  We study theoretically the transport of the one-dimensional single-channel
interacting electron gas through a strong potential barrier in the parameter
regime where the spin sector of the low-energy Luttinger liquid theory is
gapped by interaction. This phase is of particular interest since it exhibits
non-trivial interaction-induced topological properties. Using bosonization and
an expansion in the tunneling strength, we calculate the conductance through
the barrier as a function of the temperature as well as the local density of
states (LDOS) at the barrier. Our main result concerns the mechanism of
bound-state mediated tunneling. The characteristic feature of the topological
phase is the emergence of protected zero-energy bound states with fractional
spin located at the impurity position. By flipping the fractional spin the edge
states can absorb or emit spinons and thus enable single electron tunneling
across the impurity even though the bulk spectrum for these excitations is
gapped. This results in a finite LDOS below the bulk gap and in a non-monotonic
behavior of the conductance. The system represents an important physical
example of an interacting symmetry-protected topological phase---which combines
features of a topological spin insulator and a topological charge metal---in
which the topology can be probed by measuring transport properties.
