  Step edges of topological crystalline insulators can be viewed as
predecessors of higher-order topology, as they embody one-dimensional edge
channels embedded in an effective three-dimensional electronic vacuum emanating
from the topological crystalline insulator. Using scanning tunneling microscopy
and spectroscopy we investigate the behaviour of such edge channels in
Pb$_{1-x}$Sn$_{x}$Se under doping. Once the energy position of the step edge is
brought close to the Fermi level, we observe the opening of a correlation gap.
The experimental results are rationalized in terms of interaction effects which
are enhanced since the electronic density is collapsed to a one-dimensional
channel. This constitutes a unique system to study how topology and many-body
electronic effects intertwine, which we model theoretically through a
Hartree-Fock analysis.
