the rich behavior of $\left\vert \langle \sigma _{z}\rangle \right\vert $ here. For any value of $s$, [in]{} the weak-coupling regime, the corresponding ground state is the spin-down state with photonic vacuum for $\left\langle \hat{N}\right\rangle =0$, i.e., $|\downarrow \rangle |0\rangle $, thus $\langle \sigma _{z}\rangle =-1$, as just demonstrated in the upper panel of [Fig. \[GS\_quantity\]]{} for $%
s=0.3,0.7$. Once $\left\langle \hat{N}\right\rangle \neq 0$ at the first phase transition no matter whether it is of the first or the second order, the spin state in the ground state consists of both spin-up and spin-down states, which is drastically different from the spin state only including the spin-down state at $\left\langle \hat{N}\right\rangle =0$, leading to the jump of $\left\vert \langle \sigma _{z}\rangle \right\vert $ and $%
\left\vert \langle \sigma _{z}\rangle \right\vert \neq 1$, as exhibited in Fig. 2. At the later phase transitions between two finite $\left\langle \hat{%
N}\right\rangle \neq 0$, the ground states do not change drastically because both have spin-up and spin-down states, therefore $\left\vert
\langle \sigma _{z}\rangle \right\vert $ will exhibit kinks, as shown in Fig. 2 for $s=0.7$. The drop of $\left\vert \langle \sigma _{z}\rangle \right\vert $ to zero is an artefact[ due to]{} the plot using magnetization, 