Single particle properties of the 2D Hubbard model for real frequencies at weak coupling: Breakdown of the Dyson series for partial self-energy expansions

TL;DR

This paper investigates the limitations of the Dyson series in perturbative expansions of the Hubbard model, revealing that summing reducible diagrams can lead to incorrect physical predictions, especially at weak coupling.

Contribution

It demonstrates that the Dyson series may produce misleading results in the half-filled Hubbard model by summing reducible diagrams, challenging its general applicability.

Findings

Low-order expansions show gap opening, pseudogap, and Fermi-surface changes.

Matsubara data indicates insulating behavior in Green's function expansion, metallic in self-energy expansion.

Reducible diagrams summed in Dyson lead to incorrect physics, questioning the series' validity.

Abstract

We generate the perturbative expansion of the single-particle Green's function and related self-energy for a half-filled single-band Hubbard model on a square lattice. We invoke algorithmic Matsubara integration to evaluate single-particle quantities for real and Matsubara frequencies and verify results through comparison to existing data on the Matsubara axis. With low order expansions at weak-coupling we observe a number of outcomes expected at higher orders: the opening of a gap, pseudogap behavior, and Fermi-surface reconstruction. Based on low-order perturbations we consider the phase diagram that arises from truncated expansions of the self-energy and Green's function and their relation via the Dyson equation. From Matsubara axis data we observe insulating behavior in direct expansions of the Green's function, while the same order of truncation of the self-energy produces metallic behavior. This observation is supported by additional calculations for real frequencies. We attribute this difference to the order in which diagrams are implicitly summed in the Dyson series. By separating the reducible and irreducible contributions at each order we show that the reducible diagrams implicitly summed in the Dyson equation lead to incorrect physics in the half-filled Hubbard model. Our observations for this particular case lead us to question the utility of the Dyson equation for any problem that shows a disparity between reducible and irreducible contributions to the expansion of the Green's function.