semi-infinite chain with nearest-neighbor interaction  [@Friend]. Thus, Hamiltonian ([Hamiltonian2]{}) can be written as: $$\begin{aligned}
H_{\text{chain}} &=&\frac{\Delta }{2}\sigma _{z}+\frac{\epsilon }{2}\sigma
_{x}+\frac{c_{0}}{2}(b_{0}+b_{0}^{\dag })\sigma _{x}+\lambda \frac{c_{0}}{2}%
(b_{0}-b_{0}^{\dag })i\sigma _{y}  \notag \\
&&+\sum_{n=0}^{L-2}[\epsilon _{n}b_{n}^{\dag }b_{n}+t_{n}(b_{n}^{\dag
}b_{n+1}+b_{n+1}^{\dag }b_{n})],  \label{Hamitrans}\end{aligned}$$where $b_{n}^{\dag }$($b_{n}$) is the creation (annihilation) operator for a new set of boson modes in a transformed representation with $\epsilon _{n}$ describing frequency on chain site $n$, $t_{n}$ describing the nearest-neighbor hopping parameter, and $c_{0}$ describing the effective coupling strength between the spin and the new effective bath. All of the parameters mentioned above, such as $t_{n},\epsilon _{n}$ and $c_{0}$, are determined by the logarithmic discretization parameter $\Lambda $, the cutoff frequency $\omega _{c}$, and a specific form of the spectral density, which are expressed below[$$\begin{aligned}
c_{0} &=&\sqrt{\int_{0}^{\omega _{c}}\frac{J\left( \omega \right)}{\pi}
d\omega} , \\
\epsilon _{n} &=&\xi _{s}\left( A_{n}+C_{n}\right) , \\
t_{n} &=&-\xi _{s}\left( \frac{N_{n+1}}{N_{n}}\right) A_{n},\end{aligned}$$where $$\begin{aligned}
\xi _{s} &=&\frac{s+1}{s+2}\frac{1-\Lambda ^{-\left( s+2\right) }}{1-\Lambda
^{-\left( s+1\right) }}\omega _{c}, \\
A_{n} &=&\Lambda ^{-j}\frac{\left( 1-\Lambda ^{-\left( j+1+s\right) }\right)
^{2}}{\left( 1-\Lambda ^{-\left( 2j+1+s\right) }\right) \left( 1-\Lambda
^{-\left( 2j+2+s\right) }\right) }, \\
C_{n} &=&\Lambda ^{-\left( j+s\right) }\frac{\left( 1-\Lambda ^{-j}\right)
^{2}}{\left( 1-\Lambda ^{-\left( 2j+s\right) }\right) \left( 1-\Lambda
^{-\left( 2j+1+s\right) }\right) }, \\
N_{n}^{2} &=&\frac{\Lambda ^{-n\left( 1+s\right) }\left( \Lambda
^{-1}:\Lambda ^{-1}\right) _{n}^{2}}{\left( \Lambda 