  Competing scenarios for quantum critical points (QCPs) of strongly
interacting Fermi systems signaled by a divergent density of states at zero
temperature are contrasted. The conventional scenario, which enlists critical
fluctuations of a collective mode and attributes the divergence to a coincident
vanishing of the quasiparticle strength $z$, is shown to be incompatible with
identities arising from conservation laws prevailing in the fermionic medium.
An alternative scenario, in which the topology of the Fermi surface is altered
at the QCP, is found to explain the non-Fermi-liquid thermodynamic behavior
observed experimentally in Yb-based compounds close to the QCP. It is suggested
that combination of the topological scenario with the theory of quantum phase
transitions will provide a proper foundation for analysis of the extended QCP
region.
