Origin of misleading convergence in self-consistent many-electron theories: Fundamental aspects and practical implications

TL;DR

This paper investigates why self-consistent many-electron theories sometimes converge to unphysical solutions, revealing the fundamental links to multivalued functionals and proposing a stabilization method.

Contribution

It derives the mathematical conditions for the stability of physical solutions and clarifies the relation between misleading convergence and multivaluedness in many-electron theories.

Findings

Misleading convergence can occur without divergences in the irreducible vertex function.

The relation between convergence issues and multivalued functionals is fundamentally linked.

A systematic procedure for stabilizing physical solutions is proposed.

Abstract

Self-consistent approaches in many-electron problems typically converge to an unphysical solution in strongly correlated regimes. By deriving the mathematical condition for the stability of the physical solution, we unveil the precise relation between two distinct issues previously considered equivalent: the misleading convergence in self-consistent schemes and the multivaluedness of the Luttinger-Ward functional. Although these problems are fundamentally linked through the divergences of the irreducible vertex function, we show that misleading convergence can occur even in the absence of such divergences. Eventually, a systematic procedure for stabilizing the physical solution is proposed.