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Two-point functions in boundary loop models

Max Downing, Jesper Lykke Jacobsen, Rongvoram Nivesvivat, Hubert Saleur

Abstract

Using techniques of conformal bootstrap, we propose analytical expressions for a large class of two-point functions of bulk fields in critical loop models defined on the upper-half plane. Our results include the two-point connectivities in the Fortuin--Kasteleyn random cluster model with both free and wired boundary conditions. We link the continuum expressions to lattice quantities by computing universal ratios of amplitudes for the two-point connectivities, and find excellent agreement with transfer-matrix numerics.

Two-point functions in boundary loop models

Abstract

Using techniques of conformal bootstrap, we propose analytical expressions for a large class of two-point functions of bulk fields in critical loop models defined on the upper-half plane. Our results include the two-point connectivities in the Fortuin--Kasteleyn random cluster model with both free and wired boundary conditions. We link the continuum expressions to lattice quantities by computing universal ratios of amplitudes for the two-point connectivities, and find excellent agreement with transfer-matrix numerics.
Paper Structure (8 sections, 16 equations, 2 figures)

This paper contains 8 sections, 16 equations, 2 figures.

Table of Contents

  1. Motivation and goals.
  2. Critical loop models.
  3. Conformal bootstrap.
  4. Two-point connectivity $\langle V_{\left (0, 1/2 \right )} V_{\left (0, 1/2 \right )} \rangle$.
  5. Universal ratios.
  6. Transfer matrix computation.
  7. Conclusion and outlook.
  8. Acknowledgements

Figures (2)

  • Figure 1: With free BCs, two points are connected only if there is a cluster stretching between them: the probability of this decays to zero at large distance. With wired BCs instead, two points even far apart retain a finite probability of being in the same cluster, since they can do so by both being connected to the wired boundary.
  • Figure 2: Ratio $\lambda / \mu$ for the $Q$-state FK model. The BCs are free (left panel) and wired (right panel). The transfer-matrix results (blue curves using $L=5,7,9$; red curves using $L=5,7,9,11$) converge well to the bootstrap results (in green).