Exact scaling of geometric phase and fidelity susceptibility and their breakdown across the critical points

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.