transition is absent for small bath exponent $s$, such as for $s=0.3$.

Evidence for the second-order QPT by MCS variational studies
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![ (Color online) (a) The order parameter $\left\vert \left\langle%
\protect\sigma_{x}\right\rangle\right\vert$. (b) The total excitation number $%
\left\langle N \right\rangle$ as a function of the coupling strength within VMPS, $3$ED, and MCS variational approaches. (c) The difference between the VMPS (MCS) ground-state energy and that by $3$ED. $s = 0.3$, $\protect%
\lambda =1$, $\Delta =0.1$, $\protect\omega _{c}=1$, $\protect\epsilon =0$, $%
\Lambda=2$, $L=20$, $d_{opt}=12$, $D=20$, $N_c=6$.[]{data-label="VM"}](s_0d3_VMPS_VM_3Ex.eps)

Since this model at $s=0.3$ exhibits only one second-order QPT from the zero excitation ground state, we can employ the MCS variational approach to provide additional evidence, because the zero excitation can be realized and the localized phase can be also described in the trial wave function ([VM\_wave]{}). In Fig. \[VM\], we list results by MCS, VMPS, and $N=3$ED approaches for $s=0.3$. The logarithmic discretization parameter $\Lambda =2$ and $L=20$ bosonic modes are taken for all three approaches here. Note that the number of bosonic modes here is smaller than those in other figures due to the computational difficulties in the MCS approach, but it does not influence the essential results at all. The 