Advertisement

Renormalization group approach to interacting fermion systems in the two-particle-irreducible formalism

Solid and Condensed State Physics

Abstract.

We describe a new formulation of the functional renormalization group (RG) for interacting fermions within a Wilsonian momentum-shell approach. We show that the Luttinger-Ward functional is invariant under the RG transformation, and derive the infinite hierarchy of flow equations satisfied by the two-particle-irreducible (2PI) vertices. In the one-loop approximation, this hierarchy reduces to two equations that determine the self-energy and the 2PI two-particle vertex Φ(2). Susceptibilities are calculated from the Bethe-Salpeter equation that relates them to Φ(2). While the one-loop approximation breaks down at low energy in one-dimensional systems (for reasons that we discuss), it reproduces the exact results both in the normal and ordered phases in single-channel (i.e. mean-field) theories, as shown on the example of BCS theory. The possibility to continue the RG flow into broken-symmetry phases is an essential feature of the 2PI RG scheme and is due to the fact that the 2PI two-particle vertex, contrary to its 1PI counterpart, is not singular at a phase transition. Moreover, the normal phase RG equations can be directly used to derive the Ginzburg-Landau expansion of the thermodynamic potential near a phase transition. We discuss the implementation of the 2PI RG scheme to interacting fermion systems beyond the examples (one-dimensional systems and BCS superconductors) considered in this paper.

