)$ spin-boson model with the RWA described by Eq. ([Hamiltonian2]{}) for $\lambda =1$. We observe a second-order QPT in the spin-boson model under RWA, which was unnoticed in literature, to the best of our knowledge. This surprising observation is further confirmed by the MCS variational approach. The symmetry breaking was unambiguously found above the critical point in this wave function based approach. The $N$ED results are also given, which should be exact at weak coupling and can be regarded as a benchmark in this regime. Besides the second-order QPT, we also find a few first-order QPTs before the critical point of the second-order QPT for large bath exponent $s$. As $s$ decreases, the first-order QPTs disappear successively, but the second-order phase transition remains robust no matter how small $s$ is. We will discuss those phenomena based on various numerical calculations in the following subsections.

Magnetization $\left\vert \left\langle \protect\sigma %
_{x}\right\rangle \right\vert $
-------------------------------------------------------

![ (Color online) Magnetization $\left\vert \left\langle \protect%
\sigma _{x}\right\rangle \right\vert $ as a function of $\protect\alpha $ in the ground state for (a) $s=0.3$ and (b) $0.7$. Black lines with circles denote the VMPS results and the red dashed lines denote the $N=3$-ED ones. $\protect%
\lambda =1$, 