$\Delta =0.1$, $\protect\omega _{c}=1$, $\protect\epsilon =0$, $%
\Lambda =2$, $L=50$, $d_{opt}=12$, and $D=20,40$ for $s=0.3,0.7$, respectively. []{data-label="Magnetization"}](Sx_MPS_Ex.eps)

In terms of Hamiltonian (\[Hamiltonian2\]), magnetization $\left\vert
\left\langle \sigma _{x}\right\rangle \right\vert $ can be regarded as the order parameter in this model. In the second-order phase transition, due to the symmetry breaking, the order parameter changes from zero to nonzero at the critical points. In Fig. \[Magnetization\] we present the VMPS results for $\left\vert \left\langle \sigma _{x}\right\rangle \right\vert $ as a function of the coupling strength $\alpha $ for the spin-boson model in the RWA (i.e.$\ \lambda =1$) for two typical values of $s=0.3$ and $0.7$. Surprisingly, magnetization changes abruptly from zero to nonzero for both cases. The critical points of the second-order QPTs are $\alpha
_{c}=0.016,0.685$ for $s=0.3,0.7\ $ respectively.

However, the $N$ED within excitation numbers up to $N$ shows that the order parameter $\left\vert \left\langle \sigma _{x}\right\rangle
\right\vert $ remains zero for all coupling strengths. Note that the VMPS has provided convincing results for the full spin-boson model [guo2012critical]{}. $N$ED cannot yield consistent results with the VMPS ones, because the total excitation number may not conserve with $\alpha $ due to the unexpected symmetry breaking.

In the finite-size Dicke model with 