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Critical loop models are exactly solvable

by Rongvoram Nivesvivat, Sylvain Ribault, Jesper Lykke Jacobsen

Submission summary

Authors (as registered SciPost users): Jesper Lykke Jacobsen · Rongvoram Nivesvivat
Submission information
Preprint Link:  (pdf)
Code repository:
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Date submitted: 2024-04-05 09:42
Submitted by: Nivesvivat, Rongvoram
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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.

Current status:
In refereeing

Reports on this Submission

Anonymous Report 1 on 2024-5-19 (Invited Report)


1-Study of a large set of 4-pt functions in critical loop models via the bootstrap approach
2- Numerical results clearly hint to some integrable structure of these models
3-The authors provide an impressive combinatorial labeling of the solutions, supporting again the integrable structure of these family of models.
4-Highly non-trivial lattice numerical checks


1-No particular weakness


The paper represents another important step in solving critical loop models. This research is currently the center of interest for various communities, including those focused on lattice integrable models, conformal field theory, and mathematical probability theory.
In this paper is the continuation of previous papers by the same authors. Instead of counting the possible bootstrap solution, here the authors focus on specific (but still a very large number )solutions providing analytic expression for the structure constants, verified by bootstrap analysis. In particular, they provide an impressive combinatorial way of labeling the results and a convenient basis of solutions where the structure constants display a clear analytical structure. Finally, they implement a transfer matrix formalism, based on the random cluster lattice model, to study diagonal and non-diagonal 4-point correlation functions. This not only allows them to provide an independent check of their results but also offers a physical interpretation of the bootstrap correlation functions. I strongly recommend this paper.


Publish (surpasses expectations and criteria for this Journal; among top 10%)

  • validity: top
  • significance: top
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: excellent

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