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

TL;DR

This paper derives scaling expressions for free energy near quantum critical points using the Luttinger-Ward functional within the Eliashberg framework, analyzing two models: spin-density-wave and Kondo breakdown, highlighting the role of collective boson excitations.

Contribution

It systematically derives thermodynamic scaling near QCPs using LW functional and Eliashberg theory for two distinct models, incorporating self-energy corrections and collective excitations.

Findings

Scaling expressions for free energy near QCPs are obtained.

Collective boson excitations are crucial in the thermodynamics.

Two length scales characterize the Kondo breakdown QCP.

Abstract

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.