We study electron correlation effects on quantum criticalities of Lifshitz transitions at zero temperature, using the mean-field theory based on a preexisting symmetry-broken order, in two-dimensional systems. In the presence of interactions, Lifshitz transitions may become discontinuous in contrast to the continuous transition in the original proposal by Lifshitz for noninteracting systems. We focus on the quantum criticality at the endpoint of discontinuous Lifshitz transitions, which we call the marginal quantum critical point. It shows remarkable criticalities arising from its nature as a topological transition. At the point, for the canonical ensemble, the susceptibility of the order parameter χ is found to diverge as ln 1/|δΔ| when the "neck" of the Fermi surface collapses at the van Hove singularity. More remarkably, it diverges as |δΔ| -1 when the electron/hole pocket of the Fermi surface vanishes. Here δΔ is the amplitude of the mean field measured from the Lifshitz critical point. On the other hand, for the grand canonical ensemble, the discontinuous transitions appear as the electronic phase separation and the endpoint of the phase separation is the marginal quantum critical point. Especially, when a pocket of the Fermi surface vanishes, the uniform charge compressibility κ diverges as |δn|-1, instead of χ, where δn is the electron density measured from the critical point. Accordingly, Lifshitz transition induces large fluctuations represented by diverging χ and/or κ. Such fluctuations must be involved in physics of competing orders and influence diversity of strong correlation effects.
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