g_{j}\right)  \notag \\
&&+B_{n}G_{i,n}\left[ 2\left( \beta _{i,n}+\omega _{j}-E\right) h_{n,j}-g_{j}%
\right] \}=0.  \label{hh}\end{aligned}$$

In practice, these parameters can be obtained by solving the coupled equations self-consistently, which in turn give the ground-state energy and wave function. The ground state with zero excitation $\left\vert
0\right\rangle \left\vert \downarrow \right\rangle $ to Hamiltonian ([Hamiltonian2]{}) can be contained in the MCS wave function to Hamiltonian ([Hamiltonian3]{}) by setting the constrained coefficients: $A_{n}=B_{n}$ and$\
f_{n,k}=h_{n,k}=0$ for all $n$. However the state with nonzero total excitations cannot be included in the  MCS ansatz due to the finite number of coherent states. It has  been demonstrated that this wave function can describe the localized phase of the spin-boson model [@Blunden]. The number of the coherent states in the practical calculations in this paper is $N_{c}=6$, which is sufficient to judge the existence of the second-order QPT.

For the three approaches described above, discretization of the energy spectrum of the continuum bath should be performed at the very beginning in the practical calculations. The same logarithmic discretization is taken for different approaches if comparison is made in the data presented below.

Results and discussions
=======================

We first describe our main results by VMPS method to the sub-Ohmic $%
\left(0<s<1\right 