Finite-size scaling at first-order quantum transitions when boundary conditions favor one of the two phases

We investigate scaling phenomena at first-order quantum transitions, when the boundary conditions favor one of the two phases. We show that the corresponding finite-size scaling behavior, arising from the interplay between the driving parameter and the finite size of the system, is more complex than that emerging when boundary conditions do not favor any phase. We discuss this issue in the framework of the paradigmatic one-dimensional quantum Ising model, along its first-order quantum transition line driven by an external longitudinal field. Specifically, three regions with distinct scaling behaviors emerge, which correspond to different values of the field (small, intermediate, and large field), according to its capability to modify the phase favored by the boundary conditions.