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In this article, we study the high order term of the fidelity of theHeisenberg chain with next-nearest-neighbor interaction and analyze itsconnection with quantum phase transition of Beresinskii-Kosterlitz-Thoulesstype happened in the system. We calculate the fidelity susceptibility of thesystem and find that although the phase transition point can't be wellcharacterized by the fidelity susceptibility, it can be effectively picked outby the higher order of the ground-state fidelity for finite-size systems.

High-order fidelity and quantum phase transition for the Heisenberg chain with next-nearest-neighbor interaction