Fidelity susceptibility of the anisotropic $XY$ model: The exact solution

We derive several closed-form expressions for the fidelity susceptibility~(FS) of the anisotropic $XY$ model in the transverse field. The basic idea lies in a partial fraction expansion of the expression so that all the terms are related to a simple fraction or its derivative. The critical points of the model are reiterated by the FS, demonstrating its validity for characterizing the phase transitions. Moreover, the critical exponents $ u$ associated with the correlation length in both critical regions are successfully extracted by the standard finite-size scaling analysis.