Fidelity at Berezinskii-Kosterlitz-Thouless quantum phase transitions

We clarify the long-standing controversy concerning the behavior of the ground state fidelity in the vicinity of a quantum phase transition of the Berezinskii-Kosterlitz-Thouless type in one-dimensional systems. Contrary to the prediction based on the Gaussian approximation of the Luttinger liquid approach, it is shown that the fidelity susceptibility does not diverge at the transition, but has a cusp-like peak $\chi_c- \chi(\lambda)\sim \sqrt{|\lambda_c-\lambda|} $, where $\lambda$ is a parameter driving the transition, and $\chi_c$ is the peak value at the transition point $\lambda=\lambda_c$. Numerical claims of the logarithmic divergence of fidelity susceptibility with the system size (or temperature) are explained by logarithmic corrections due to marginal operators, which is supported by numerical calculations for large systems.