Nonexistence of the Luttinger-Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models

Abstract

The Luttinger-Ward functional Φ [G], which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function G, is found to be ill-defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy Σ[G] ∝ δΦ[G]/δG is not a single-valued functional of G: in addition to the physical solution for Σ [G], there exists at least one qualitatively distinct unphysical branch. This result is demonstrated for several models: the Hubbard atom, the Anderson impurity model, and the full two-dimensional Hubbard model. Despite this pathology, the skeleton Feynman diagrammatic series for Σ in terms of G is found to converge at least for moderately low temperatures. However, at strong interactions, its convergence is to the unphysical branch. This reveals a new scenario of breaking down of diagrammatic expansions. In contrast, the bare series in terms of the non-interacting Green's function G0 converges to the correct physical branch of Σ in all cases currently accessible by diagrammatic Monte Carlo. Besides their conceptual importance, these observations have important implications for techniques based on the explicit summation of diagrammatic series.

Nonperturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

“…While, as mentioned before, a direct analytical proof of this assumption is not easily feasible for the AL, its validity has been demonstrated in Ref. [30] by explicit numerical calculations following a scheme analogous to the one proposed in Eqs. (26).…”

“…To this end, along the lines of Ref. [30], we rewrite Eq. (15) in two different ways so that they represent iterative schemes for the determination of Σ from a given G (at a fixed value of W ):…”

“…First specific reports about unexpected theoretical consequences induced by the breakdown of the many-body perturbation expansion have been recently presented by several groups. Such effects range from the occurrence [24][25][26][27] of low-frequency divergences of two-particle irreducible vertex functions in the Hubbard 28 and Falicov-Kimball 29 models to the multivaluedness [30][31][32][33][34][35][36] of the electronic self-energy expressed as a functional of the (interacting) Green's function or, equivalently, of the Luttinger-Ward functional. All these manifestations of non-perturbativeness are interconnected and represent different aspects of the same problem, as suggested in some of the abovementioned works.…”

On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

“…In addition to the choice of topologies to be included, the diagrams can be evaluated with bare propagators G 0 or interacting propagators G. If the self-energy is derived from a functional of the self-consistently computed G, the approximation can be shown to satisfy certain conservation laws [30]. In practice, however, bare expansions in terms of G 0 are often found to be more reliable [31,32]. (MBPT approaches which involve both bare and interacting propagators have also been proposed [33], but will not be considered here.…”

“…By varying N * and monitoring the convergence of the selfenergy, the accessible parameter regime can be determined and the errors can be controlled. We use an expansion in terms of bare propagators which is typically found to converge towards the correct solution in the weak-coupling regime U 4t, wherever numerically exact benchmarks are available [32,40].…”