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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.
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