The nonperturbative feats of local electronic correlation: The physics of irreducible vertex divergences

Abstract

The theoretical study of strongly correlated electron systems is both a fascinating and, at the same time, also a very challenging task. In particular, within the quantum field theoretic description of interacting many-electron systems, no small parameter can be defined a priori according to which a perturbation expansion could be safely formulated. As it turns out, already in situations of intermediate coupling among the electrons, this can have remarkable consequences: the breakdown of self-consistent perturbation theory, which manifests itself in the divergences of the two-particle irreducible vertex functions as well as the associated crossings of the physical with unphysical solutions of the (intrinsically multivalued) Luttinger-Ward functional. The occurrence of these nonperturbative effects poses considerable challenges to the state-of-the-art many-body theory and to the numerical applicability of several forefront approximation schemes.In this thesis, one aspect of the breakdown of perturbation theory, namely the divergences of the irreducible vertex functions, is analyzed in different respects.In particular, as a first step, the appearance of these nonperturbative manifestations throughout the parameter spaces of several fundamental many-electron models is investigated and systematically discussed for cases with and without particle-hole symmetry.Thereafter, on a more fundamental level, the physical origin of the divergences of the irreducible vertex functions is unveiled. To this end, the way the formation of the local magnetic moment and its Kondo screening impact the generalized susceptibility in the charge channel is carefully analyzed. This study reveals the emergence of characteristic structures in Matsubara frequency space that originate vanishing eigenvalues, which are associated with the appearance of irreducible vertex divergences. As a remarkable byproduct of this analysis, an alternative criterion for the determination of the Kondo temperature on the two-particle level is identified. Further, the physical implications of the occurrence of irreducible vertex divergences are studied.As it turns out, the sign change of the associated eigenvalues of the generalized susceptibility in specific scattering channels can lead to effectively attractive contributions in these sectors. Ultimately, these contributions are responsible for the divergence of the isothermal compressibility, observed in the proximity of the Mott metal-to-insulator transition of the Hubbard model solved by the dynamical mean-field theory. From a more methodological perspective, different algorithmic strategies for circumventing the computational problems posed by the breakdown of self-consistent perturbation theory are reviewed. Because of its potential for future method development, a particular emphasis is given to the multiloop functional renormalization group (mfRG) scheme for the fundamental case of the Anderson impurity model, where its performance and physical content are studied from weak- to strong-coupling.Finally, the multifaceted perspectives for future studies, inspired by the main results presented in this thesis, are outlined in the final chapter.