Keywords

Phase Transition Renormalization Group Normal Phase Thermodynamic Potential Group Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.M. Luttinger, J.C. Ward, Phys. Rev. 118, 1417 (1960) CrossRefADSMATHMathSciNetGoogle Scholar
  2. G. Baym, L.P. Kadanoff, Phys. Rev. 124, 287 (1961) CrossRefADSMATHMathSciNetGoogle Scholar
  3. G. Baym, Phys. Rev. 127, 1391 (1962) CrossRefADSMATHMathSciNetGoogle Scholar
  4. C. De Dominicis, P.C. Martin, J. Math. Phys. 5, 14 and 31 (1964) CrossRefADSGoogle Scholar
  5. The 2PI formalism has been generalized to relativistic field theories by J. M. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. D 10, 2428 (1974) CrossRefADSMATHGoogle Scholar
  6. In bosonic systems, the functional Γ depends also on the expectation value of the bosonic field Google Scholar
  7. Φ-derivable approximations satisfy conservation laws and Ward identities in fermion systems Baym62. More generally, expectation values of the Noether currents corresponding to global symmetries are conserved, but Ward identities for the self-energy and higher-order vertex functions can be violated, e.g. in bosonic systems with broken gauge invariance (see Ref. VanHees02c) Google Scholar
  8. N.E. Bickers, D.J. Scalapino, S.R. White, Phys. Rev. Lett. 62, 961 (1989) CrossRefADSGoogle Scholar
  9. N.E. Bickers, D.J. Scalapino, Annals of Physics 193, 206 (1989) CrossRefADSGoogle Scholar
  10. Y. Vilk, A.-M.S. Tremblay, J. Phys. I 7, 1309 (1997) CrossRefGoogle Scholar
  11. C. Wetterich, e-print arXiv:cond-mat/0208361 Google Scholar
  12. C. Bourbonnais, in Les Houches, Session LVI (1991), Strongly interacting fermions and high-Tc superconductivity, edited by B. Douçot, J. Zinn-Justin (Elsevier Science, Amsterdam, 1995), p. 307 Google Scholar
  13. R. Duprat, C. Bourbonnais, Eur. Phys. J. B 21, 219 (2001) CrossRefADSGoogle Scholar
  14. C. Bourbonnais, R. Duprat, J. Phys. IV France 114, 3 (2004) CrossRefGoogle Scholar
  15. J.C. Nickel, R. Duprat, C. Bourbonnais, N. Dupuis, Phys. Rev. Lett. 95, 247001 (2005) CrossRefADSGoogle Scholar
  16. J.C. Nickel, R. Duprat, C. Bourbonnais, N. Dupuis, e-print arXiv:cond-mat/0510744 Google Scholar
  17. Y. Fuseya, Y. Suzumura, J. Phys. Soc. Jpn 74, 1264 (2005) ADSGoogle Scholar
  18. G. Abramovici, J.C. Nickel, M. Heritier, Phys. Rev. B 72, 045120 (2005) CrossRefADSGoogle Scholar
  19. K.-M. Tam, S.-W. Tsai, D.K. Campbell, e-print arXiv:cond-mat/0505396 Google Scholar
  20. D. Zanchi, H.J. Schulz, Europhys. Lett. 44, 235 (1998) CrossRefADSGoogle Scholar
  21. D. Zanchi, H.J. Schulz, Phys. Rev. B 61, 13609 (2000) CrossRefADSGoogle Scholar
  22. C.J. Halboth, W. Metzner, Phys. Rev. B 61, 7364 (2000) CrossRefADSGoogle Scholar
  23. C. Honerkamp, M. Salmhofer, N. Furukawa, T.M. Rice, Phys. Rev. B 63, 035109 (2001) CrossRefADSGoogle Scholar
  24. C. Honerkamp, M. Salmhofer, Phys. Rev. B 64, 184516 (2001) CrossRefADSGoogle Scholar
  25. M. Salmhofer, C. Honerkamp, Prog. Theor. Phys. 105, 1 (2001) CrossRefADSMATHMathSciNetGoogle Scholar
  26. C. Honerkamp, M. Salmhofer, T.M. Rice, Eur. Phys. J. B 27, 127 (2002) ADSGoogle Scholar
  27. C. Honerkamp, D. Rohe, S. Andergassen, T. Enss, Phys. Rev. B 70, 235115 (2004) CrossRefADSGoogle Scholar
  28. M. Salmhofer, C. Honerkamp, W. Metzner, O. Lauscher, Prog. Theor. Phys. 112, 943 (2005) CrossRefADSMathSciNetGoogle Scholar
  29. P. Kopietz, T. Busche, Phys. Rev. B 64, 155101 (2001) CrossRefADSGoogle Scholar
  30. T. Busche, L. Bartosch, P. Kopietz, J. Phys.: Cond. Matt. 14, 8513 (2002) CrossRefADSGoogle Scholar
  31. S. Ledowski, P. Kopietz, J. Phys.: Cond. Matt. 15, 4779 (2003) CrossRefADSGoogle Scholar
  32. F. Schütz, L. Bartosch, P. Kopietz, Phys. Rev. B 72, 035107 (2005) CrossRefADSGoogle Scholar
  33. B. Binz, D. Baeriswyl, B. Douçot, Eur. Phys. J. B 25, 69 (2002) ADSGoogle Scholar
  34. B. Binz, D. Baeriswyl, B. Douçot, Ann. Phys. 12, 704 (2003) CrossRefMATHGoogle Scholar
  35. A.A. Katanin, A.P. Kampf, Phys. Rev. B 68, 195101 (2003) CrossRefADSGoogle Scholar
