  We study the fixed point structure of the Higgs-Yukawa model, with its scalar
being non-minimally coupled to the asymptotically safe gravity, using the
functional renormalization group. We have obtained the renormalization group
equations for the cosmological and Newton constants, the scalar mass-squared
and quartic coupling constant, and the Yukawa and non-minimal coupling
constants, taking into account all the scalar, fermion, and graviton loops. We
find that switching on the fermionic quantum fluctuations makes the non-minimal
coupling constant irrelevant around the Gaussian-matter fixed point with the
asymptotically safe gravity.
