  Lifshitz transitions are topological transitions of a Fermi surface, whose
signatures typically appear in the conduction properties of a host metal. Here,
we demonstrate, using an extended Falicov- Kimball model of a two-flavor
fermion system, that a Lifshitz transition which occurs in the noninteracting
limit impacts interaction-induced insulating phases, even though they do not
host Fermi surfaces. For strong interactions we find a first order transition
between states of different polarization This transition line ends in a very
unusual quantum critical endpoint, whose presence is stabilized by the onset of
inter-flavor coherence. We demonstrate that the surfaces of maximum coherence
in these states reflect the distinct Fermi surface topologies of the states
separated by the non-interacting Lifshitz transition. The phenomenon is shown
to be independent of the band topologies involved. Experimental realizations of
our results are discussed for both electronic and optical lattice systems.
