Communications in Mathematical Physics

, Volume 256, Issue 3, pp 681–735 | Cite as

Anomalous Universality in the Anisotropic Ashkin–Teller Model

  • A. GiulianiEmail author
  • V. Mastropietro


The Ashkin–Teller (AT) model is a generalization of Ising 2–d to a four states spin model; it can be written in the form of two Ising layers (in general with different couplings) interacting via a four–spin interaction. It was conjectured long ago (by Kadanoff and Wegner, Wu and Lin, Baxter and others) that AT has in general two critical points, and that universality holds, in the sense that the critical exponents are the same as in the Ising model, except when the couplings of the two Ising layers are equal (isotropic case). We obtain an explicit expression for the specific heat from which we prove this conjecture in the weakly interacting case and we locate precisely the critical points. We find the somewhat unexpected feature that, despite universality, holds for the specific heat, nevertheless nonuniversal critical indexes appear: for instance the distance between the critical points rescale with an anomalous exponent as we let the couplings of the two Ising layers coincide (isotropic limit); and so does the constant in front of the logarithm in the specific heat. Our result also explains how the crossover from universal to nonuniversal behaviour is realized.


Neural Network Complex System Nonlinear Dynamics Explicit Expression Weakly Interact 
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.
    Ashkin, J., Teller, E.: Statistics of Two-Dimensional Lattices with Four Components. Phys. Rev. 64, 178–184 (1943)CrossRefGoogle Scholar
  2. 2.
    Baxter, R.J.: Eight-Vertex Model in Lattice Statistics. Phys. Rev. Lett. 26, 832–833 (1971)CrossRefGoogle Scholar
  3. 3.
    Baxter, R.: Exactly solved models in statistical mechanics. London-New York: Academic Press, 1982Google Scholar
  4. 4.
    Benfatto, G., Gallavotti, G.: Perturbation Theory of the Fermi Surface in Quantum Liquid. A General Quasiparticle Formalism and One-Dimensional Systems. J. Stat. Phys. 59, 541–664 (1990)Google Scholar
  5. 5.
    Benfatto, G., Gallavotti, G.: Renormalization group. Physics notes 1, Princeton, NJ: Princeton University Press 1995Google Scholar
  6. 6.
    Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: Beta function and Schwinger functions for a Many Fermions System in One Dimension. Commun. Math. Phys. 160, 93–171 (1994)Google Scholar
  7. 7.
    Benfatto, G., Mastropietro, V.: Renormalization group, hidden symmetries and approximate Ward identities in the XYZ model. Rev. Math. Phys. 13 no 11, 1323–143 (2001); Commun. Math. Phys. 231, 97–134 (2002)CrossRefGoogle Scholar
  8. 8.
    Bonetto, F., Mastropietro, V.: Beta function and Anomaly of the Fermi Surface for a d=1 System of interacting Fermions in a Periodic Potential. Commun. Math. Phys. 172, 57–93 (1995)Google Scholar
  9. 9.
    Badehdah, M., et al.: Physica B, 291, 394 (2000)Google Scholar
  10. 10.
    Bezerra, C.G., Mariz, A.M.: The anisotropic Ashkin-Teller model: a renormalization Group study. Physica A 292, 429–436 (2001)MathSciNetGoogle Scholar
  11. 11.
    Bartelt, N.C., Einstein, T.L., et al.: Phys. Rev. B 40, 10759 (1989)CrossRefGoogle Scholar
  12. 12.
    Bekhechi, S., et al.: Physica A 264, 503 (1999)Google Scholar
  13. 13.
    Benfatto, G., Mastropietro, V.: Ward identities and Dyson equations in interacting Fermi systems. To appear in J. Stat. Phys.Google Scholar
  14. 14.
    Domany, E., Riedel, E.K.: Phys. Rev. Lett. 40, 562 (1978)CrossRefGoogle Scholar
  15. 15.
    Gentile, G., Mastropietro, V.: Renormalization group for one-dimensional fermions. A review on mathematical results. Phys. Rep. 352(4–6), 273–243 (2001)Google Scholar
