On four-point connectivities in the critical 2d Potts model

TL;DR

This paper uses Monte Carlo simulations to study four-point connectivities in the critical 2D Potts model for non-integer Q, comparing results with conformal field theory predictions and exploring analytic continuations.

Contribution

It provides the first detailed comparison between Monte Carlo results and CFT predictions for non-integer Q in the 2D Potts model, suggesting exact agreement only at specific Q values.

Findings

Three connectivity combinations match CFT results to 2-4 digits.

Agreement with CFT is exact at Q=0, 3, 4.

Proposes analytic continuation of the Potts model in the central charge.

Abstract

We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states $Q\in (0,4)$ that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the $2$ to $4$ significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases $Q=0, 3, 4$. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane $\{\Re c <13\}$, where $c$ is the central charge of the Virasoro symmetry algebra.