Entropic $F$-function of 3D Ising conformal field theory via the fuzzy sphere regularization

TL;DR

This paper computes the non-perturbative entropic $F$-function of the 3D Ising conformal field theory using fuzzy sphere regularization, confirming the $F$-theorem and aligning with previous estimates.

Contribution

First non-perturbative calculation of the 3D Ising $F$-function via fuzzy sphere regularization and entanglement entropy.

Findings

Estimated $F_{Ising} \\approx 0.0612$, slightly below free scalar value.

Results are consistent with the $F$-theorem and $4-\\epsilon$ expansion.

Demonstrates the effectiveness of fuzzy sphere regularization for non-local quantities.

Abstract

The $F$-function, the three-dimensional counterpart of the central charge in the 2D conformal field theory, measures the effective number of degrees of freedom in 3D quantum field theory, and it is monotonically decreasing under the renormalization group flow. However, unlike the 2D central charge, the $F$-function is a non-local quantity and cannot be computed using correlators of local operators. Utilizing the recently proposed fuzzy sphere regularization, we have performed the first non-perturbative computation of the $F$-function for the paradigmatic 3D Ising conformal field theory through entanglement entropy. Our estimate yields $F_{\text{Ising}} \approx 0.0612(5)$, slightly smaller than the $F$-function of a real free scalar, $F_{\text{free}} = \frac{\log 2}{8} - \frac{3\zeta(3)}{16\pi^2} \approx 0.0638$, consistent with the $F$-theorem, and close to the $4-\epsilon$ expansion estimates of $F_{\text{Ising}} \approx 0.0610 \sim 0.0623$.