Critical loop models are exactly solvable

Abstract

In two-dimensional critical loop models, including the $O(n)$ and Potts models, the spectrum is exactly known, as are a few structure constants or ratios thereof. Using numerical conformal bootstrap methods, we study $235$ of the simplest 4-point structure constants. For each structure constant, we find an analytic expression as a product of two factors: 1) a universal function of conformal dimensions, built from Barnes' double Gamma function, and 2) a polynomial function of loop weights, whose degree obeys a simple upper bound. We conjecture that all structure constants are of this form. For a few 4-point functions, we build corresponding observables in a lattice loop model. From numerical lattice results, we extract amplitude ratios that depend neither on the lattice size nor on the lattice coupling. These ratios agree with the corresponding ratios of 4-point structure constants.

68 pages, v3: corrected Table (2.68), added comments on rank 3 Jordan cells

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• Correlation functions in loop models
• Correlation functions of twist operators applied to single self-avoiding loops
• Snowmass White Paper: The Analytic Conformal Bootstrap
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Cited by in corpus (5)

• Coupled minimal models revisited II: Constraints from permutation symmetry
• Fusion rules and structure constants of E-series minimal models
• Critical spin chains and loop models with $PSU(n)$ symmetry
• Logarithmic operators in $c=0$ bulk CFTs
• On the Wilson-Fisher fixed point in the limit of integer spacetime dimensions