Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra

TL;DR

This work reexamines the connectivity four-point functions of the 2D Q-state Potts model in the FK representation, combining lattice algebras (FK join-detach and TL/JTL) with cylinder transfer-matrix techniques to extract the s-channel operator spectra and their amplitudes. The authors demonstrate a substantially richer spectrum than proposed in Ribault, including infinite families with h_{r,s} where r is dense on the real axis, and show that Ribault’s bootstrap misses many relevant fields for these geometrical observables. They validate their generating-function framework against exact results and perform two complementary numerical methods to map lattice data to continuum CFT data, including careful finite-size extrapolations. The findings challenge the applicability of the Ribault bootstrap to Potts geometrical correlators and suggest that logarithmic or indecomposable CFT structures may play a role in the full description of critical Potts models.

Abstract

We revisit in this paper the problem of connectivity correlations in the Fortuin-Kasteleyn cluster representation of the two-dimensional $Q$-state Potts model conformal field theory. In a recent work [M. Picco, S. Ribault and R. Santachiara, SciPost Phys. 1, 009 (2016); arXiv:1607.07224], results for the four-point functions were obtained, based on the bootstrap approach, combined with simple conjectures for the spectra in the different fusion channels. In this paper, we test these conjectures using lattice algebraic considerations combined with extensive numerical studies of correlations on infinite cylinders. We find that the spectra in the scaling limit are much richer than those proposed in [arXiv:1607.07224]: they involve in particular fields with conformal weight $h_{r,s}$ where $r$ is dense on the real axis.

Figures (12)

• Figure 1: Four-point functions in the cylinder geometry.
• Figure 2: FK cluster configurations that contribute to the correlation functions $P_{abab}$ and $P_{abba}$.
• Figure 3: The ratio $A_{\Phi_{h_{1/4,-4},h_{1/4,4}}}/A_{\Phi_{h_{1/2,-2},h_{1/2,2}}}$ as a function of $Q$ for $L=5,6,7,8$ (blue, orange, green and red dots respectively). This ratio is generically non zero. It exhibits (in finite size) simple poles at $Q=4 \cos^2{\pi\over 8}$, $Q=4\cos^2{4\pi\over 8}$, and vanishes exactly (in finite size) for $Q=0,3,4$.
• Figure 4: Sub-module structure of $\mathcal{W}_{2,-1}$ for $\mathfrak{q}=e^{3i\pi/8}$. Note the appearance of sub-modules isomorphic to $\mathcal{W}_{4,\pm i}$, which lead to glueing of standard modules into bigger, indecomposable modules, and Jordan cells for the transfer matrix.
• Figure 5: A sample of the full spectrum in the s-channel for $P_{abab}-P_{abba}$ represented by the pairs $(r,s)$ of the $h_{r,s}$ exponents ($r$ is on the y-axis, and $s$ on the x-axis. The spectrum considered in Ribault, depicted as crosses, is seen to be a tiny subset of the full spectrum: the projections of the dots on the y-axis in fact should cover it densely (represented here are exponents for $M=2,4,6,8,10,12$ only).

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Example 1
• Example 2