  Recently universal dynamic scaling is observed in several systems, which
exhibit a spatiotemporal self-similar scaling behavior, analogous to the
spatial scaling near phase transitions. The latter arises from the emergent
continuous scaling symmetry due to the divergent correlation length. Motivated
by this, we investigate the relation between the scaling dynamics and
continuous scaling symmetry. We derive a theorem that the scaling invariance of
the quenched Hamiltonian and the initial density matrix can lead to the
universal dynamic scaling in quench dynamics. It is demonstrated both in the
two-body problem analytically and in the many-body problem numerically. For the
latter one, we calculate the dynamics of quantum gases quenched from
noninteracting to finite interaction in the framework of non-equilibrium
high-temperature virial expansion. A dynamic scaling of the momentum
distribution appears in certain momentum-time windows at unitarity as well as
in the weak interacting limit. Remarkably, this universal scaling dynamics
persists approximately with smaller scaling exponents even if the scaling
symmetry is fairly broken. Our findings may offer a new perspective to
interpret the related experiments. We also calculate the Contact dynamics in
the BEC-BCS crossover. Surprisingly, the half-way time displays a maximum near
unitarity while some damping oscillations occur on the BEC side due to the
dimer state, which can be used to detect possible two-body bound states in
experiments.
