  We show here that numerous examples abound where changing topology does not
necessarily close the bulk insulating charge gap as demanded in the standard
non-interacting picture. From extensive determinantal and dynamical cluster
quantum Monte Carlo simulations of the half-filled and quarter-filled
Kane-Mele-Hubbard model, we show that for sufficiently strong interactions at
either half- or quarter-filling, a transition between topological and trivial
insulators occurs without the closing of a charge gap. To shed light on this
behavior, we illustrate that an exactly solvable model reveals that while the
single-particle gap remains, the many-body gap does in fact close. These two
gaps are the same in the non-interacting system but depart from each other as
the interaction turns on. We purport that for interacting systems, the proper
probe of topological phase transitions is the closing of the many-body rather
than the single-particle gap.
