  The study of quantum phase transitions requires the preparation of a
many-body system near its ground state, a challenging task for many
experimental systems. The measurement of quench dynamics, on the other hand, is
now a routine practice in most cold atom platforms. Here we show that
quintessential ingredients of quantum phase transitions can be probed directly
with quench dynamics in integrable and nearly integrable systems. As a
paradigmatic example, we study global quench dynamics in a transverse-field
Ising model with either short-range or long-range interactions. When the model
is integrable, we discover a new dynamical critical point with a non-analytic
signature in the short-range correlators. The location of the dynamical
critical point matches that of the quantum critical point and can be identified
using a finite-time scaling method. We extend this scaling picture to systems
near integrability and demonstrate the continued existence of a dynamical
critical point detectable at prethermal time scales. Therefore, our method can
be used to approximately locate the quantum critical point. The scaling method
is also relevant to experiments with finite time and system size, and our
predictions are testable in near-term experiments with trapped ions and Rydberg
atoms.
