  In a recent paper [Phys. Rev. Lett. 129, 120601] we have shown that the
dynamics of interfaces, in the symmetry-broken phase of the two-dimensional
ferromagnetic quantum Ising model, displays a robust form of ergodicity
breaking. In this paper, we elaborate more on the issue. First, we discuss two
classes of initial states on the square lattice, the dynamics of which is
driven by complementary terms in the effective Hamiltonian and may be solved
exactly: (a) strips of consecutive neighbouring spins aligned in the opposite
direction of the surrounding spins, and (b) a large class of initial states,
characterized by the presence of a well-defined "smooth" interface separating
two infinitely extended regions with oppositely aligned spins. The evolution of
the latter states can be mapped onto that of an effective one-dimensional
fermionic chain, which is integrable in the infinite-coupling limit. In this
case, deep connections with noteworthy results in mathematics emerge, as well
as with similar problems in classical statistical physics. We present a
detailed analysis of the evolution of these interfaces both on the lattice and
in a suitable continuum limit, including the interface fluctuations and the
dynamics of entanglement entropy. Second, we provide analytical and numerical
evidence supporting the conclusion that the observed non-ergodicity -- arising
from Stark localization of the effective fermionic excitations -- persists away
from the infinite-Ising-coupling limit, and we highlight the presence of a
timescale $T\sim e^{c L\ln L}$ for the decay of a region of large linear size
$L$. The implications of our work for the classic problem of the decay of a
false vacuum are also discussed.
