Abstract
We derive the exact actions of the Q-state Potts model valid on any graph, first for the spin degrees of freedom, and second for the Fortuin-Kasteleyn clusters. In both cases the field is a traceless Q-component scalar field Φα. For the Ising model (Q = 2), the field theory for the spins has upper critical dimension \({d}_{{\text{c}}}^{{\text{spin}}}\) = 4, whereas for the clusters it has \({d}_{{\text{c}}}^{{\text{cluster}}}\) = 6. As a consequence, the probability for three points to be in the same cluster is not given by mean-field theory for d within 4 < d < 6. We estimate the associated universal structure constant as \(C=\sqrt{6-d}+\mathcal{O}{\left(6-d\right)}^{3/2}\). This shows that some observables in the Ising model have an upper critical dimension of 4, while others have an upper critical dimension of 6. Combining perturbative results from the ϵ = 6 – d expansion with a non-perturbative treatment close to dimension d = 4 allows us to locate the shape of the critical domain of the Potts model in the whole (Q, d) plane.
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Acknowledgments
We thank Mikhail Kompaniets and Andrey Pikelner for providing us with the 5-loop perturbative expansion for the Q-state Potts model, and John Cardy, Bernard Julia, Mehran Kardar, Adam Nahum and Slava Rychkov for stimulating discussions. We are most indebted to Andrei Fedorenko for generously sharing his expertise on the NPRG, and for questioning all our assumptions. This work was supported by the French Agence Nationale de la Recherche (ANR) under grant ANR-21-CE40-0003 (project CONFICA).
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Wiese, K.J., Jacobsen, J.L. The two upper critical dimensions of the Ising and Potts models. J. High Energ. Phys. 2024, 92 (2024). https://doi.org/10.1007/JHEP05(2024)092
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DOI: https://doi.org/10.1007/JHEP05(2024)092