Skip to main content
SpringerLink
Log in

Exact Solution of a 2D Interacting Fermion Model

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study an exactly solvable quantum field theory (QFT) model describing interacting fermions in 2+1 dimensions. This model is motivated by physical arguments suggesting that it provides an effective description of spinless fermions on a square lattice with local hopping and density-density interactions if, close to half filling, the system develops a partial energy gap. The necessary regularization of the QFT model is based on this proposed relation to lattice fermions. We use bosonization methods to diagonalize the Hamiltonian and to compute all correlation functions. We also discuss how, after appropriate multiplicative renormalizations, all short- and long distance cutoffs can be removed. In particular, we prove that the renormalized two-point functions have algebraic decay with non-trivial exponents depending on the interaction strengths, which is a hallmark of Luttinger-liquid behavior.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mattis D.C.: Implications of infrared instability in a two-dimensional electron gas. Phys. Rev. B 36, 745 (1987)

    Article  ADS  Google Scholar 

  2. Langmann E.: A two dimensional analogue of the Luttinger model. Lett. Math. Phys. 92, 109 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Langmann E.: A 2D Luttinger model. J. Stat. Phys. 141, 17 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. de Woul J., Langmann E.: Partially gapped fermions in 2D. J. Stat. Phys. 139, 1033 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Tomonaga S.: Remarks on Bloch’s method of sound waves applied to many-fermion problems. Prog. Theor. Phys. 5, 544 (1950)

    Article  MathSciNet  ADS  Google Scholar 

  6. Thirring W.: A soluble relativistic field theory. Ann. Phys. 3, 91 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Luttinger J.M.: An exactly soluble model of a many-fermion system. J. Math. Phys. 4, 1154 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  8. Mattis D.C., Lieb E.H.: Exact solution of a many-fermion system and its associated boson field. J. Math. Phys. 6, 304 (1965)

    Article  MathSciNet  ADS  Google Scholar 

  9. Haldane F.D.M.: “Luttinger liquid theory” of one-dimensional quantum fluids: I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas. J. Phys. C 14, 2585 (1981)

    Article  ADS  Google Scholar 

  10. Anderson P.W.: “Luttinger-liquid” behavior of the normal metallic state of the 2D Hubbard model. Phys. Rev. Lett. 64, 1839 (1990)

    Article  ADS  Google Scholar 

  11. Furukawa N., Rice T.M., Salmhofer M.: Truncation of a two-dimensional Fermi surface due to quasiparticle gap formation at the saddle points. Phys. Rev. Lett. 81, 3195 (1998)

    Article  ADS  Google Scholar 

  12. Honerkamp C., Salmhofer M., Furukawa N., Rice T.M.: Breakdown of the Landau-Fermi liquid in two dimensions due to umklapp scattering. Phys. Rev. B 63, 035109 (2001)

    Article  ADS  Google Scholar 

  13. Hlubina R.: Luttinger liquid in a solvable two-dimensional model. Phys. Rev. B 50, 8252 (1994)

    Article  ADS  Google Scholar 

  14. Luther A.: Interacting electrons on a square Fermi surface. Phys. Rev. B 50, 11446 (1994)

    Article  ADS  Google Scholar 

  15. Syljuåsen O.F., Luther A.: Adjacent face scattering and stability of the square Fermi surface. Phys. Rev. B 72, 165105 (2005)

    Article  ADS  Google Scholar 

  16. Schotte K.D., Schotte U.: Tomonaga’s model and the threshold singularity of X-ray spectra of metals. Phys. Rev. 182, 479 (1969)

    Article  MathSciNet  ADS  Google Scholar 

  17. Luther A., Peschel I.: Single-particle states, Kohn anomaly, and pairing fluctuations in one dimension. Phys. Rev. B 9, 2911 (1974)

    Article  ADS  Google Scholar 

  18. Fjærestad J.O., Sudbø A., Luther A.: Correlation functions for a two-dimensional electron system with bosonic interactions and a square Fermi surface. Phys. Rev. B 60, 13361 (1999)

    Article  ADS  Google Scholar 

  19. Haldane F.D.M.: Coupling between charge and spin degrees of freedom in the one-dimensional Fermi gas with backscattering. J. Phys. C 12, 4791 (1979)

    Article  ADS  Google Scholar 

  20. Heidenreich R., Seiler R., Uhlenbrock D.A.: The Luttinger Model. J. Stat. Phys 22, 27 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  21. von Delft J., Schoeller H.: Bosonization for beginners - refermionization for experts. Ann. Phys. (Leipzig) 7, 225 (1998)

    Article  ADS  MATH  Google Scholar 

  22. Frenkel I.B.: Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. F. Funct. Anal. 44, 259 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  23. Carey A.L., Hurst C.A.: A note on the boson-fermion correspondence and infinite-dimensional groups. Commun. Math. Phys. 98, 435 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Carey A.L., Ruijsenaars S.N.M.: On fermion gauge groups, current algebras and Kac-Moody algebras. Acta Appl. Mat. 10, 1 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kac, V.: Vertex Algebras for Beginners. University Lecture Series, 10 (2nd ed.), Providence, RI: Amer. Math. Soc., 1998

