Journal of Statistical Physics

, Volume 177, Issue 6, pp 1089–1130 | Cite as

Conformal Invariance and Vector Operators in the O(N) Model

  • Gonzalo De Polsi
  • Matthieu Tissier
  • Nicolás WscheborEmail author


It is widely expected that, for a large class of models, scale invariance implies conformal invariance. A sufficient condition for this to happen is that there exists no integrated vector operator, invariant under all internal symmetries of the model, with scaling dimension \(-1\). In this article, we compute the scaling dimensions of vector operators with lowest dimensions in the O(N) model. We use three different approximation schemes: \(\epsilon \) expansion, large N limit and third order of the derivative expansion of Non-Perturbative Renormalization Group equations. We find that the scaling dimensions of all considered integrated vector operators are always much larger than \(-1\). This strongly supports the existence of conformal invariance in this model. For the Ising model, an argument based on correlation functions inequalities was derived, which yields a lower bound for the scaling dimension of the vector perturbations. We generalize this proof to the case of the O(N) model with \(N\in \left\{ 2,3,4 \right\} \).


Conformal symmetry Critical phenomena O(N) model 



The authors thank B. Delamotte, G. Tarjus, T. Morris and Y. Nakayama for fruitful discussions. The authors acknowledge financial support from the ECOS-Sud France-Uruguay Program U11E01. N. W. and D. P. thanks PEDECIBA (Programa de desarrollo de las Ciencias Básicas, Uruguay) and acknowledges funding through Grant from the Comisión Sectorial de Investigación Científica de la Universidad de la República, Project I+D 2016 (cod 412).

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Authors and Affiliations

  • Gonzalo De Polsi
    • 1
  • Matthieu Tissier
    • 2
  • Nicolás Wschebor
    • 3
    Email author
  1. 1.Instituto de Física, Facultad de CienciasUniversidad de la RepúblicaMontevideoUruguay
  2. 2.CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMCSorbonne UniversitéParisFrance
  3. 3.Instituto de Física, Facultad de IngenieríaUniversidad de la RepúblicaMontevideoUruguay

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