Practical consequences of Luttinger-Ward functional multivaluedness for cluster DMFT methods

TL;DR

This paper explores how the multivaluedness of the Luttinger-Ward functional causes failures in cluster DMFT methods, leading to unphysical solutions in strongly correlated regimes.

Contribution

It demonstrates the impact of LWF multivaluedness on computational methods like NCS in the Hubbard model, revealing unphysical stationary points.

Findings

Unphysical stationary points can dominate solutions in cluster DMFT.

Divergence of the irreducible vertex function indicates problematic solutions.

Standard iterative methods may converge to unphysical branches in strongly correlated regimes.

Abstract

The Luttinger-Ward functional (LWF) has been a starting point for conserving approximations in many-body physics for 50 years. The recent discoveries of its multivaluedness and the associated divergence of the two-particle irreducible vertex function $\Gamma$ have revealed an inherent limitation of this approach. Here we demonstrate how these undesirable properties of the LWF can lead to a failure of computational methods based on an approximation of the LWF. We apply the Nested Cluster Scheme (NCS) to the Hubbard model and observe the existence of an additional stationary point of the self-consistent equations, associated with an unphysical branch of the LWF. In the strongly correlated regime, starting with the first divergence of $\Gamma$, this unphysical stationary point becomes attractive in the standard iterative technique used to solve DMFT. This leads to an incorrect solution, even in the large cluster size limit, for which we discuss diagnostics.