  Dynamical phase transitions extend the notion of criticality to
non-stationary settings and are characterized by sudden changes in the
macroscopic properties of time-evolving quantum systems. Investigations of
dynamical phase transitions combine aspects of symmetry, topology, and
non-equilibrium physics, however, progress has been hindered by the notorious
difficulties of predicting the time evolution of large, interacting quantum
systems. Here, we tackle this outstanding problem by determining the critical
times of interacting many-body systems after a quench using Loschmidt
cumulants. Specifically, we investigate dynamical topological phase transitions
in the interacting Kitaev chain and in the spin-1 Heisenberg chain. To this
end, we map out the thermodynamic lines of complex times, where the Loschmidt
amplitude vanishes, and identify the intersections with the imaginary axis,
which yield the real critical times after a quench. For the Kitaev chain, we
can accurately predict how the critical behavior is affected by strong
interactions, which gradually shift the time at which a dynamical phase
transition occurs. We also discuss the experimental perspectives of predicting
the first critical time of a quantum many-body system by measuring the energy
fluctuations in the initial state, and we describe the prospects of
implementing our method on a near-term quantum computer with a limited number
of qubits. Our work demonstrates that Loschmidt cumulants are a powerful tool
to unravel the far-from-equilibrium dynamics of strongly correlated many-body
systems, and our approach can immediately be applied in higher dimensions.
