Four-point geometrical correlation functions in the two-dimensional $Q$-state Potts model: connections with the RSOS models

Abstract

The "bootstrap determination" of the geometrical correlation functions in the two-dimensional Potts model proposed in a paper [arXiv:1607.07224] was later shown in [arXiv:1809.02191] to be incorrect, the actual spectrum of the model being considerably more complex than initially conjectured. We provide in this paper a geometrical interpretation of the four-point functions built in [arXiv:1607.07224], and explain why the results obtained by these authors, albeit incorrect, appeared so close to those of their numerical simulations of the Potts model. Our strategy is based on a cluster expansion of correlation functions in RSOS minimal models, and a subsequent numerical and algebraic analysis of the corresponding $s$-channel spectrum, in full analogy with our early work on the Potts model [arXiv:1809.02191]. Remarkable properties of the lattice amplitudes are uncovered, which explain in particular the truncation of the spectrum of [arXiv:1809.02191] to the much simpler one of the RSOS models, and which will be used in a forthcoming paper to finally determine the geometric four-point functions of the Potts model itself.

arXiv comment47 pages, 12 figures

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• Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters
• The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$
• The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
• Critical loop models are exactly solvable
• Topological effects and conformal invariance in long-range correlated random surfaces
• Global symmetry and conformal bootstrap in the two-dimensional $Q$-state Potts model
• On the analytical continuation of lattice Liouville theory
• Logarithmic operators in $c=0$ bulk CFTs