Interplay between local response and vertex divergences in many-fermion systems with on-site attraction

TL;DR

This paper explores the nature of divergences in two-particle vertex functions of the attractive Hubbard model, revealing their impact on physical susceptibilities and challenging previous interpretations through DMFT calculations.

Contribution

It provides a detailed analysis of vertex divergences in the attractive Hubbard model, linking them to local responses and symmetries, and clarifying their role in many-fermion systems.

Findings

Divergences occur in magnetic and density channels.

Vertex divergences can affect dominant scattering sectors.

Symmetry and spectral analysis clarify the relation between divergences and local responses.

Abstract

We investigate the divergences appearing in the two-particle irreducible vertex functions of many-fermion systems with attractive on-site interactions. By means of dynamical mean-field theory calculations, we determine the location of singularity lines in the phase diagram of the attractive Hubbard model at half-filling, where the local Bethe-Salpeter equations are non invertible. We find that divergences appear both in the magnetic and in the density scattering channels. The former affect a sector of suppressed fluctuations and comply with the mapping of the physical susceptibilities of the repulsive case. At the same time, the appearance of singularities in the density channel of the attractive model demonstrates that vertex divergences can also plague the dominant scattering sectors associated with enhanced local susceptibilities. This constitutes a counterexample to previously proposed interpretations. Eventually, by exploiting the underlying physical symmetries and a spectral representation of the susceptibilities, we clarify the relation between vertex divergences and the local response of the system in different channels.