MCS approach is used here to account for the existence of the second-order QPT qualitatively, not for the precise location of the critical points.

Before the critical points, the results by all three methods are the same. Nonzero order parameter $|\left\langle \sigma _{x}\right\rangle |$ by MCS approach appears above the critical points, providing further convincing evidence of the spontaneous symmetry breaking. The MCS ground-state energies are lower than those by $N$ED method after the critical point, indicating that the state with broken symmetry is more stable. The total excitation number by MCS method increases suddenly above the critical point, because of no limitation of total excitation number in the coherent state. All these findings in the MCS variational study provide strong evidence of the second-order QPT in the spin-boson model with RWA. As found recently by Blunden-Codd et al. [@Blunden] that a very accurate wave function can be only obtained by at least $100$ coherent states, by $N_{c}=6$ coherent states the MCS results for the order parameter and energy still slightly deviate from those by VMPS above the critical points.

The critical exponent for magnetization
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![ (Color online) The log-log plot of the magnetization $\left\vert
\left\langle\protect\sigma_{x}\right\rangle\right\vert$ as a function 