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We address the out-of-equilibrium dynamics of a many-body system when one ofits Hamiltonian parameters is driven across a first-order quantum transition(FOQT). In particular, we consider systems subject to fixed boundaryconditions, favoring one of the two phases separated by the FOQT: moreprecisely, boundary conditions that favor the same magnetized phase (EFBC) oropposite phases (OFBC) at the two ends of the chain. These issues areinvestigated within the paradigmatic one-dimensional quantum Ising model, inwhich FOQTs are driven by the longitudinal magnetic field h. We study thedynamic behavior for an instantaneous quench and for a protocol in which h isslowly varied across the FOQT. We develop a dynamic finite-size scaling theoryfor both EFBC and OFBC, which displays some remarkable differences with respectto the case of neutral boundary conditions. The corresponding relevant timescale shows a qualitative different size dependence in the two cases: itincreases exponentially with the size in the case of EFBC, and as a power ofthe size in the case of OFBC.

Scaling properties of the dynamics at first-order quantum transitions when boundary conditions favor one of the two phases