Note also the following study from zero temperature [62].

  15. 15.

    This formula is similar to one that appeared in [63].

  16. 16.

    For comparisons with paramagnon theory see [65].

  17. 17.

    See also conclusion of [29].

  18. 18.

    Such tails tend to disappear in more recent laser ARPES measurements on hole-doped compounds [99].

  19. 19.

    See contribution of Randeria in this volume.

  20. 20.

    See contribution of Sénéchal in this volume.

  21. 21.

    For a pseudogap in the single-particle spectral weight, it is important not to assume a Migdal theorem [138,  139], and include vertex corrections [7].

  22. 22.

    See contributions of Vollhardt, Sénéchal, Potthoff and Jarrell in this volume.

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Acknowledgments

I am indebted to Yury Vilk who was at the origin of most of the ideas on TPSC and to Bumsoo Kyung who has collaborated on numerous TPSC project. In addition, many students, postdocs and colleagues over the years have worked extremely hard and made use of their creative powers and originality to extend and apply this approach to a large number of problems. Students and postdocs include in chronological order Liang Chen, Alain Veilleux, Anne-Marie Daré, Steve Allen, Hugo Touchette, Samuel Moukouri, Bumsoo Kyung, François Lemay, David Poulin, Jean-Sébastien Landry, Vasyl Hankevych, Bahman Davoudi, Syed Raghib Hassan, Sébastien Roy, Charles Brillon and Dominic Bergeron. I am also indebted to my colleagues David Sénéchal, Claude Bourbonnais, and collaborators Gilbert Albinet and Anne-Marie Daré. I am grateful to D. Sénéchal and D. Bergeron for critical comments on this work. This work was partially supported by NSERC (Canada) and by the Tier I Canada Research Chair Program.

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  1. Département de physique, and RQMP, Université de Sherbrooke, Sherbrooke, QC, Canada, J1K 2R1

    André-Marie S. Tremblay

  2. Canadian Institute for Advanced Research, Toronto, ON, Canada

    André-Marie S. Tremblay

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  1. André-Marie S. Tremblay
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Correspondence to André-Marie S. Tremblay .

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  1. Dipartimento di Fisica "E.R. Caianiello", Università degli Studi di Salerno, Via Ponte don Melillo, Fisciano (SA), 84084, Italy

    Adolfo Avella

  2. Dipartimento di Fisica "E.R. Caianiello", Università degli Studi di Salerno, Via Ponte don Melillo, Fisciano (SA), 84084, Italy

    Ferdinando Mancini

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Tremblay, AM.S. (2012). Two-Particle-Self-Consistent Approach for the Hubbard Model. In: Avella, A., Mancini, F. (eds) Strongly Correlated Systems. Springer Series in Solid-State Sciences, vol 171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21831-6_13

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