Quantum and Topological Criticalities of Lifshitz Transition in   Two-Dimensional Correlated Electron Systems
  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 chi is found to diverge as ln 1/|delta Delta| when the ``neck'' of the Fermi surface collapses at the van Hove singularity. More remarkably, it diverges as 1/|delta Delta| when the electron/hole pocket of the Fermi surface vanishes. Here delta Delta 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 kappa diverges as 1/|delta n|, instead of chi, where delta n is the electron density measured from the critical point. Accordingly, Lifshitz transition induces large fluctuations represented by diverging chi and/or kappa. Such fluctuations must be involved in physics of competing orders and influence diversity of strong correlation effects. 
Youhei Yamaji, Takahiro Misawa, and Masatoshi Imada