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Physical review. B./Physical review. B

Normalization of exact quasiparticle wave functions in the Green's function method guaranteed by the Ward identity

Takeru Nakashima, Hannes Raebiger, Kaoru Ohno

Published Nov 29, 2021• 6 citations
Publisher

Abstract

It has been a long-standing problem in $N$-electron systems as to how to solve the Dyson equation with the quasiparticle wave functions defined as $\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Psi}}}_{0}^{N}\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}{\ensuremath{\psi}}_{s}(\mathbit{r})\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Psi}}}_{\ensuremath{\nu}}^{N+1}\phantom{\rule{0.16em}{0ex}}\ensuremath{\rangle}$ and $\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Psi}}}_{\ensuremath{\mu}}^{N\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}{\ensuremath{\psi}}_{s}(\mathbit{r})\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Psi}}}_{0}^{N}\phantom{\rule{0.16em}{0ex}}\ensuremath{\rangle}$, which are not mutually orthogonal and have a norm less than 1. We show that quasiparticle wave functions without multiple excitations can exactly be normalized to unity owing to the Ward identity and the vertex function in $(\mathbit{q},{\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{-}\ensuremath{\omega})\ensuremath{\rightarrow}0$, although they are not necessarily mutually orthogonal. Since they satisfy an eigenvalue equation with the hermitized self-energy due to Baym-Kadanoff's conservation laws, the present theory can be regarded as an extension of the Kohn-Sham density functional theory, with correlated, interacting, one-electron orbitals.

DOI
10.1103/physrevb.104.l201116

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We propose a framework to construct the ground-state energy and density matrix of an NNN-electron system by solving a self-consistent set of single-particle equations. The method can be viewed as a nontrivial extension of the Kohn-Sham scheme (which is embedded as a special case). It is based on separating the Green's function into a quasiparticle part and a background part, and expressing only the background part as a functional of the density matrix. The calculated single-particle energies and wave functions have a clear physical interpretation as quasiparticle energies and orbitals.
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) (van Schilfgaarde et al 2006 Phys. Rev. Lett. 96 226402-4) 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.
In a nonequilibrium system, quasiparticles are defined, consistently with the construction of quantum transport equations, as one-electron excitations whose propagator possesses a time-local optical potential. This potential is constructed as a solution of the phase equation, a self-consistency condition reducing the renormalized Dyson equation to a formerly proposed multiplicative composition rule for the propagators. The Dyson equation for the particle correlation function is transformed to a renormalized quasiparticle reconstruction equation, whose leading term represents the quasiparticle Kadanoff-Baym ansatz [Physica E 29, 196 (2005)], an improvement over the standard generalized Kadanoff-Baym ansatz.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
Ward identities and Dyson equation for d=1d=1d=1 interacting Fermi systems