  Long-range interacting spin systems are ubiquitous in physics and exhibit a
variety of ground state disorder-to-order phase transitions. We consider a
prototype of infinite-range interacting models known as the
Lipkin-Meshkov-Glick (LMG) model describing the collective interaction of $N$
spins, and investigate the dynamical properties of fluctuations and
correlations after a sudden quench of the Hamiltonian. Specifically, we focus
on critical quenches, where the initial state and/or the quench Hamiltonian are
critical. Depending on the type of quench, we identify three distinct behaviors
where both the short-time dynamics and the stationary state at long times are
effectively thermal, quantum, and genuinely non-equilibrium, characterized by
distinct universality classes and static and dynamical critical exponents.
These behaviors can be identified by an infrared effective temperature that is
finite, zero, and infinite (the latter scaling with the system size as
$N^{1/3}$), respectively. The quench dynamics is studied through a combination
of exact numerics and analytical calculations utilizing the non-equilibrium
Keldysh field theory. Our results are amenable to realization in experiments
with trapped-ion experiments where long-range interactions naturally arise.
