  The Loschmidt echo (LE) is a purely quantum-mechanical quantity whose
determination for large quantum many-body systems requires an exceptionally
precise knowledge of all eigenstates and eigenenergies. One might therefore be
tempted to dismiss the applicability of any approximations to the underlying
time evolution as hopeless. However, using the fully connected transverse-field
Ising model (FC-TFIM) as an example, we show that this indeed is not the case,
and that a simple semiclassical approximation to systems well described by
mean-field theory (MFT) is in fact in good quantitative agreement with the
exact quantum-mechanical calculation. Beyond the potential to capture the
entire dynamical phase diagram of these models, the method presented here also
allows for an intuitive geometric interpretation of the fidelity return rate at
any temperature, thereby connecting the order parameter dynamics and the
Loschmidt echo in a common framework. Videos of the post-quench dynamics
provided in the supplemental material visualize this new point of view.
