  It was shown via numerical simulations that geometric phase (GP) and fidelity
susceptibility (FS) in some quantum models exhibit universal scaling laws
across phase transition points. Here we propose a singular function expansion
method to determine their exact form across the critical points as well as
their corresponding constants. For the models such as anisotropic XY model
where the energy gap is closed and reopened at the special points ($k_0 = 0,
\pi$), scaling laws can be found as a function of system length $N$ and
parameter deviation $\lambda - \lambda_c$ (where $\lambda_c$ is the critical
parameter). Intimate relations for the coefficients in GP and FS have also been
determined. However in the extended models where the gap is not closed and
reopened at these special points, the scaling as a function of system length
$N$ breaks down. We also show that the second order derivative of GP also
exhibits some intriguing scaling laws across the critical points. These exact
results can greatly enrich our understanding of GP and FS in the
characterization of quantum phase transitions.
