  Adiabatic and periodic variation of the lattice parameters can make it
possible to transport charge through a system even without net external
electric or magnetic fields, known as Thouless charge pumping. The amount of
charge pumped in a cycle is quantized and entirely determined by the system's
topology, which is robust against perturbations such as disorder and
interactions. However, coupling to the environment may play a vital role in
topological transport in many-body systems. We study the topological Thouless
pumping, where the charge carriers interact with local optical phonons. The
semi-classical multi-trajectory Ehrenfest method is employed to treat the
phonon trajectories classically and charge carriers quantum mechanically. We
find a breakdown of the quantized charge transport in the presence of phonons.
It happens for any finite electron-phonon coupling strength at the resonance
condition when the pumping frequency matches the phonon frequency, and it takes
finite phonon coupling strength away from the resonance. Moreover, there exist
parameter regimes with non-quantized negative and positive charge transport.
The modified effective pumping path due to electron-phonon coupling accurately
explains the underlying physics. In the large coupling regime where the pumping
disappears, the phonons are found to eliminate the staggering of the onsite
potentials, which is necessary for the pumping protocol. Finally, we present a
stability diagram of quantized pumping as a function of time period of pumping
and phonon coupling strength.
