the sub-Ohmic spin-boson model. In the present paper, we will extend it to study the RWA spin-boson model, mainly focusing on phase transitions in the sub-Ohmic baths, namely $\lambda =1$, $0<s<1$. For the data presented below, we typically choose the model parameters as $\Delta =0.1$, $\omega _{c}=1$, and $\epsilon =0$, if there are no special statements. Some parameters of orthogonal polynomials transformation and VMPS are the same as those in Ref. [guo2012critical]{}, e.g.  the logarithmic discretization parameter $\Lambda
=2$, the length of the semi-infinite chain $L=50$, and optimized truncation numbers $%
d_{opt}=12$. In addition, we adjust the bond dimension to achieve better convergence of the results. In this paper, we choose $D_{max}=20,40$ for $%
s=0.3,0.7$, respectively, which is sufficient to obtain the converged results, as demonstrated in Appendix A in detail.

*Exact diagonalization in truncated Hilbert space.-*. It is well known that the spin-boson model with the RWA possesses $U(1)$ symmetry, and the total excitation number $\hat{N}$ is conserved because $\left[ \hat{N},H%
\right] =0$. It has been reported in Ref. [@Tong] that the total excitation number $N$in the ground state of the RWA spin-boson model jumps from $%
0 $ to $1$ at a critical coupling strength. This instability can 