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Scaling expressions for the free energy are derived, using the Luttinger-Ward(LW) functional approach in the Eliashberg framework, for two different modelsof 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 spinfluctuations. The dynamic of the latter are described by a dynamical exponent$z$ depending on the nature of the transition. Second, we consider the Kondobreakdown model for QCP's, one possible scenario for heavy-fermion quantumtransitions, for which the effective theory is given by a gauge theory in termsof conduction electrons, spinons for localized spins, holons for hybridizationfluctuations, and gauge bosons for collective spin excitations. For bothmodels, we construct the thermodynamic potential, in the whole phase diagram,including all kinds of self-energy corrections in a self-consistent way, at theone loop level. We show how Eliashberg framework emerges at this level and usethe resulting Eliashberg equations to simplify the LW expression for freeenergy . it is found that collective boson excitations play a central role. Thescaling expression for the singular part of the free energy near the Kondobreakdown QCP is characterized by two length scales : one is the correlationlength for hybridization fluctuations, and the other is that for gaugefluctuations, analogous to the penetration depth in superconductors.

Luttinger-Ward functional approach in the Eliashberg framework : A systematic derivation of scaling for thermodynamics near a quantum critical point