Non-scalar operators for the Potts model in arbitrary dimension

Abstract

We investigate the operator content of the Q-state Potts model in arbitrary dimension, using the representation theory of the symmetric group. In particular we construct all possible tensors acting on N spins, corresponding to given symmetries under $S_Q$ and $S_N$, in terms of representations involving any Young diagram. These operators transform non-trivially under the group of spatial rotations, with a definite conformal spin. The two-point correlation functions are then computed, and their physical interpretation is given in terms of Fortuin-Kasteleyn clusters propagating between two neighbourhoods of each N spins. In two dimensions, we obtain analytically the critical exponent corresponding to each operator. In the simplest and physically most relevant cases, we confirm the values of the critical exponent and the conformal spin by numerical measurements, using both Monte Carlo simulations and transfer matrix diagonalisations. Our classification partially provides the structure of Jordan cells of the dilatation operator in arbitrary dimensions, which in turn gives rise to logarithmic correlation functions.

39 pages, 3 figures

References in corpus (4)

• Associative-algebraic approach to logarithmic conformal field theories
• Logarithmic observables in critical percolation
• Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0

Cited by in corpus (15)

• Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$
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• Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model
• 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
• Spaces of states of the two-dimensional O(n) and Potts models
• A multicritical Landau-Potts field theory
• Four-point geometrical correlation functions in the two-dimensional $Q$-state Potts model: connections with the RSOS models
• Numerical estimation of discharge probability in GEM-based detectors
• Observation of non-scalar and logarithmic correlations in 2D and 3D percolation
• $N$-cluster correlations in four- and five-dimensional percolation
• Topological effects and conformal invariance in long-range correlated random surfaces
• Universality of closed nested paths in two-dimensional percolation
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice
• Logarithmic operators in $c=0$ bulk CFTs