  In this article, we study the high order term of the fidelity of the
Heisenberg chain with next-nearest-neighbor interaction and analyze its
connection with quantum phase transition of Beresinskii-Kosterlitz-Thouless
type happened in the system. We calculate the fidelity susceptibility of the
system and find that although the phase transition point can't be well
characterized by the fidelity susceptibility, it can be effectively picked out
by the higher order of the ground-state fidelity for finite-size systems.
