Operator content of the critical Potts model in d dimensions and logarithmic correlations

Abstract

Using the symmetric group $S_Q$ symmetry of the $Q$-state Potts model, we classify the (scalar) operator content of its underlying field theory in arbitrary dimension. In addition to the usual identity, energy and magnetization operators, we find fields that generalize the $N$-cluster operators well-known in two dimensions, together with their subleading counterparts. We give the explicit form of all these operators -- up to non-universal constants -- both on the lattice and in the continuum limit for the Landau theory. We compute exactly their two- and three-point correlation functions on an arbitrary graph in terms of simple probabilities, and give the general form of these correlation functions in the continuum limit at the critical point. Specializing to integer values of the parameter $Q$, we argue that the analytic continuation of the $S_Q$ symmetry yields logarithmic correlations at the critical point in arbitrary dimension, thus implying a mixing of some scaling fields by the scale transformation generator. All these logarithmic correlation functions are given a clear geometrical meaning, which can be checked in numerical simulations. Several physical examples are discussed, including bond percolation, spanning trees and forests, resistor networks and the Ising model. We also briefly address the generalization of our approach to the $O(n)$ model.

35 pages, 6 figures

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Cited by in corpus (26)

• Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory
• Measurement-induced phase transitions in $(d+1)$-dimensional stabilizer circuits
• Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap
• Parafermionic conformal field theory on the lattice
• The ABC (in any D) of Logarithmic CFT
• Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap
• Bootstrapping hypercubic and hypertetrahedral theories in three dimensions
• Short-range correlations in percolation at criticality
• The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$
• Geometric properties of the Fortuin-Kasteleyn representation of the Ising model
• Quantum Spectral Curve of $\gamma$-twisted $\mathcal{N}=4$ SYM theory and fishnet CFT
• Non-scalar operators for the Potts model in arbitrary dimension
• Exact logarithmic four-point functions in the critical two-dimensional Ising model
• Four-point geometrical correlation functions in the two-dimensional $Q$-state Potts model: connections with the RSOS models
• A multicritical Landau-Potts field theory
• Q-colourings of the triangular lattice: Exact exponents and conformal field theory
• $N$-cluster correlations in four- and five-dimensional percolation
• Observation of non-scalar and logarithmic correlations in 2D and 3D percolation
• Bridges in the random-cluster model
• Crossover exponents, fractal dimensions and logarithms in Landau-Potts field theories
• Two-point boundary correlation functions of dense loop models
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• Logarithmic Correlations in Quantum Hall Plateau Transitions
• Universality of closed nested paths in two-dimensional percolation
• Logarithmic operators in $c=0$ bulk CFTs
• Faithfulness of Real-Space Renormalization Group Maps