The two upper critical dimensions of the Ising and Potts models

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_c^spin = 4, whereas for the clusters it has d_c^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=√(6-d)+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 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.