  Scaling expressions for the free energy are derived, using the Luttinger-Ward
(LW) functional approach in the Eliashberg framework, for two different models
of quantum critical point (QCP). First, we consider the spin-density-wave (SDW)
model for which the effective theory is the Hertz-Moriya-Millis (HMM) theory,
describing the interaction between itinerant electrons and collective spin
fluctuations. The dynamic of the latter are described by a dynamical exponent
$z$ depending on the nature of the transition. Second, we consider the Kondo
breakdown model for QCP's, one possible scenario for heavy-fermion quantum
transitions, for which the effective theory is given by a gauge theory in terms
of conduction electrons, spinons for localized spins, holons for hybridization
fluctuations, and gauge bosons for collective spin excitations. For both
models, we construct the thermodynamic potential, in the whole phase diagram,
including all kinds of self-energy corrections in a self-consistent way, at the
one loop level. We show how Eliashberg framework emerges at this level and use
the resulting Eliashberg equations to simplify the LW expression for free
energy . it is found that collective boson excitations play a central role. The
scaling expression for the singular part of the free energy near the Kondo
breakdown QCP is characterized by two length scales : one is the correlation
length for hybridization fluctuations, and the other is that for gauge
fluctuations, analogous to the penetration depth in superconductors.
