How to stay on the physical branch in self-consistent many-electron approaches

The self-consistent solution of many-electron problems typically switches to a non-perturbative branch of the Luttinger-Ward functional in strongly correlated regimes. We demonstrate that even when the correct branch is captured by a self-consistent theory, the solution can become unstable, ultimately converging to an unphysical result. By deriving the mathematical condition for this instability, we unveil the distinction and the underlying connection between two 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 for self-consistent methods like bold-expansion, parquet, nested-cluster will be illustrated.