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

Scaling expressions for the free energy are derived, using the Luttinger–Ward (LW) functional approach in the Eliashberg framework, for two different models of the quantum critical point (QCP). First, we consider the spin-density-wave model for which the effective theory is the Hertz–Moriya–Millis theory, describing the interaction between itinerant electrons and collective spin fluctuations. The dynamics of the latter are described using a dynamical exponent z depending on the nature of the transition. Second, we consider the Kondo breakdown model for QCPs, 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 the Eliashberg framework emerges at this level and use the resulting Eliashberg equations to simplify the LW expression for the 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 for superconductors.