the RWA [@Buzek; @Zhangyy], the system usually undergoes the first-order phase transition, i.e., sequence of instabilities, among the phases within different conserved excitation numbers as the coupling strength increases. In the limit $\Delta /\omega
\rightarrow \infty $, the second-order quantum phase transitions have been observed in the quantum Rabi model with the RWA [@JC_Plenio]. It has been also reported that the Dicke model under the RWA displays a second-order QPT in the thermodynamic limit [@AnistropicDicke]. Contrary to the quantum Rabi (Dicke) model in the RWA, the spin-boson model under the RWA could undergo the second-order QPT even for a finite value of $\Delta /\omega $ ( one qubit).

[ ]{}

Magnetization $\left\vert \left\langle \protect\sigma %
_{z}\right\rangle \right\vert $ and the ground-state energy
-----------------------------------------------------------

![ (Color online) Magnetization $\left\vert \left\langle\protect\sigma%
_{z}\right\rangle\right\vert$ and the ground-state energy difference (lower panel) by the VMPS and $N=3$ED for $s=0.3$ (left panel) and $0.7$ (right panel). $\protect\lambda =1$, $\Delta =0.1$, $\protect\omega _{c}=1$, $%
\protect\epsilon =0$, $\Lambda=2$, $L=50$, $d_{opt}=12$, and $D=20,40$ for $%
s=0.3,0.7 $ respectively. The inset in the right-upper panel shows the enlarged view of the same plot where the kink can be clearly visible.[]{data-label="GS_quantity"}](Sz_disE_MPS_Ex.eps)

[ In the original spin-boson Hamiltonian, $\sigma _{z}$ describes a 