  We address the out-of-equilibrium dynamics of many-body systems subject to
slow time-dependent round-trip protocols across quantum and classical (thermal)
phase transitions. We consider protocols where one relevant parameter w is
slowly changed across its critical point wc = 0, linearly in time with a large
time scale ts, from wi < 0 to wf > 0 and then back to wi < 0, thus entailing
multiple passages through the critical point. Analogously to the one-way
Kibble-Zurek protocols across a critical point, round-trip protocols develop
dynamic scaling behaviors at both classical and quantum transitions, put
forward within renormalization-group frameworks. The scaling scenario is
analyzed within some paradigmatic models undergoing quantum and classical
transitions belonging to the two-dimensional Ising universality class, such as
one-dimensional quantum Ising models and fermionic wires, and two-dimensional
classical Ising models (supplemented with a purely relaxational dynamics).
While the dynamic scaling frameworks are similar for classical and quantum
systems, substantial differences emerge due to the different nature of their
dynamics, which is purely relaxational for classical systems (implying
thermalization in the large-time limit at fixed model parameters), and unitary
in the case of quantum systems. In particular, when the critical point
separates two gapped (short-ranged) phases and the extreme value wf > 0 is kept
fixed in the large-ts limit of the round-trip protocol, we observe
hysteresis-like scenarios in classical systems, while quantum systems do not
apparently develop a sufficiently robust scaling limit along the return way,
due to the presence of rapidly oscillating relative phases among the relevant
quantum states.
