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We examine the nature of phase transitions occurring in strongly correlated Fermi systems at the quantum critical point (QCP) associated with a divergent effective mass. Conventional scenarios for the QCP involving collective degrees of freedom are shown to have serious shortcomings. Working within the original Landau quasiparticle picture, we propose an alternative topological scenario for the QCP, in systems that obey standard Fermi liquid (FL) theory in advance of the QCP. Applying the technique of Poincaré mapping, we analyze the sequence of iterative maps generated by the Landau equation for the single-particle spectrum at zero temperature. It is demonstrated that the Fermi surface is subject to rearrangement beyond the QCP. If the sequence of maps converges, a multi-connected Fermi surface is formed. If it fails to converge, the Fermi surface swells into a volume that provides a measure of entropy associated with formation of an exceptional state of the system characterized by partial occupation of single-particle states and dispersion of their spectrum proportional to temperature. Based on this dual scenario, the thermodynamics of Fermi systems beyond the QCP exhibits striking departures from the predictions of standard FL theory. Mechanisms for the release of the entropy excess of the exceptional state are discussed.

Thermodynamic characteristics of Fermi systems are investigated in the vicinity of a phase transition where the effective mass diverges and the single-particle spectrum becomes flat. It is demonstrated that at extremely low temperatures T , the flattening of the spectrum is reflected in non-Fermi-liquid behavior of the inverse susceptibility χ −1 (T ) ∼ T α and the specific heat C(T )/T ∼ T −α , with the critical index α = 2/3. In the presence of an external static magnetic field H, both these quantities are found to exhibit a scaling behavior, e.g. χ −1 (T, H) = χ −1 (T, 0) + T 2/3 F (H/T ), in agreement with available experimental data.