Holstein polaron transport from numerically "exact" real-time quantum dynamics simulations

Numerically “exact” methods addressing the dynamics of coupled electron–phonon systems have been intensively developed. Nevertheless, the corresponding results for the electron mobility ${\mu }_{\mathrm{d}\mathrm{c}}$ are scarce, even for the one-dimensional (1d) Holstein model. Building on our recent progress on single-particle properties, here, we develop the momentum-space hierarchical equations of motion (HEOM) method to evaluate real-time two-particle correlation functions of the 1d Holstein model at finite temperature. We compute numerically “exact” dynamics of the current–current correlation function up to real times sufficiently long to capture the electron’s diffusive motion and provide reliable results for ${\mu }_{\mathrm{d}\mathrm{c}}$ in a wide range of model parameters. In contrast to the smooth ballistic-to-diffusive crossover in the weak-coupling regime, we observe a temporally limited slow-down of the electron on intermediate time scales already in the intermediate-coupling regime, which translates to a finite-frequency peak in the optical response. Our momentum-space formulation lowers the numerical effort with respect to existing HEOM-method implementations, while we remove the numerical instabilities inherent to the undamped-mode HEOM by devising an appropriate hierarchy closing scheme. Still, our HEOM remains unstable at too low temperatures, for too strong electron–phonon coupling, and for too fast phonons.