  In a scenario of spontaneous symmetry breaking in finite time, topological
defects are generated at a density that scale with the driving time according
to the Kibble-Zurek mechanism (KZM). Signatures of universality beyond the KZM
have recently been unveiled: The number distribution of topological defects has
been shown to follow a binomial distribution, in which all cumulants inherit
the universal power-law scaling with the quench rate, with cumulant rations
being constant. In this work, we analyze the role of boundary conditions in the
statistics of topological defects. In particular, we consider a lattice system
with nearest-neighbor interactions subject to soft anti-periodic, open, and
periodic boundary conditions implemented by an energy penalty term. We show
that for fast and moderate quenches, the cumulants of the kink number
distribution present a universal scaling with the quench rate that is
independent of the boundary conditions except by an additive term, that becomes
prominent in the limit of slow quenches, leading to the breaking of power-law
behavior. We test our theoretical predictions with a one-dimensional scalar
theory on a lattice.
