  We study the boundary physics of bulk insulators by considering three coupled
Hubbard chains in a linear confining potential. In the Hartree-Fock
approximation, the ground state at and slightly off the particle-hole symmetric
point remains insulating even at large slopes of the confining potential. By
contrast, accounting for quantum fluctuations and correlations through a
combination of bosonization and RG methods, we find that there is always a
gapless dipole mode, but it does not contribute to a finite compressibility.
Moreover, when increasing the slope of the confining potential at half filling,
we find a quantum phase transition between an insulating and a metallic state,
indicating the formation of a soft edge state of non-topological origin. Away
from half filling, a similar transition takes place.
