Quantum critical point associated with the electronic topological transition in a two-dimensional electron system as a driving force for anomalies in underdoped high-Tccuprates

We study an electronic topological transition (ETT) (the transition due to change in topology of Fermi surface) that occurs in a two-dimensional electron system on a square lattice with hopping beyond nearest neighbors under a change of electronic concentration n (or of hole doping $\delta =1\mathrm{}-n).\mathrm{}$ We show that the ETT point $(\delta ={\delta }_c,\hspace{0.5em}T=0)$ is an exotic quantum critical point (QCP) with several aspects of criticality. The first trivial one is related to singularities in thermodynamic properties. A nontrivial and never considered aspect concerns a behavior of the electron-hole response function: the ETT point is a quantum multicritical point, the end point of the critical line $(T=0,\hspace{0.5em}\delta >{\delta }_c)$ of static Kohn singularities. We show that the existence of this QCP results in global anomalies of the system in the presence of interaction. (We consider the interaction corresponding to the ${t-t}^{\prime }-J$ model.) The anomalies take place on one side of the ETT $(\delta <{\delta }_c$ in the case of $t^{\prime }/t<\mathrm{}0\mathrm{}$ corresponding to the high-$T_c$ cuprates) and have a striking similarity with the anomalies observed in the high-$T_c$ cuprates in the underdoped regime. The most important consequence of the ETT (besides the appearance of superconducting instability of the d-wave symmetry) is the existence of the pseudocritical zone, $T^{*}\left(\delta \right)\propto {\delta }_c-\delta ,$ which grows from the point $\delta ={\delta }_c$ on the side $\delta <{\delta }_c.$ Below and around this zone the metal state is anomalous. Some anomalies are considered in the present paper and compared with experiment (NMR, neutron scattering).