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.

arXiv comment31 pages, 10 figures

References in corpus (28)

• Optimised Renormalisation Group Flows
• The nonperturbative functional renormalization group and its applications
• An Introduction to the Nonperturbative Renormalization Group
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$
• Three Loop Analysis of the Critical $O(N)$ Models in $6-\epsilon$ Dimensions
• Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le 4$
• Five loop renormalization of ${\varphi }^3$ theory with applications to the Lee-Yang edge singularity and percolation theory
• On three-point connectivity in two-dimensional percolation
• Three-point functions in c <= 1 Liouville theory and conformal loop ensembles
• Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator
• Random field spin models beyond one loop: a mechanism for decreasing the lower critical dimension
• Stability of Random-Field and Random-Anisotropy Fixed Points at large N
• Partition function of two- and three-dimensional Potts ferromagnets for arbitrary values of q>0
• Self-avoiding Tethered Membranes at the Tricritical Point
• Simulation of Potts models with real q and no critical slowing down
• Shadow of complex fixed point: Approximate conformality of Q>4 Potts model
• New Renormalization Group Results for Scaling of Self-Avoiding Tethered Membranes
• Critical exponents from five-loop scalar theory renormalization near six-dimensions
• A multicritical Landau-Potts field theory
• Equivalent-neighbor Potts models in two dimensions
• Functional RG approach to the Potts model
• Renormalization-Group Behavior of ${\varphi }^3$ Theories in $d=6$ Dimensions
• Hidden Critical Points in the Two-Dimensional $O(n>2)$ model: Exact Numerical Study of a Complex Conformal Field Theory
• The $q-$state Potts model from the Nonperturbative Renormalization Group
• Six dimensional Landau-Ginzburg-Wilson theory
• Metastability in the Potts model: exact results in the large q limit
• Crossover exponents, fractal dimensions and logarithms in Landau-Potts field theories

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