  In the present work, we investigate the intrinsic relation between quantum
fidelity susceptibility (QFS) and the dynamical structure factor. We give a
concise proof of the QFS beyond the perturbation theory. With the QFS in the
Lehmann representation, we point out that the QFS actually the
negative-two-power moment of dynamical structure factor, and illuminate the
inherent relation between physical quantities in the linear response theory.
Moreover, we discuss the generalized fidelity susceptibility (GFS) of any
quantum relevant operator, that may not be coupled to the driving parameter,
present similar scaling behaviors. Finally, we demonstrate that the QFS can not
capture the fourth-order quantum phase transition in a spin-1/2 anisotropic XY
chain in the transverse alternating field, while a lower-order GFS can seize
the criticalities.
