Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Pertubation theory around nonnested fermi surfaces. I. Keeping the fermi surface fixed

Abstract

The perturbation expansion for a general class of many-fermion systems with a nonnested, nonspherical Fermi surface is renormalized to all orders. In the limit as the infrared cutoff is removed, the counterterms converge to a finite limit which is differentiable in the band structure. The map from the renormalized to the bare band structure is shown to be locally injective. A new classification of graphs as overlapping or nonoverlapping is given, and improved power counting bounds are derived from it. They imply that the only subgraphs that can generater factorials in ther th order of the renormalized perturbation series are indeed the ladder graphs and thus give a precise sense to the statement that “ladders are the most divergent diagrams.” Our results apply directly to the Hubbard model at any filling except for half-filling. The half-filled Hubbard model is treated in another place.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    O. Bratteli and D. W. Robinson,Operator Algebras and Quantum Statistical Mechanics II (Springer, New York, 1979).

  2. 2.

    J. Feldman and E. Trubowitz, Perturbation theory for many-fermion systems,Helv. Phys. Acta 63:156 (1990).

  3. 3.

    J. Feldman and E. Trubowitz, The flow of an electron-phonon system to the superconducting state,Helv. Phys. Acta 64:213 (1991).

  4. 4.

    E. H. Lieb, The Hubbard model: Some rigorous results and open problems, inAdvances in Dynamical Systems and Quantum Physics (World Scientific, Singapore, 1995).

  5. 5.

    J. Feldman, J. Magnen, V. Rivasseau, and E. Trubowitz, An infinite volume expansion for many fermion Green's functions,Helv. Phys. Acta 65:679 (1992).

  6. 6.

    A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,Methods of Quantum Field Theory in Statistical Mechanics (Dover, New York, 1975).

  7. 7.

    J. Feldman, H. Knörrer, D. Lehmann, and E. Trubowitz, Fermi liquids in two space dimensions, inConstructive Physics, V. Rivasseau, ed. (Springer, New York, 1995).

  8. 8.

    J. Dieudonné,Foundations of Modern Analysis, Vol. 1 (Academic Press, New York, 1969).

  9. 9.

    J. Glimm and A. Jaffe,Quantum Physics (Springer, New York, 1987).

  10. 10.

    J. Feldman, M. Salmhofer, and E. Trubowitz, Perturbation theory around non-nested Fermi surfaces II, to appear.

Download references

Author information

Additional information

This paper is dedicated to the memory of Ansgar Schnizer.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Feldman, J., Salmhofer, M. & Trubowitz, E. Pertubation theory around nonnested fermi surfaces. I. Keeping the fermi surface fixed. J Stat Phys 84, 1209–1336 (1996). https://doi.org/10.1007/BF02174132

Download citation

Key Words

  • Many-fermion systems
  • perturbation theory
  • renormalization nonspherical Fermi surfaces
  • Hubbard model
  • overlapping graphs