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.

Figure 1

(Color online) RFS as a function of $h$ at $\gamma =1/2$ and $\tau =1/2$. The peaks approach the critical point and become sharper and sharper with the increase in $N$.

Figure 2

(Color online) A comparison between (a) $\eta$ and (b) $\mathcal{E}$ as a function of $h$ at $\gamma =1/2$, $\tau =1/2$ for various system sizes. At the critical point, $\eta$ tends to 1 while $\mathcal{E}$ is divergent.

Figure 3

(Color online) $\eta$ as a function of $\tau$ with $\gamma =1/2$, at (a) $h=0.6$, (b) 0.9, (c) 1.0, and (d) 1.1. We see that $\eta$ is independent of $N$ when $h$ is away from the critical region.