results for the parity. It is interesting to find that in this regime, the value of parity for all values of $s<1$ is either $1$ or $-1$, which corresponds to even or odd parity respectively. It follows that the symmetry is not broken in this regime. Nevertheless, the average parity $\left\langle
\hat{\Pi}\right\rangle $ becomes zero due to quantum fluctuations above the critical points. It is not the eigenstate of the parity operator, indicating spontaneous parity symmetry breaking. Note also that the parity jumps for a few times before the critical coupling for $s=0.7$, and remains unchanged for $s=0.3$.

We also present the average value of the total excitation number $%
\left\langle \hat{N}\right\rangle $ in the ground-state as a function of the coupling strength$\ \alpha $ in the lower panel of Fig. \[parity\]. Below the critical points, both approaches give the same $\left\langle \hat{N}%
\right\rangle $, which jumps between different plateaus with different integers, just like in the JC model [@JC_Plenio]. For $s=0.3$, $%
\left\langle \hat{N}\right\rangle=0 $ remains until the second-order critical point $\alpha_c$, while, for $s=0.7$, $\left\langle \hat{N}%
\right\rangle $ increases from $0$ to $2$ one by one before $\alpha _{c}$. It follows that the lowest energies in different coupling 