Communications in Mathematical Physics

, Volume 330, Issue 1, pp 217–282 | Cite as

Universality of One-Dimensional Fermi Systems, II. The Luttinger Liquid Structure

  • G. Benfatto
  • P. Falco
  • V. MastropietroEmail author


We complete the proof started in Benfatto et al. (2014) of the universal Luttinger liquid relations for a general model of spinning fermions on a lattice, by making use of the Ward identities due to asymptotically emerging symmetries. This is done by introducing an effective model verifying extra symmetries and by relating its critical exponents to those of the fermion lattice gas by suitable fine tuning of the parameters.


Hubbard Model Ward Iden Effective Model Grassmann Variable Luttinger Liquid 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benfatto, G., Falco, P., Mastropietro, V.: Universality of one-dimensional Fermi systems, I. Response functions and critical exponents. Comm. Math. Phys. (2014). doi: 10.1007/s00220-014-2008-y
  2. 2.
    Lieb E.H., Wu F.Y.: Absence of Mott transition in the 1D Hubbard model. Phys. Rev. Lett. 20, 1445–1449 (1968)ADSCrossRefGoogle Scholar
  3. 3.
    Haldane F.D.M.: General relation of correlarion exponents and application to the anisotropic S = 1/2 Heisenberg chain. Phys. Rev. Lett. 45, 1358–1362 (1980)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Kadanoff L.P., Wegner F.J.: Some critical properties of the eight-vertex model. Phys. Rev. B 4, 3989–3993 (1971)ADSCrossRefGoogle Scholar
  5. 5.
    Luther A., Peschel I.: Calculation of critical exponents in two dimensions from quantum field theory in one dimension. Phys. Rev. B 12, 3908–3917 (1975)ADSCrossRefGoogle Scholar
  6. 6.
    Mattis D., Lieb E.H.: Exact solution of a many fermion system and its associated boson field. J. Math. Phys. 6, 304–3129 (1965)ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kadanoff L.P.: Connections between the critical behavior of the planar model and that of the eight-vertex model. Phys. Rev. Lett. 39, 903–905 (1977)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    Benfatto G., Mastropietro V.: On the density–density critical indices in interacting Fermi systems. Commun. Math. Phys. 231, 97–134 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Benfatto G., Falco P., Mastropietro V.: Functional integral construction of the massive Thirring model: verification of axioms and massless limit. Commun. Math. Phys. 273, 67–118 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Benfatto G., Mastropietro V.: Ward identities and chiral anomaly in the Luttinger liquid. Commun. Math. Phys. 258, 609–655 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Benfatto G., Mastropietro V.: Ward identities and vanishing of the beta function for d = 1 interacting Fermi systems. J. Stat. Phys. 115, 143–184 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Benfatto G., Gallavotti G., Procacci A., Scoppola B.: Beta functions and Schwinger functions for a many fermions system in one dimension. Commun. Math. Phys. 160, 93–171 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Benfatto G., Falco P., Mastropietro V.: Massless sine-Gordon and massive Thirring models: proof of Coleman’s equivalence. Commun. Math. Phys. 285, 713–762 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Benfatto G., Falco P., Mastropietro V.: Extended scaling relations for planar lattice models. Commun. Math. Phys. 292, 569–605 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lesniewski A.: Effective action for the Yukawa2 quantum field theory. Commun. Math. Phys. 108, 437–467 (1987)ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mattis D.: Band theory of magnetism in metals in context of exactly soluble models. Physics 1, 183–193 (1964)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly
  2. 2.Department of MathematicsCalifornia State UniversityNorthridgeUSA
  3. 3.Dipartimento di Matematica F. EnriquezUniversità di MilanoMilanItaly

Personalised recommendations