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It was shown via numerical simulations that geometric phase (GP) and fidelitysusceptibility (FS) in some quantum models exhibit universal scaling lawsacross phase transition points. Here we propose a singular function expansionmethod to determine their exact form across the critical points as well astheir corresponding constants. For the models such as anisotropic XY modelwhere 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 $N$ andparameter deviation $\lambda - \lambda_c$ (where $\lambda_c$ is the criticalparameter). Intimate relations for the coefficients in GP and FS have also beendetermined. However in the extended models where the gap is not closed andreopened 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 alsoexhibits some intriguing scaling laws across the critical points. These exactresults can greatly enrich our understanding of GP and FS in thecharacterization of quantum phase transitions.

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