Out-of-equilibrium scaling behavior arising during round-trip protocols across a quantum first-order transition

We investigate the nonequilibrium dynamics of quantum spin chains during a round-trip protocol that slowly drives the system across a quantum first-order transition. Out-of-equilibrium scaling behaviors \`a la Kibble-Zurek for the single-passage protocol across the first-order transition have been previously determined. Here, we show that such scaling relations persist when the driving protocol is inverted and the transition is approached again by a far-from-equilibrium state. This results in a quasi-universality of the scaling functions, which keep some dependence on the details of the protocol at the inversion time. We explicitly determine such quasi-universal scaling functions by employing an effective two-level description of the many-body system near the transition. We discuss the validity of this approximation and how this relates to the observed scaling regime. Although our results apply to generic systems, we focus on the prototypical example of a $1D$ transverse field Ising model in the ferromagnetic regime, which we drive across the first-order transitions through a time-dependent longitudinal field.