increase of the coupling strength. We will discuss this issue based on parity symmetry breaking and total excitation numbers in the next subsection.]{}

Parity symmetry breaking
------------------------

![ (Color online) Parity (upper panel) and total excitations (lower panel) in the ground-state by the three-excitation NED and the present VMPS for $s=0.3$ and $0.7$. $\protect\lambda =1$, $\Delta =0.1$, $\protect\omega %
_{c}=1$, $\protect\epsilon =0$, $\Lambda=2$, $L=50$, $d_{opt}=12$, and $%
D=20,40$ for $s=0.3,0.7$ respectively. []{data-label="parity"}](Pi_N_MPS_3Ex.eps)

As stated before, the spin-boson model with (without) the RWA possesses a $%
U(1)$ ($Z_{2}$) symmetry. $U(1)$ is a higher symmetry than $Z_{2}$, so in the RWA spin-boson model, the system also has $Z_{2}$ symmetry, i.e., parity symmetry, like in the full spin-boson model. In this section, we study the behavior of the expectation value of the parity $\hat{\Pi}=\exp
\left( i\pi \hat{N}\right) $.

The finite order parameter above the critical points obtained by VMPS in the previous subsection displays spontaneous symmetry breaking, while parity is generally just the criterion to determine whether symmetry is broken. The upper panel in Fig. \[parity\] gives the expectation value of parity $%
\left\langle \hat{\Pi}\right\rangle \ $obtained by both methods for $%
s=0.3,0.7$. Before the critical points $\alpha_c$, both methods yield the same 