Critical loop models are exactly solvable

TL;DR

This paper uses numerical conformal bootstrap to derive exact formulas for 4-point structure constants in 2D critical loop models, revealing a universal structure involving special functions and polynomial factors.

Contribution

It provides an analytic expression for 235 structure constants in critical loop models, combining conformal bootstrap with conjectured universal formulas.

Findings

Exact formulas for 235 structure constants involving Barnes' double Gamma function.

Universal polynomial dependence on loop weights with bounded degree.

Lattice computations confirm ratios match conformal bootstrap predictions.

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.