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

Intrinsic arrested nanoscale phase separation near a topological Lifshitz transition in strongly correlated two-band metals