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The question of whether there exists a finite mobility in the standardHolstein model with one vibrational mode on each site remains unclear. In thisletter, we approach this problem by employing the hierarchical equation ofmotion (HEOM) method to simulate model systems where the vibrational modes aredissipative. It is found that, as the friction becomes smaller, the chargecarrier mobility increases significantly and a friction free limit can not beobtained. The current autocorrelation functions are also calculated for thefriction free Holstein model, and converged results can not be obtained withthe increase of the number of sites. Based on these observations, we concludethat a finite mobility can not be defined for the standard Holstein model,which is consistent with the recent finding by Kolss {\it et al.} (Phys. Rev.Lett. 123, 126601).

Is there a finite mobility for the one vibrational mode Holstein model? Implications from real time simulations