Entropic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> function of three-dimensional Ising conformal field theory via fuzzy sphere regularization
Liangdong Hu, Wei Zhu, Yin-Chen He
Abstract
The $F$ function, the three-dimensional counterpart of the central charge in the two-dimensional (2D) conformal field theory, measures the effective number of degrees of freedom in three-dimensional (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 nonlocal quantity and cannot be computed using correlators of local operators. Utilizing the recently proposed fuzzy sphere regularization, we have performed the first nonperturbative computation of the $F$ function for the paradigmatic 3D Ising conformal field theory through entanglement entropy. Our computation yields ${F}_{\text{Ising}}=0.0612(5)$, slightly smaller than the $F$ function of a real free scalar, ${F}_{\text{free}}=\frac{log2}{8}\ensuremath{-}\frac{3\ensuremath{\zeta}(3)}{16{\ensuremath{\pi}}^{2}}\ensuremath{\approx}0.0638$, consistent with the $F$ theorem, and close to the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion estimates of ${F}_{\text{Ising}}\ensuremath{\approx}0.061\ensuremath{\sim}0.062$.