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arXiv:1805.00989 · Physical review. B./Physical review. B

Analytical investigation of singularities in two-particle irreducible vertex functions of the Hubbard atom

Patrik Thunström, O. Gunnarsson, Sergio Ciuchi, G. Rohringer

Published Dec 3, 2018• 63 citations
arXivHTMLPDF

Abstract

The local two-particle Green's function and its generalized susceptibility are central building blocks in state-of-the-art theoretical descriptions of strongly correlated many-electron systems. The unexpected discovery of singular points in the generalized susceptibility, resulting in divergences of the two-particle irreducible vertex function and in multivaluedness of the Luttinger-Ward functional, has therefore attracted considerable interest. Unfortunately, almost no analytical results for irreducible vertices are available for correlated electrons. Here, the authors fill this gap by presenting fully analytical expressions for all irreducible vertex function of a simple, yet prototypical, correlated model, the single-orbital Hubbard atom. This gives highly relevant insights into the energy dependence of these functions.

DOI
10.1103/physrevb.98.235107

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The Moldavian Journal of the Physical Sciences•Jun 1, 2003•0 citations
The local two-particle Green's function and its generalized susceptibility are central building blocks in state-of-the-art theoretical descriptions of strongly correlated many-electron systems. The unexpected discovery of singular points in the generalized susceptibility, resulting in divergences of the two-particle irreducible vertex function and in multivaluedness of the Luttinger-Ward functional, has therefore attracted considerable interest. Unfortunately, almost no analytical results for irreducible vertices are available for correlated electrons. Here, the authors fill this gap by presenting fully analytical expressions for all irreducible vertex function of a simple, yet prototypical, correlated model, the single-orbital Hubbard atom. This gives highly relevant insights into the energy dependence of these functions.
Electronic-correlated systems are often well described by dynamical mean field theory (DMFT). While DMFT studies have mainly focused hitherto on one-particle properties, valuable information is also enclosed into local two-particle Green's functions and vertices. They represent the main ingredient to compute momentum-dependent response functions at the DMFT level and to treat nonlocal spatial correlations at all length scales by means of diagrammatic extensions of DMFT. The aim of this paper is to present a DMFT analysis of the local reducible and irreducible two-particle vertex functions for the Hubbard model in the context of a unified diagrammatic formalism. An interpretation of the observed frequency structures is also given in terms of perturbation theory, of the comparison with the atomic limit, and of the mapping onto the attractive Hubbard model.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
Starting from the (Hubbard) model of an atom, we demonstrate that the uniqueness of the mapping from the interacting to the noninteracting Green function, G→G_{0}, is strongly violated, by providing numerous explicit examples of different G_{0} leading to the same physical G. We argue that there are indeed infinitely many such G_{0}, with numerous crossings with the physical solution. We show that this rich functional structure is directly related to the divergence of certain classes of (irreducible vertex) diagrams, with important consequences for traditional many-body physics based on diagrammatic expansions. Physically, we ascribe the onset of these highly nonperturbative manifestations to the progressive suppression of the charge susceptibility induced by the formation of local magnetic moments and/or resonating valence bond (RVB) states in strongly correlated electron systems.
In this work, we analyze in detail the occurrence of divergences in the irreducible vertex functions for one of the fundamental models of many-body physics: the Anderson impurity model (AIM). These divergences, a surprising hallmark of the breakdown of many-electron perturbation theory, have been recently observed in several contexts, including the dynamical mean-field solution of the Hubbard model. The numerical calculations for the AIM presented in this work, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the properties of vertex divergences in a particularly interesting context, the correlated metallic regime at low temperatures. Specifically, our analysis (i) rules out explicitly the transition to a Mott-insulating phase, but not the more general suppression of charge fluctuations (proposed in [O. Gunnarsson et al., Phys. Rev. B 93, 245102 (2016)]), as a necessary condition for the occurrence of vertex divergences, (ii) clarifies their relation with the underlying Kondo physics, and, eventually, (iii) individuates which divergences might also appear on the real-frequency axis in the limit of zero temperature, through the discovered scaling properties of the singular eigenvectors.
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 $\mathrm{\ensuremath{\Gamma}}$ 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 $\mathrm{\ensuremath{\Gamma}}$, 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.
The asymptotic expression for the one-particle Green's function in the strong coupling limit of the double degenerate Hubbard model containing multiplet terms is obtained for arbitrary band filling within the irreducible Green's function approach. An explicit analytic expression is obtained for the mean-field Greens's function, conserving the sum rules for the first few spectral moments. Stability of homogeneous phases is analised in terms of a single flip spin and orbital excitations. It is shown that degeneracy has an additional enhancing effect on correlation driven interband transfer of spectral weight. Discovery of superconductivity in fullerides and of materials with colossal magnetoresistance during recent years have increased the interest for electron systems with physical properties being essentially dependent on orbital degeneracy and strong correlation effects. In many cases the orbitally degenerate Hubbard model (DHM) incorporates the key features of such materials [1], although there are even fewer exact results established for DHM than for the nondegenerate Hubbard model (NHM). In view of complexity of correlated systems, corrections to the atomic limit of one-particle excitation spectrum and spectral weights of correlated bands found by Harris and Lange for NHM [2] by means of perturbation theory based on canonical transformations provide an important exact result which serves as a test for different approximations. A similar derivation for DHM has not been carried out yet. In the present paper the Green's function for the strong coupling limit of the DHM is obtained by means of irreducible Green's function approach [3] - [6] which is closely related to canonical perturbation theory because of its spectral moment conserving construction. As was explicitly proved in [7] for the NHM, the first couples of spectral moments of correlated bands obtained within mean-field decoupling are identical to those of canonical perturbation theory to first order in / tU ( t is the n.n. hopping amplitude and U the Coulomb repulsion).