  We show that there exist two dimensional (2D) time reversal invariant
fractionalized insulators with the property that both their boundary with the
vacuum and their boundary with a topological insulator can be fully gapped
without breaking time reversal or charge conservation symmetry. This result
leads us to an apparent paradox: we consider a geometry in which a disk-like
region made up of a topological insulator is surrounded by an annular strip of
a fractionalized insulator, which is in turn surrounded by the vacuum. If we
gap both boundaries of the strip, we naively obtain an example of a gapped
interface between a topological insulator and the vacuum that does not break
any symmetries -- an impossibility. The resolution of this paradox is that this
system spontaneously breaks time reversal symmetry in an unusual way, which we
call \emph{weak symmetry breaking}. In particular, we find that the only order
parameters that are sensitive to the symmetry breaking are nonlocal operators
that describe quasiparticle tunneling processes between the two edges of the
strip; expectation values of local order parameters vanish exponentially in the
limit of a wide strip. Also, we find that the symmetry breaking in our system
comes with a ground state degeneracy, but this ground state degeneracy is
topologically protected, rather than symmetry protected. We show that this kind
of symmetry breaking can also occur at the edge of 2D fractional topological
insulators.
