Analytical investigation of singularities in two-particle irreducible vertex functions of the Hubbard atom

TL;DR

This paper provides an analytical study of divergences in two-particle irreducible vertex functions within the Hubbard atom, revealing their classification, origins, and connection to self-energy multivaluedness in strongly correlated systems.

Contribution

It derives explicit formulas for all two-particle irreducible vertices of the Hubbard atom and analyzes their divergences, linking them to eigenvalues, eigenvectors, and the self-energy functional.

Findings

Divergences occur in all scattering channels of the Hubbard atom.

Eigenvalue analysis classifies the divergences and their origins.

Approximate methods capture the qualitative divergence structure.

Abstract

Two-particle generalized susceptibilities and their irreducible vertex functions play a prominent role in the quantum many-body theory for correlated electron systems. They act as basic building blocks in the parquet formalism which provides a flexible scheme for the calculation of spectral and response functions. The irreducible vertices themselves have recently attracted increased attention as unexpected divergences in these functions have been identified. Remarkably, such singularities appear already for one of the simplest strongly interacting systems: the atomic limit of the half-filled Hubbard model (Hubbard atom). In this paper, we calculate the analytical expressions for all two-particle irreducible vertex functions of the Hubbard atom in all scattering channels as well as the fully irreducible two-particle vertices. We discuss their divergences and classify them by the eigenvalues and eigenvectors of the corresponding generalized susceptibilities. In order to establish a connection to the recently found multivaluedness of the exact self-energy functional $\Sigma[G]$, we show that already an approximation akin to iterated perturbation theory is sufficient to capture, qualitatively, the divergent structure of the vertex functions. Finally, we show that the localized divergences in the disordered binary mixture model are directly linked to a minimum in the single-particle Matsubara Green's function.