  16. 16.
    Gentile, G., Scoppola, B.: Renormalization Group and the ultraviolet problem in the Luttinger model. Commun. Math. Phys. 154, 153–179 (1993)Google Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Kadanoff, L.P., Wegner, F.J.: Phys. Rev. B 4, 3989–3993 (1971)CrossRefGoogle Scholar
  19. 19.
    Kasteleyn, P.W.: Dimer Statistics and phase transitions. J. Math.Phys. 4, 287 (1963)CrossRefGoogle Scholar
  20. 20.
    Fan, C.: On critical properties of the Ashkin-Teller model. Phys. Lett. 39A, 136–138 (1972)CrossRefGoogle Scholar
  21. 21.
    Hurst, C.: New approach to the Ising problem. J.Math. Phys. 7(2), 305–310 (1966)CrossRefGoogle Scholar
  22. 22.
    Itzykson, C., Drouffe, J.: Statistical field theory: 1, Cambridge: Cambridge Univ. Press, 1989Google Scholar
  23. 23.
    Lesniewski, A.: Effective action for the Yukawa 2 quantum field Theory. Commun. Math. Phys. 108, 437–467 (1987)CrossRefGoogle Scholar
  24. 24.
    Lieb, H.: Exact solution of the problem of entropy of two-dimensional ice. Phys. Rev. Lett. 18, 692–694 (1967)CrossRefGoogle Scholar
  25. 25.
    Luther, A., Peschel, I.: Calculations of critical exponents in two dimension from quantum field theory in one dimension. Phys. Rev. B 12, 3908–3917 (1975)Google Scholar
  26. 26.
    Mastropietro, V.: Ising models with four spin interaction at criticality. Commun. Math. Phys 244, 595–642 (2004)Google Scholar
  27. 27.
    Mattis, D., Lieb, E.: Exact solution of a many fermion system and its associated boson field. J. Math. Phys. 6, 304–312 (1965)Google Scholar
  28. 28.
    McCoy, B., Wu, T.: The two-dimensional Ising model. Cambridge, MA: Harvard Univ. Press, 1973Google Scholar
  29. 29.
    Montroll, E., Potts, R., Ward, J.: Correlation and spontaneous magnetization of the two dimensional Ising model. J. Math. Phys. 4, 308 (1963)Google Scholar
  30. 30.
    den Nijs, M.P.M.: Derivation of extended scaling relations between critical exponents in two dimensional models from the one dimensional Luttinger model. Phys. Rev. B 23(11), 6111–6125 (1981)Google Scholar
  31. 31.
    Onsager, L.: Critical statistics. A two dimensional model with an order-disorder transition. Phys. Rev. 56, 117–149 (1944)Google Scholar
  32. 32.
    Pruisken, A.M.M., Brown, A.C.: Universality for the critical lines of the eight vertex, Ashkin-Teller and Gaussian models. Phys. Rev. B, 23(3), 1459–1468 (1981)Google Scholar
  33. 33.
    Pinson, H., Spencer, T.: Universality in 2D critical Ising model. To appear in Commun. Math. Phys.Google Scholar
  34. 34.
    Samuel, S.: The use of anticommuting variable integrals in statistical mechanics. J. Math. Phys. 21 2806 (1980)Google Scholar
  35. 35.
    Sutherland, S.B.: Two-Dimensional Hydrogen Bonded Crystals. J. Math. Phys. 11, 3183–3186 (1970)Google Scholar
  36. 36.
    Spencer, T.: A mathematical approach to universality in two dimensions. Physica A 279, 250–259 (2000)Google Scholar
  37. 37.
    Schultz, T., Mattis, D., Lieb, E.: Two-dimensional Ising model as a soluble problem of many Fermions. Rev. Mod. Phys. 36, 856 (1964)Google Scholar
  38. 38.
    Wegner, F.J.: Duality relation between the Ashkin-Teller and the eight-vertex model. J. Phys. C 5, L131–L132 (1972)Google Scholar
  39. 39.
    Wu, F.Y., Lin, K.Y.: Two phase transitions in the Ashkin-Teller model. J. Phys. C 5, L181–L184 (1974)Google Scholar
  40. 40.
    Wu, F.W.: The Ising model with four spin interaction. Phys. Rev. B 4, 2312–2314 (1971)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Dipartimento di FisicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  3. 3.INFNRomaItaly

Personalised recommendations