Communications in Mathematical Physics

, Volume 314, Issue 1, pp 1–56 | Cite as

Exact Solution of a 2D Interacting Fermion Model

  • Jonas de Woul
  • Edwin Langmann


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.


Fermi Surface Mattis Model Vertex Operator Algebra Lattice Fermion Boson Operator 
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.


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  1. 1.
    Mattis D.C.: Implications of infrared instability in a two-dimensional electron gas. Phys. Rev. B 36, 745 (1987)ADSCrossRefGoogle Scholar
  2. 2.
    Langmann E.: A two dimensional analogue of the Luttinger model. Lett. Math. Phys. 92, 109 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Langmann E.: A 2D Luttinger model. J. Stat. Phys. 141, 17 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    de Woul J., Langmann E.: Partially gapped fermions in 2D. J. Stat. Phys. 139, 1033 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Tomonaga S.: Remarks on Bloch’s method of sound waves applied to many-fermion problems. Prog. Theor. Phys. 5, 544 (1950)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Thirring W.: A soluble relativistic field theory. Ann. Phys. 3, 91 (1958)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Luttinger J.M.: An exactly soluble model of a many-fermion system. J. Math. Phys. 4, 1154 (1963)MathSciNetADSCrossRefGoogle Scholar
  8. 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)MathSciNetADSCrossRefGoogle Scholar
  9. 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)ADSCrossRefGoogle Scholar
  10. 10.
    Anderson P.W.: “Luttinger-liquid” behavior of the normal metallic state of the 2D Hubbard model. Phys. Rev. Lett. 64, 1839 (1990)ADSCrossRefGoogle Scholar
  11. 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)ADSCrossRefGoogle Scholar
  12. 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)ADSCrossRefGoogle Scholar
  13. 13.
    Hlubina R.: Luttinger liquid in a solvable two-dimensional model. Phys. Rev. B 50, 8252 (1994)ADSCrossRefGoogle Scholar
  14. 14.
    Luther A.: Interacting electrons on a square Fermi surface. Phys. Rev. B 50, 11446 (1994)ADSCrossRefGoogle Scholar
  15. 15.
    Syljuåsen O.F., Luther A.: Adjacent face scattering and stability of the square Fermi surface. Phys. Rev. B 72, 165105 (2005)ADSCrossRefGoogle Scholar
  16. 16.
    Schotte K.D., Schotte U.: Tomonaga’s model and the threshold singularity of X-ray spectra of metals. Phys. Rev. 182, 479 (1969)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Luther A., Peschel I.: Single-particle states, Kohn anomaly, and pairing fluctuations in one dimension. Phys. Rev. B 9, 2911 (1974)ADSCrossRefGoogle Scholar
  18. 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)ADSCrossRefGoogle Scholar
  19. 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)ADSCrossRefGoogle Scholar
  20. 20.
    Heidenreich R., Seiler R., Uhlenbrock D.A.: The Luttinger Model. J. Stat. Phys 22, 27 (1980)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    von Delft J., Schoeller H.: Bosonization for beginners - refermionization for experts. Ann. Phys. (Leipzig) 7, 225 (1998)ADSzbMATHCrossRefGoogle Scholar
  22. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Carey A.L., Hurst C.A.: A note on the boson-fermion correspondence and infinite-dimensional groups. Commun. Math. Phys. 98, 435 (1985)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Carey A.L., Ruijsenaars S.N.M.: On fermion gauge groups, current algebras and Kac-Moody algebras. Acta Appl. Mat. 10, 1 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kac, V.: Vertex Algebras for Beginners. University Lecture Series, 10 (2nd ed.), Providence, RI: Amer. Math. Soc., 1998Google Scholar
  26. 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. 45–94Google Scholar
  27. 27.
    Mastropietro V.: Luttinger liquid fixed point for a two-dimensional flat Fermi surface. Phys. Rev. B 77, 195106 (2008)ADSCrossRefGoogle Scholar
  28. 28.
    Zheleznyak A.T., Yakovenko V.M., Dzyaloshinskii I.E.: Parquet solution for a flat Fermi surface. Phys. Rev. B 55, 3200 (1997)ADSCrossRefGoogle Scholar
  29. 29.
    Houghton A., Kwon H.-J., Marston J.B.: Multidimensional bosonization. Adv. Phys. 49, 141 (2000)ADSCrossRefGoogle Scholar
  30. 30.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis. New York: Academic Press, 1980Google Scholar
  31. 31.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness. New York: Academic Press, 1975Google Scholar
  32. 32.
    Grosse H., Langmann E.: A superversion of quasi-free second quantization. I. Charged particles. J. Math. Phys. 33, 1032 (1992)MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Mastropietro V.: Schwinger functions in Thirring and Luttinger models. Nuovo Cim. B. 108, 1095 (1993)ADSCrossRefGoogle Scholar
  34. 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)MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    Osterwalder K., Schrader R.: Axioms for Euclidean Green’s Functions II. Commun. math. Phys. 42, 281 (1975)MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Grosse H., Langmann E., Raschhofer E.: On the Luttinger-Schwinger model. Annals of Phys. (N.Y.) 253, 310 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 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)ADSCrossRefGoogle Scholar
  38. 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)ADSCrossRefGoogle Scholar
  39. 39.
    Vishwanath A., Carpentier D.: Two-dimensional anisotropic non-Fermi-liquid phase of coupled Luttinger liquids. Phys. Rev. Lett. 86, 676 (2001)ADSCrossRefGoogle Scholar
  40. 40.
    Salmhofer, M.: Renormalization: an Introduction. Heidelberg: Springer, 1999Google Scholar
  41. 41.
    Mastropietro, V.: Non-perturbative Renormalization. Singapore: World Scientific, 2008Google Scholar
  42. 42.
    Fogedby H.C.: Correlation functions for the Tomonaga model. J. Phys. C: Sol. Stat. Phys. 9, 3757 (1976)ADSCrossRefGoogle Scholar
  43. 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)ADSCrossRefGoogle Scholar
  44. 44.
    Kopietz P., Hermisson J., Schönhammer K.: Bosonization of interacting fermions in arbitrary dimension beyond the Gaussian approximation. Phys. Rev. B 52, 10877 (1995)ADSCrossRefGoogle Scholar
  45. 45.
    Salmhofer M.: Continuous renormalization for fermions and Fermi liquid theory. Commun. Math. Phys. 194, 249 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  46. 46.
    Salmhofer M.: Improved power counting and Fermi surface renormalization. Rev. Math. Phys. 10, 553 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Disertori M., Rivasseau V.: A rigorous proof of Fermi liquid behavior for jellium two-dimensional interacting fermions. Phys. Rev. Lett. 85, 361 (2000)ADSCrossRefGoogle Scholar
  48. 48.
    Benfatto G., Gallavotti G., Mastropietro V.: Renormalization group and the Fermi surface in the Luttinger model. Phys. Rev. B 45, 5468 (1992)ADSCrossRefGoogle Scholar
  49. 49.
    Kahn P.: Mathematical Methods for Scientists and Engineers: Linear and Nonlinear Systems. Wiley-Interscience, New York (1996)Google Scholar
  50. 50.
    Abramowitz, M., Stegun, I.A. (eds): Handbook of Mathematical Functions. Dover Publications, New York (1965)Google Scholar
  51. 51.
    Khveshchenko D.V.: Bosonization of current-current interactions. Phys. Rev. B 49, 16893 (1994)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsRoyal Institute of Technology (KTH)StockholmSweden

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