Two-particle Irreducible Vertex Research Articles

Multiloop flow equations for single-boson exchange fRG

The recently introduced single-boson exchange (SBE) decomposition of the four-point vertex of interacting fermionic many-body systems is a conceptually and computationally appealing parametrization of the vertex. It relies on the notion of reducibility of vertex diagrams with respect to the bare interaction U, instead of a classification based on two-particle reducibility within the widely used parquet decomposition. Here, we re-derive the SBE decomposition in a generalized framework (suitable for extensions to, e.g., inhomogeneous systems or real-frequency treatments) following from the parquet equations. We then derive multiloop functional renormalization group (mfRG) flow equations for the ingredients of this SBE decomposition, both in the parquet approximation, where the fully two-particle irreducible vertex is treated as an input, and in the more restrictive SBE approximation, where this role is taken by the fully U-irreducible vertex. Moreover, we give mfRG flow equations for the popular parametrization of the vertex in terms of asymptotic classes of the two-particle reducible vertices. Since the parquet and SBE decompositions are closely related, their mfRG flow equations are very similar in structure.Graphic abstract

Dynamical vertex approximation for many-electron systems with spontaneously broken SU(2) symmetry

We generalize the formalism of the dynamical vertex approximation (D$\Gamma$A) -- a diagrammatic extension of the dynamical mean-field theory (DMFT)-- to treat magnetically ordered phases. To this aim, we start by concisely illustrating the many-electron formalism for performing ladder resummations of Feynman diagrams in systems with broken SU(2)-symmetry, associated to ferromagnetic (FM) or antiferromagnetic (AF) order. We then analyze the algorithmic simplifications introduced by taking the local approximation of the two-particle irreducible vertex functions in the Bethe-Salpeter equations, which defines the ladder implementation of D$\Gamma$A for magnetic systems. The relation of this assumption with the DMFT limit of large coordination-number/ high-dimensions is explicitly discussed. As a last step, we derive the expression for the ladder D$\Gamma$A self-energy in the FM- and AF-ordered phases of the Hubbard model. The physics emerging in the AF-ordered case is explicitly illustrated by means of approximated calculations based on a static mean-field input for the D$\Gamma$A equations. The results obtained capture fundamental aspects of both metallic and insulating ground states of two-dimensional antiferromagnets, providing a reliable compass for future, more extensive applications of our approach. Possible routes to further develop diagrammatic-based treatments of magnetic phases in correlated electron systems are briefly outlined in the conclusions.

Interplay between local response and vertex divergences in many-fermion systems with on-site attraction

We investigate the divergences appearing in the two-particle irreducible vertex functions of many-fermion systems with attractive on-site interactions. By means of dynamical mean-field theory calculations, we determine the location of singularity lines in the phase diagram of the attractive Hubbard model at half-filling, where the local Bethe-Salpeter equations are non invertible. We find that divergences appear both in the magnetic and in the density scattering channels. The former affect a sector of suppressed fluctuations and comply with the mapping of the physical susceptibilities of the repulsive case. At the same time, the appearance of singularities in the density channel of the attractive model demonstrates that vertex divergences can also plague the dominant scattering sectors associated with enhanced local susceptibilities. This constitutes a counterexample to previously proposed interpretations. Eventually, by exploiting the underlying physical symmetries and a spectral representation of the susceptibilities, we clarify the relation between vertex divergences and the local response of the system in different channels.

Fermi-liquid theory and divergences of the two-particle irreducible vertex in the periodic Anderson lattice

Here we analyze the divergences of the irreducible vertex function in dynamical mean field theory, which may indicate either a non-physical breakdown of the perturbation theory or a response to some physical phenomenon. To investigate this question, we construct a quasiparticle vertex from the diverging irreducible vertex functions. This vertex describes the scattering between quasiparticles and quasiholes in a Fermi liquid. We show that the quasparticle vertex \textit{does not} diverge in the charge channel, wherein the irreducible vertex \textit{does} diverge; and we show that the quasiparticle vertex does diverge in the spin channel, wherein the irreducible vertex does not diverge. This divergence occurs at the Mott transition wherein the Fermi liquid theory breaks down. Both Hubbard and Anderson lattices are investigated. In general, our results support that the divergences of the irreducible vertex function do not indicate a non-physical failure of the perturbation theory. Instead, the divergences are the mathematical consequence of inverting a matrix (the local charge susceptibility) which accumulates increasingly negative diagonal elements as the Hubbard interaction suppresses charge fluctuations. Indeed, we find that the first symmetric divergences of the irreducible vertex in both Hubbard and Anderson lattices occurs near the maximum magnitude of the (negative) vertex-connected part of the charge susceptibility, which suppresses charge fluctuations.

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

Two-particle generalized susceptibilities and their irreducible vertex functions play a prominent role in the quantum many-body theory for correlated electron systems. They act as basic building blocks in the parquet formalism which provides a flexible scheme for the calculation of spectral and response functions. The irreducible vertices themselves have recently attracted increased attention as unexpected divergences in these functions have been identified. Remarkably, such singularities appear already for one of the simplest strongly interacting systems: the atomic limit of the half-filled Hubbard model (Hubbard atom). In this paper, we calculate the analytical expressions for all two-particle irreducible vertex functions of the Hubbard atom in all scattering channels as well as the fully irreducible two-particle vertices. We discuss their divergences and classify them by the eigenvalues and eigenvectors of the corresponding generalized susceptibilities. In order to establish a connection to the recently found multivaluedness of the exact self-energy functional $\Sigma[G]$, we show that already an approximation akin to iterated perturbation theory is sufficient to capture, qualitatively, the divergent structure of the vertex functions. Finally, we show that the localized divergences in the disordered binary mixture model are directly linked to a minimum in the single-particle Matsubara Green's function.

