2000 character limit reached
The two upper critical dimensions of the Ising and Potts models (2311.01529v2)
Published 2 Nov 2023 in hep-th and cond-mat.stat-mech
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 $d_{\rm c}{\rm spin}=4$, whereas for the clusters it has $d_{\rm c}{\rm cluster}=6$. 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 $C=\sqrt{6-d}+ {\cal O}(6-d){3/2}$. 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.
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