  The arrested nanoscale phase separation in a two-band Hubbard model for
strongly correlated charge carriers is shown to occur in a particular range in
vicinity of the topological Lifshitz transition, where the Fermi energy crosses
the bottom of the narrow band and a new sheet of the Fermi surface related to
the charge carriers of the second band comes into play. We determine the phase
separation diagram of this two-band Hubbard model as a function of two
variables, the charge carrier density and the energy shift between the chemical
potential and the bottom of the second band. In this phase diagram, we first
determine a line of quantum critical points for the Lifshitz transition and
find criteria for the electronic phase separation resulting in an inhomogeneous
charge distribution. Finally, we identify the critical point in presence of a
variable long-range Coulomb interaction where the scale invariance of the
coexisting phases with different charge densities appears. We argue that this
point is relevant for the regime of scale invariance of the nanoscale phase
separation in cuprates like it was first observed in La$_2$CuO$_{4.1}$.
