  We reveal universal dynamical scaling behavior across adiabatic quantum phase
transitions (QPTs) in networks ranging from traditional spatial systems (Ising
model) to fully connected ones (Dicke and Lipkin-Meshkov-Glick models). Our
findings, which lie beyond traditional critical exponent analysis and adiabatic
perturbation approximations, are applicable even where excitations have not yet
stabilized and hence provide a time-resolved understanding of QPTs encompassing
a wide range of adiabatic regimes. We show explicitly that even though two
systems may traditionally belong to the same universality class, they can have
very different adiabatic evolutions. This implies more stringent conditions
need to be imposed than at present, both for quantum simulations where one
system is used to simulate the other, and for adiabatic quantum computing
schemes.