  36. A.A. Katanin, A.P. Kampf, Phys. Rev. Lett. 93, 106406 (2004) CrossRefADSGoogle Scholar
  37. A. Schwenk, B. Friman, G.E. Brown, Nucl. Phys. A 713, 191 (2003) CrossRefADSMATHGoogle Scholar
  38. T. Baier, E. Bick, C. Wetterich, Phys. Rev. B 70, 125111 (2004) CrossRefADSMathSciNetGoogle Scholar
  39. T. Baier, E. Bick, C. Wetterich, Phys. Lett. B 605, 144 (2005) CrossRefADSGoogle Scholar
  40. H. Freire, E. Corrêa, A. Ferraz, Phys. Rev. B 71, 165113 (2005) CrossRefADSGoogle Scholar
  41. The 1PI RG scheme was first introduced in quantum field theory and statistical physics. For a review, see J. Berges, N. Tetradis, C. Wetterich, Phys. Rep. 363, 223 (1993) Google Scholar
  42. Y. Fuseya, H. Kohno, K. Miyake, J. Phys. Soc. Jpn 74, 722 (2005) CrossRefADSGoogle Scholar
  43. W. Metzner, J. Reiss, D. Rohe, e-print arXiv:cond-mat/0509412 Google Scholar
  44. T.M. Morris, Int. J. Mod. Phys. A 9, 2411 (1994) CrossRefADSMATHGoogle Scholar
  45. R. Shankar, Rev. Mod. Phys. 66, 129 (1994) CrossRefADSMathSciNetGoogle Scholar
  46. H. Van Hees, J. Knoll, Phys. Rev. D 65, 105005 (2002) CrossRefADSGoogle Scholar
  47. H. Van Hees, J. Knoll, Phys. Rev. D 65, 025010 (2002) CrossRefADSGoogle Scholar
  48. H. Van Hees, J. Knoll, Phys. Rev. D 66, 025028 (2002) CrossRefADSGoogle Scholar
  49. J.P. Blaizot, E. Iancu, U. Reinosa, Nucl. Phys. A 736, 149 (2004) CrossRefADSGoogle Scholar
  50. Note that due to the antisymmetry of G and J, the chain rule for derivation includes additional \(\frac{1}{2}\) factors, e.g. \(\frac{\delta W[J]}{\delta G_\gamma} = \frac{1}{2} \sum_{\gamma'} \frac{\delta J_{\gamma'}}{\delta G_\gamma} \frac{\delta W[J]}{\delta J_{\gamma'}}\). For the same reason, we have \(\frac{\delta G_{\gamma_1}}{\delta G_{\gamma_2}}= \frac{\delta J_{\gamma_1}}{\delta J_{\gamma_{2}}}=I_{\gamma_{1} \gamma_{2}}\) where I is the unit matrix defined in equation (12 Google Scholar
  51. R. Haussmann, Self-consistent quantum field theory and bosonization for strongly correlated electron systems (Springer, New York, 1999) Google Scholar
  52. Note that our derivation of the universality of the LW functional is non perturbative as it does not rely on any diagrammatic expansion. For a related discussion, see M. Potthoff, e-print arXiv:cond-mat/0406671 Google Scholar
  53. The functionals F[Σ] and Γ[Σ] have been introduced in the context of the self-energy functional approach by M. Potthoff, Eur. Phys. J. B 32, 429 (2003) CrossRefADSGoogle Scholar
  54. Equation (92) follows from (82) and (85). To avoid ambiguities due to factors Θ(0), we use \(\partial_\Lambda = \partial_\Lambda|_\Delta + \dot\Delta\partial_\Delta\), and carry out the sums over momentum before applying \(\partial_\Lambda|_\Delta\). The same trick is used to calculate the ground state condensation energy in Section 3.2.1 Google Scholar
  55. P.W. Anderson, Phys. Rev. 112, 1900 (1958) CrossRefADSMathSciNetGoogle Scholar
  56. N.N. Bogoliubov, Sov. Phys. JETP 34, 41 and 51 (1958) MathSciNetGoogle Scholar
  57. N. Dupuis, unpublished Google Scholar
  58. A pairing instability in a high angular momentum channel can occur when the normal phase remains stable down to extremely low temperatures (Kohn-Luttinger effect). Such an instability would however not be experimentally observable Google Scholar
  59. T. Vuletic, P. Auban-Senzier, C. Pasquier, S. Tomic, D. Jérome, M. Héritier, K. Bechgaard, Eur. Phys. J. B 25, 319 (2002) CrossRefADSGoogle Scholar
  60. I.J. Lee, S.E. Brown, W. Yu, M.J. Naughton, P.M. Chaikin, Phys. Rev. Lett. 94, 197001 (2005) CrossRefADSGoogle Scholar
  61. J.P. Pouget, S. Ravy, J. Phys. I France 6, 1501 (1996) CrossRefGoogle Scholar
  62. S. Kagoshima, Y. Saso, M. Maesato, R. Kondo, T. Hasegawa, Sol. State. Comm. 110, 479 (1999) CrossRefADSGoogle Scholar
  63. N. Dupuis, Int. J. Mod. Phys. B 14, 379 (2000) CrossRefADSGoogle Scholar
  64. A.A. Katanin, Phys. Rev. B 70, 115109 (2004) CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2AZUK and Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-SudOrsayFrance

Personalised recommendations