  26. Carey, A.L., Langmann, E.: Loop groups and quantum fields. In: Geometric Analysis and Applications to Quantum Field Theory Progress in Mathematics, Vol. 205, P. Bouwknegt, S. Wu (eds.). Boston: Birkhauser, 2002, pp. 45–94

  27. Mastropietro V.: Luttinger liquid fixed point for a two-dimensional flat Fermi surface. Phys. Rev. B 77, 195106 (2008)

    Article  ADS  Google Scholar 

  28. Zheleznyak A.T., Yakovenko V.M., Dzyaloshinskii I.E.: Parquet solution for a flat Fermi surface. Phys. Rev. B 55, 3200 (1997)

    Article  ADS  Google Scholar 

  29. Houghton A., Kwon H.-J., Marston J.B.: Multidimensional bosonization. Adv. Phys. 49, 141 (2000)

    Article  ADS  Google Scholar 

  30. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis. New York: Academic Press, 1980

  31. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness. New York: Academic Press, 1975

  32. Grosse H., Langmann E.: A superversion of quasi-free second quantization. I. Charged particles. J. Math. Phys. 33, 1032 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  33. Mastropietro V.: Schwinger functions in Thirring and Luttinger models. Nuovo Cim. B. 108, 1095 (1993)

    Article  ADS  Google Scholar 

  34. Carey A.L., Ruijsenaars S.N.M., Wright J.D.: The massless Thirring model: Positivity of Klaiber’s n-point functions. Commun. Math. Phys. 99, 347 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  35. Osterwalder K., Schrader R.: Axioms for Euclidean Green’s Functions II. Commun. math. Phys. 42, 281 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Grosse H., Langmann E., Raschhofer E.: On the Luttinger-Schwinger model. Annals of Phys. (N.Y.) 253, 310 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. Boies D., Bourbonnais C., Tremblay A.-M.S.: One-particle and two-particle instability of coupled Luttinger liquids. Phys. Rev. Lett. 74, 968 (1995)

    Article  ADS  Google Scholar 

  38. Kopietz P., Meden V., Schönhammer K.: Crossover between Luttinger and Fermi-liquid behavior in weakly coupled metallic chains. Phys. Rev. B 56, 7232 (1997)

    Article  ADS  Google Scholar 

  39. Vishwanath A., Carpentier D.: Two-dimensional anisotropic non-Fermi-liquid phase of coupled Luttinger liquids. Phys. Rev. Lett. 86, 676 (2001)

    Article  ADS  Google Scholar 

  40. Salmhofer, M.: Renormalization: an Introduction. Heidelberg: Springer, 1999

  41. Mastropietro, V.: Non-perturbative Renormalization. Singapore: World Scientific, 2008

  42. Fogedby H.C.: Correlation functions for the Tomonaga model. J. Phys. C: Sol. Stat. Phys. 9, 3757 (1976)

    Article  ADS  Google Scholar 

  43. Fröhlich J., Götschmann R., Marchetti P.A.: Bosonization of Fermi systems in arbitrary dimension in terms of gauge forms. J. Phys. A: Math. Gen. 28, 1169 (1995)

    Article  ADS  Google Scholar 

  44. Kopietz P., Hermisson J., Schönhammer K.: Bosonization of interacting fermions in arbitrary dimension beyond the Gaussian approximation. Phys. Rev. B 52, 10877 (1995)

    Article  ADS  Google Scholar 

  45. Salmhofer M.: Continuous renormalization for fermions and Fermi liquid theory. Commun. Math. Phys. 194, 249 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. Salmhofer M.: Improved power counting and Fermi surface renormalization. Rev. Math. Phys. 10, 553 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  47. Disertori M., Rivasseau V.: A rigorous proof of Fermi liquid behavior for jellium two-dimensional interacting fermions. Phys. Rev. Lett. 85, 361 (2000)

    Article  ADS  Google Scholar 

  48. Benfatto G., Gallavotti G., Mastropietro V.: Renormalization group and the Fermi surface in the Luttinger model. Phys. Rev. B 45, 5468 (1992)

    Article  ADS  Google Scholar 

  49. Kahn P.: Mathematical Methods for Scientists and Engineers: Linear and Nonlinear Systems. Wiley-Interscience, New York (1996)

    Google Scholar 

  50. Abramowitz, M., Stegun, I.A. (eds): Handbook of Mathematical Functions. Dover Publications, New York (1965)

    Google Scholar 

  51. Khveshchenko D.V.: Bosonization of current-current interactions. Phys. Rev. B 49, 16893 (1994)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Physics, Royal Institute of Technology (KTH), 106 91, Stockholm, Sweden

    Jonas de Woul & Edwin Langmann

Authors
  1. Jonas de Woul
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edwin Langmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edwin Langmann.

Additional information

Communicated by G. Gallavotti

Rights and permissions

Reprints and permissions

About this article

Cite this article

de Woul, J., Langmann, E. Exact Solution of a 2D Interacting Fermion Model. Commun. Math. Phys. 314, 1–56 (2012). https://doi.org/10.1007/s00220-012-1518-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1518-8

Keywords

Search

Navigation