In weak coupling regime plotted in Fig. (g) and 3(h), the difference in correlation functions calculated via different methods begins to disappear, and VPT performs better than LF. Considering the approximations adopted in polaron transformation approaches , these differences imply the limitation of LF/VPT in intermediate coupling region and the necessity of more accurate numerical methods such as TD-DMRG. The dependence of transport coefficients on the doping ratio 𝑐𝑐 is plotted in Fig. . As shown in Fig. (a), the conductivity increases with increasing 𝑐𝑐 first, and then decreases. The maximum value of conductivity is reached when 𝑐𝑐 = 0.5 , corresponding to the half-filling case where most charge carriers are involved in charge transport. In the hopping limit, the conductivity satisfies 𝜎𝜎 ∝ 𝜏𝜏 2 𝑐𝑐(1 -𝑐𝑐) when 𝜏𝜏 ≪ 𝜆𝜆 (Supporting Information 5 and 6), which is plotted in Fig.