The two upper critical dimensions of the Ising and Potts models

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 ${\Phi }^{\alpha }$. For the Ising model ($Q=2$), the field theory for the spins has upper critical dimension $d_{\mathrm{c}}^{\mathrm{spin}}=4$, whereas for the clusters it has $d_{\mathrm{c}}^{\mathrm{cluster}}=6$. 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 $C=\sqrt{6-d}+\mathcal{O}(6-d{)}^{3/2}$. 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 $\epsilon =6-d$ 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.

31 pages, 10 figures

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Cited by in corpus (2)

• New renormalization group study of the 3-state Potts model and related statistical models
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories