Many-body reduced fidelity susceptibility in Lipkin-Meshkov-Glick model

We study the reduced fidelity susceptibility chi(r) for an M-body subsystem of an N-body Lipkin-Meshkov-Glick model with tau=M/N fixed. The reduced fidelity susceptibility can be viewed as the response of subsystem to a certain parameter. In noncritical region, the inner correlation of the system is weak, and chi(r) behaves similar with the global fidelity susceptibility chi(g), the ratio eta=chi(r)/chi(g) depends on tau but not on N. However, at the critical point, the inner correlation tends to be divergent, and we find chi(r) approaches chi(g) with increasing the N. It is interesting to note that, eta=1 in the thermodynamic limit, which means the susceptibilities of the local and global system are the same. Finally, we make numerical computations, and they are in perfect agreement with the analytical predictions.