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

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.

Mean-field approximation for thermodynamic and spectral functions of correlated electrons: Strong coupling and arbitrary band filling

We present a construction of a mean-field theory for thermodynamic and spectral properties of correlated electrons reliable in the strong-coupling limit. We introduce an effective interaction determined self-consistently from the reduced parquet equations. It is a static local approximation of the two-particle irreducible vertex, the kernel of a potentially singular Bethe-Salpeter equation. The effective interaction enters the Ward identity from which a thermodynamic self-energy, renormalizing the one-electron propagators, is determined. The dynamical Schwinger-Dyson equation with the thermodynamic propagators is then used to calculate the spectral properties. The thermodynamic and spectral properties of correlated electrons are in this way determined on the same footing and in a consistent manner. Such a mean-field approximation is analytically controllable and free of unphysical behavior and spurious phase transitions. We apply the construction to the asymmetric Anderson impurity and the Hubbard models in the strong-coupling regime.

Ab initio dynamical vertex approximation

Diagrammatic extensions of dynamical mean field theory (DMFT) such as the dynamical vertex approximation (D$\Gamma$A) allow us to include non-local correlations beyond DMFT on all length scales and proved their worth for model calculations. Here, we develop and implement an AbinitioD$\Gamma$A approach for electronic structure calculations of materials. Starting point is the two-particle irreducible vertex in the two particle-hole channels which is approximated by the bare non-local Coulomb interaction and all local vertex corrections. From this we calculate the full non-local vertex and the non-local self-energy through the Bethe-Salpeter equation. The AbinitioD$\Gamma$A approach naturally generates all local DMFT correlations and all non-local $GW$ contributions, but also further non-local correlations beyond: mixed terms of the former two and non-local spin fluctuations. We apply this new methodology to the prototypical correlated metal SrVO$_3$.

Hole-s± State Induced by Coexisting Ferro- and Antiferromagnetic and Antiferro-orbital Fluctuations in Iron Pnictides

The multi-orbital Hubbard model is investigated in order to clarify the electron correlation effects and the superconductivity in the iron-based superconductor. The renormalization effects on the self-energy and the two-particle irreducible vertex function are calculated on the basis of the dynamical mean field theory. We find that the vertex function exhibits a strong renormalization with the significant orbital dependence as compared with the renormalization factor, when the antiferromagnetic and the antiferro-orbital fluctuations are comparably enhanced due to the electron and the hole Fermi surfaces nesting effects. The orbital-dependent vertex function together with the $q\sim0$ nesting between the two-hole Fermi surfaces results in the inter-orbital ferromagnetic fluctuation gradually enhanced which is expected to be observed in LiFeAs. We show that the hole-$s_{\pm}$-wave superconductivity with the sign change of the two-hole Fermi surfaces is realized by the enhanced ferromagnetic fluctuation accompanied by the antiferromagnetic fluctuation and the antiferro-orbital fluctuation.

Conserving approximations for response functions of the Fermi gas in a random potential

We scrutinize the diagrammatic perturbation theory of noninteracting electrons in a random potential with the aim to accomplish a consistent comprehensive theory of quantum diffusion. Ward identity between the one-electron self-energy and the two-particle irreducible vertex is generally not guaranteed in the perturbation theory with only elastic scatterings. We show how the Ward identity can be established in practical approximations and how the functions from the perturbation expansion should be used to obtain a fully consistent conserving theory. We derive the low-energy asymptotics of the conserving full two-particle vertex from which we find an exact representation of the diffusion pole and of the static diffusion constant in terms of Green functions of the perturbation expansion. We illustrate the construction on the leading vertex corrections to the mean-field diffusion due to maximally-crossed diagrams responsible for weak localization.

Renormalization group flow of the Luttinger-Ward functional: Conserving approximations and application to the Anderson impurity model

We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle irreducible vertex functions, given a cut-off in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to non-conserving approximations introduced in an earlier publication ctuation exchange approximation as well as to nonconserving approximations introducen [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single impurity Anderson model in thermal equilibrium at vanishing temperature. Numerical results for the effective mass, the spin susceptibility, the charge susceptibility, and the linear conductance reflect the similarity of the methods to the fluctuation exchange approximation. We find the majority of the approximations to deviate stronger from the exact results than one-particle irreducible functional renormalization group schemes. However, we identify a simple static two-particle irreducible flow scheme which performs remarkably well and produces an exponential Kondo-like scale in the renormalized level position.

Critical metal-insulator transition and divergence in a two-particle irreducible vertex in disordered and interacting electron systems

We use the dynamical mean-field approximation to study singularities in the self-energy and a two-particle irreducible vertex induced by the metal-insulator transition of the disordered Falicov-Kimball model. We set general conditions for the existence of a critical metal-insulator transition caused by a divergence of the imaginary part of the self-energy. We calculate explicitly the critical behavior of the self-energy for the symmetric and asymmetric disorder distributions. We demonstrate that the metal-insulator transition is preceded by a pole in a two-particle irreducible vertex. We show that unlike the singularity in the self-energy the divergence in the irreducible vertex does not lead to non-analyticities in measurable physical quantities. We reveal universal features of the critical metal-insulator transition that are transferable also to the Mott-Hubbard transition in the models of the local Fermi liquid.