  Correlations in topological states of matter provide a rich phenomenology,
including a reduction in the topological classification of the interacting
system compared to its non-interacting counterpart. This happens when two
phases that are topologically distinct on the non-interacting level become
adiabatically connected once interactions are included. We use a quantum Monte
Carlo method to study such a reduction. We consider a 2D charge-conserving
analog of the Levin-Gu superconductor whose classification is reduced from
$\mathbb{Z}$ to $\mathbb{Z}_4$. We may expect any symmetry-preserving
interaction that leads to a symmetric gapped ground state at strong coupling,
and consequently a gapped symmetric surface, to be sufficient for such
reduction. Here, we provide a counter example by considering an interaction
which (i) leads to a symmetric gapped ground state at sufficient strength and
(ii) does not allow for any adiabatic path connecting the trivial phase to the
topological phase with $w=4$. The latter is established by numerically mapping
the phase diagram as a function of the interaction strength and a parameter
tuning the topological invariant. Instead of the adiabatic connection, the
system exhibits an extended region of spontaneous symmetry breaking separating
the topological sectors. Frustration reduces the size of this long-range
ordered region until it gives way to a first order phase transition. Within the
investigated range of parameters, there is no adiabatic path deforming the
formerly distinct free fermion states into each other. We conclude that an
interaction which trivializes the surface of a gapped topological phase is
necessary but not sufficient to establish an adiabatic path within the reduced
classification. In other words, the class of interactions which trivializes the
surface is different from the class which establishes an adiabatic connection
in the bulk.
