  Topological phases of matter can support fractionalized quasi-particles
localized at topological defects. The current understanding of these exotic
excitations, based on the celebrated bulk-defect correspondence, typically
relies on crude approximations where such defects are replaced by a static
classical background coupled to the matter sector. In this work, we explore the
strongly-correlated nature of symmetry-protected topological defects by
focusing on situations where such defects arise spontaneously as dynamical
solitons in intertwined topological phases, where symmetry breaking coexists
with topological symmetry protection. In particular, we focus on the
$\mathbb{Z}_2$ Bose-Hubbard model, a one-dimensional chain of interacting
bosons coupled to $\mathbb{Z}_2$ fields, and show how solitons with
$\mathbb{Z}_n$ topological charges appear for particle/hole dopings about
certain commensurate fillings, extending the results of [1] beyond half
filling. We show that these defects host fractionalized bosonic
quasi-particles, forming bound states that travel through the system unless
externally pinned, and repel each other giving rise to a fractional soliton
lattice for sufficiently high densities. Moreover, we uncover the topological
origin of these fractional bound excitations through a pumping mechanism, where
the quantization of the inter-soliton transport allows us to establish a
generalized bulk-defect correspondence. This in-depth analysis of dynamical
topological defects bound to fractionalized quasi-particles, together with the
possibility of implementing our model in cold-atomic experiments, paves the way
for further exploration of exotic topological phenomena in strongly-correlated
systems.
