Exploring Replica-Potts CFTs in Two Dimensions

TL;DR

This paper conducts a non-perturbative numerical conformal bootstrap analysis of two-dimensional CFTs with specific permutation symmetries, focusing on coupled Potts models, and confirms previous theoretical predictions.

Contribution

It provides the first independent non-perturbative bootstrap analysis of coupled 3-state Potts models with permutation symmetry, validating earlier results and establishing groundwork for future precision studies.

Findings

Confirmed the allowed parameter space for scaling dimensions matches previous predictions.

Derived bounds on scaling dimensions consistent with earlier results.

Supported the non-integrable nature of the studied CFTs through bootstrap constraints.

Abstract

We initiate a numerical conformal bootstrap study of CFTs with $S_n \ltimes (S_Q)^n$ global symmetry. These include CFTs that can be obtained as coupled replicas of two-dimensional critical Potts models. Particular attention is paid to the special case $S_3 \ltimes (S_3)^3$, which governs the critical behaviour of three coupled critical 3-state Potts models, a multi-scalar realisation of a (potentially) non-integrable CFT in two dimensions. The model has been studied in earlier works using perturbation theory, transfer matrices, and Monte Carlo simulations. This work represents an independent non-perturbative analysis. Our results are in agreement with previous determinations: we obtain an allowed peninsula within parameter space for the scaling dimensions of the three lowest-lying operators in the theory, which contains the earlier predictions for these scaling dimensions. Additionally, we derive numerous bounds on admissible scaling dimensions in the theory, which are compatible with earlier results. Our work sets the necessary groundwork for a future precision study of these theories in the conformal bootstrap.