Skip to main content
SpringerLink
Log in

The \(\epsilon \) Expansion and Universality in Three Dimensions

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

It has been observed that the classification into universality classes of critical behavior, as established by perturbative renormalization group in the vicinity of four or six dimensions of space by the epsilon expansion, remains valid down to three dimensions in all known cases, even when perturbative renormalization group fails in lower dimensions. In this paper we argue that this classification into universality classes remains true in lower dimensions of space, even when perturbative renormalization group fails, because of the well known phenomenon of eigenvalue repulsion.

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. Wilson, K.G., Kogut, J.: The renormalization group and the \( \epsilon \) expansion. Phys. Rep. 12, 75 (1974)

    Article  ADS  Google Scholar 

  2. Wilson, K.G.: The renormalization group: critical phenomena and the Kondo problem Rev. Mod. Phys. 47, 773 (1975)

    Article  ADS  Google Scholar 

  3. Wilson, K.G.: Non-Lagrangian models of current algebra. Phys. Rev. 179, 1499 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  4. Wilson, K.G., Fisher, M.E.: Critical exponents in 3.99 dimensions. Phys. Rev. Lett. 28, 240 (1972)

    Article  ADS  Google Scholar 

  5. Guida, R., Zinn-Justin, J.: Critical exponents of the N-vector model. J. Phys. A: Math. Gen. 31, 8130 (1988)

    Google Scholar 

  6. Edwards, S.F., Anderson, P.W.: Theory of spin glasses. J. Phys. F: Met. Phys. 5(5), 965 (1975)

    Article  ADS  Google Scholar 

  7. Young, A.P.: On the lowering of dimensionality in phase transitions with random fields. J. Phys. 10, L257 (1977)

    ADS  Google Scholar 

  8. Parisi, G., Sourlas, N.: Random magnetic fields, supersymmetry and negative dimensions. Phys. Rev. Lett. 43, 744 (1979)

    Article  ADS  Google Scholar 

  9. Imbrie, J.Z.: Lower critical dimensions of the random-field Ising model. Phys. Rev. Lett. 53, 1747 (1984)

    Article  ADS  Google Scholar 

  10. Bricmont, J., Kupianen, A.: Lower critical dimensions of the random-field Ising model. Phys. Rev. Lett. 59, 1829 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  11. Fytas, N.G., Martin-Mayor, V., Picco, M., Sourlas, N.: Phase transitions in disordered systems: the example of the random-field Ising model in four dimensions. Phys. Rev. Lett. 116, 227201 (2016)

    Article  ADS  Google Scholar 

  12. Fytas, N.G., Martin-Mayor, V.: Universality in the three-dimensional random-field Ising model. Phys. Rev. Lett. 110, 227201 (2013)

    Article  ADS  Google Scholar 

  13. Picco, M., Sourlas, N.: Diluted antiferromagnetic 3D Ising model in a field EPL. Europhys. Lett. 109, 37001 (2015)

    Article  ADS  Google Scholar 

  14. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics. Pergamon Press, Oxford (1965)

    MATH  Google Scholar 

  15. von Neumann, J., Wigner, E.: Ueber das Verhalten von Eigenwerten bei adiabatischen Prozessen. Physicalische Zeitschrift 30, 467–470 (1929)

    ADS  MATH  Google Scholar 

  16. Fytas, N.G., Martin-Mayor, V., Picco, M., Sourlas, N.: Restoration of dimensional reduction in the random-field Ising model at five dimensions. Phys. Rev. E 95, 042117 (2017)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Théorique de l’ENS, École Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC Univ. Paris 06, CNRS, 75005, Paris, France

    Nicolas Sourlas

Authors
  1. Nicolas Sourlas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Sourlas.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sourlas, N. The \(\epsilon \) Expansion and Universality in Three Dimensions. J Stat Phys 172, 673–677 (2018). https://doi.org/10.1007/s10955-018-2002-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-018-2002-4

Keywords

Search

Navigation