Justifying quasiparticle self-consistent schemes via gradient optimization in Luttinger-Ward theory

TL;DR

This paper justifies quasiparticle self-consistent schemes within Luttinger-Ward theory by showing that gradient optimization of the total energy functional leads to self-consistent equations identical to QSGW, applicable broadly and with practical implications.

Contribution

It provides a theoretical foundation linking gradient minimization to quasiparticle self-consistency, generalizing beyond GW approximation and emphasizing the importance of diagonal Dyson equation solutions.

Findings

Gradient minimization yields QSGW-like self-consistent equations.

Diagonal Dyson equation solutions are most important in quasiparticle basis.

Choosing solutions with largest quasiparticle weight when multiple solutions exist.

Abstract

The question of which non-interacting Green's function "best" describes an interacting many-body electronic system is both of fundamental interest as well as of practical importance in describing electronic properties of materials in a realistic manner. Here, we study this question within the framework of Baym-Kadanoff theory, an approach where one locates the stationary point of a total energy functional of the one-particle Green's function in order to find the total ground-state energy as well as all one-particle properties such as the density matrix, chemical potential, or the quasiparticle energy spectrum and quasiparticle wave functions. For the case of the Klein functional, our basic finding is that minimizing the length of the gradient of the total energy functional over non-interacting Green's functions yields a set of self-consistent equations for quasiparticles that is identical to those of the Quasiparticle Self-Consistent GW (QSGW) [1] approach, thereby providing an a priori justification for such an approach to electronic structure calculations. In fact, this result is general, applies to any self-energy operator, and is not restricted to any particular approximation, e.g. the GW approximation for the self-energy. The approach also shows that, when working in the basis of quasiparticle states, solving the diagonal part of the self-consistent Dyson equation is of primary importance while the off-diagonals are of secondary importance, a common observation in the electronic structure literature of self-energy calculations. Finally, numerical tests and analytical arguments show that when the Dyson equation produces multiple quasiparticle solutions corresponding to a single non-interacting state, minimizing the length of the gradient translates into choosing the solution with largest quasiparticle weight.