Some consequences of electronic topological transition in 2D system on a square lattice: Excitonic ordered states

We study in a mean-field approximation the ordered "excitonic" states which develop around the quantum critical point. (QCP) associated with the elect ronic topological transition (ETT) in a 2D electron system on a square latt ice. We consider the case of hopping beyond nearest neighbors when ETT has an unusual character. We show that the amplitude of the order parameter (OP ) and of the gap in the electron spectrum increase with increasing the dist ance from the QCP, delta(c) - delta, where delta = 1 - n and n is an electr on concentration. Such a behavior is different from the ordinary case when OP and the gap decrease when going away from the point which is a motor for instability. WE show that the chemical potential lies always inside the ga p fur wavevectors k in a proximity of (0, pi) whatever is the doping concen tration. The spectrum gets a characteristic flat shape as a result of hybri dization effect in the vicinity of two different SP's. The shape of the spe ctrum as a function of k and the angle dependence of the gap have a strikin g similarity with the features observed in the normal state of the underdop ed high-T-c cuprates. We discuss also details about the phase diagram and t he behaviour of the density of states.