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

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 $F_{\text{Ising}} \approx 0.0610 \sim 0.0623$.