Lifshitz quantum phase transitions and Fermi surface transformation with hole doping in high-$T_c$ superconductors

We study the doping evolution of the electronic structure in the normal phase of high-$T_c$ cuprates. Electronic structure and Fermi surface of cuprates with single CuO${}_2$ layer in the unit cell like La${}_{2-x}$Sr${}_x$CuO${}_4$ have been calculated by the LDA+GTB method in the regime of strong electron correlations (SEC) and compared to ARPES and quantum oscillations data. We have found two critical concentrations, $x_{c1}$ and $x_{c2}$, where the Fermi surface topology changes. Following I.M. Lifshitz ideas of the quantum phase transitions (QPT) of the 2.5-order we discuss the concentration dependence of the low temperature thermodynamics. The behavior of the electronic specific heat $\delta (C\mathrm{/}T)\sim (x-x_c{)}^{1\mathrm{/}2}$ is similar to the Loram and Cooper experimental data in the vicinity of $x_{c1}\approx 0.15$.