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arXiv:1909.02793 · Physical review. B./Physical review. B

Parquetlike equations for the Hedin three-leg vertex

Friedrich Krien, Angelo Valli

Published Dec 26, 2019• 42 citations
arXivHTMLPDF

Abstract

Taking the competition and the mutual screening of various bosonic fluctuations in correlated electron systems into account requires an unbiased approach to the many-body problem. One such approach is the self-consistent solution of the parquet equations, whose numerical treatment in lattice systems is however prohibitively expensive. In a recent article it was shown that there exists an alternative to the parquet decomposition of the four-point vertex function, which classifies the vertex diagrams according to the principle of single-boson exchange (SBE) [Krien, Valli, and Capone, Phys. Rev. B 100, 155149 (2019)]. Here we show that the SBE decomposition leads to a closed set of equations for the Hedin three-leg vertex, the polarization, and the electronic self-energy, which sums self-consistently the diagrams of the Maki-Thompson type. This circumvents the calculation of four-point vertex functions and the inversion of the Bethe-Salpeter equations, which are the two major bottlenecks of the parquet equations. The convergence of the calculation scheme starting from a fully irreducible vertex is demonstrated for the Anderson impurity model.

DOI
10.1103/physrevb.100.245147

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Taking the competition and the mutual screening of various bosonic fluctuations in correlated electron systems into account requires an unbiased approach to the many-body problem. One such approach is the self-consistent solution of the parquet equations, whose numerical treatment in lattice systems is however prohibitively expensive. In a recent article it was shown that there exists an alternative to the parquet decomposition of the four-point vertex function, which classifies the vertex diagrams according to the principle of single-boson exchange (SBE) [Krien, Valli, and Capone, Phys. Rev. B 100, 155149 (2019)]. Here we show that the SBE decomposition leads to a closed set of equations for the Hedin three-leg vertex, the polarization, and the electronic self-energy, which sums self-consistently the diagrams of the Maki-Thompson type. This circumvents the calculation of four-point vertex functions and the inversion of the Bethe-Salpeter equations, which are the two major bottlenecks of the parquet equations. The convergence of the calculation scheme starting from a fully irreducible vertex is demonstrated for the Anderson impurity model.
We present a decomposition of the two-particle vertex function of the single-band Anderson impurity model which imparts a physical interpretation of the vertex in terms of the exchange of bosons of three flavors. We evaluate the various components of the vertex for an impurity model corresponding to the half-filled Hubbard model within dynamical mean-field theory. For small values of the interaction almost the entire information encoded in the vertex function corresponds to single-boson exchange processes, which can be represented in terms of the Hedin three-leg vertex and the screened interaction. Also for larger interaction, the single-boson exchange still captures scatterings between electrons and the dominant low-energy fluctuations and provides a unified description of the vertex asymptotics. The proposed decomposition of the vertex does not require the matrix inversion of the Bethe-Salpeter equation. Therefore, it represents a computationally lighter and hence more practical alternative to the parquet decomposition.
Abstract The parquet equations present a cornerstone of some of the most important diagrammatic many-body approximations and methods currently on the market for strongly correlated materials: from non-local extensions of the dynamical mean-field theory to the functional renormalization group. The recently introduced single-boson exchange decomposition of the vertex presents an alternative set of equivalent equations in terms of screened interactions, Hedin vertices, and rest functions. This formulation has garnered much attention for several reasons: opening the door to new approximations, for avoiding vertex divergences associated with local moment formation plaguing the traditional parquet decomposition, and for its interpretative advantage in its built-in diagrammatic identification of bosons without resorting to Hubbard–Stratonovich transformations. In this work, we show how the fermionic diagrams of the particle–particle and particle–hole polarizations in the SBE formalism can be mapped to diagrammatics of a bosonic self-energy of two respective bosonic theories with pure bosonic constituents, solidifying the identification of the screened interaction with a bosonic propagator. Resorting to a spin-diagonalized basis for the bosonic fields and neglecting the coupling between singlet and triplet components are shown to recover the trace log theory known from Hubbard–Stratonovich transformations. Armed with this concrete mapping, we revisit a conjecture claiming that universal aspects of the parquet approximation coincide with those of the self-consistent screening approximation for a bosonic O ( N ) model. We comment on the role of the self-energy and crossing symmetry in enforcing the Hohenberg–Mermin–Wagner theorem in parquet-related approaches. Graphic Abstract
Abstract 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
We present a reformulation of the functional renormalization group (fRG) for many-electron systems, which relies on the recently introduced single-boson exchange (SBE) representation of the parquet equations [F. Krien, A. Valli, and M. Capone, Phys. Rev. B 100, 155149 (2019)]. The latter exploits a diagrammatic decomposition, which classifies the contributions to the full scattering amplitude in terms of their reducibility with respect to cutting one interaction line, naturally distinguishing the processes mediated by the exchange of a single boson in the different channels. We apply this idea to the fRG by splitting the one-loop fRG flow equations for the vertex function into SBE contributions and a residual four-point fermionic vertex. Similarly as in the case of parquet solvers, recasting the fRG algorithm in the SBE representation offers both computational and interpretative advantages: The SBE decomposition not only significantly reduces the numerical effort of treating the high-frequency asymptotics of the flowing vertices, but it also allows for a clear physical identification of the collective degrees of freedom at play. We illustrate the advantages of an SBE formulation of fRG-based schemes, by computing through the merger of dynamical mean-field theory and fRG the susceptibilities and the Yukawa couplings of the two-dimensional Hubbard model from weak to strong coupling, for which we also present an intuitive physical explanation of the results. The SBE formulation of the one-loop flow equations paves a promising route for future multiboson and multiloop extensions of fRG-based algorithms.
Starting with Hedin's equations, simple expressions for the irreducible self-energy are derived. The derivation with vertex effects included in the self-energy results in a number of terms beyond G W such as second-order screened exchange (the term that also gives rise to vertex/excitonic effects in the polarization) and an infinite series describing correlations between the added-particle (removed-hole) and the excited electrons and holes: the screened T -matrix channels. Two-body correlated propagation is considered, with the third propagating freely, however, three-body correlations are discussed and these can be added hierarchically to the method. For electron-hole propagation, the reducible polarization is calculated, which results in an expression for the self-energy that can be solved analytically without the need for widely used expensive numerical methods for frequency integration or having to adopt the plasmon-pole approximation. The method requires diagonalization of the usual particle-hole matrix—derived from the Bethe-Salpter equation—and also a particle-particle + hole -hole matrix that has a similar structure; with the second-order exchange present in all channels. The method goes beyond current widely used methods that adopt the G W approximation, whilst being in nature and requiring relatively modest extensions to existing functionality, e.g., it could be implemented in the sophisticated quasiparticle self-consistent framework. It is anticipated the method will significantly improve on the accuracy of current state-of-the-art calculations, improving our understanding of processes in simple and strongly correlated systems, with the additional contributions demanding a similar cost to methods that include vertex effects in the polarization only. Published by the American Physical Society 2024
We present and implement a parquet approximation within the dual-fermion formalism based on a partial bosonization of the dual vertex function which substantially reduces the computational cost of the calculation. The method relies on splitting the vertex exactly into single-boson exchange contributions and a residual four-fermion vertex, which physically embody, respectively, long- and short-range spatial correlations. After recasting the parquet equations in terms of the residual vertex, these are solved using the truncated-unity method of Eckhardt et al. [Phys. Rev. B 101, 155104 (2020)], which allows for a rapid convergence with the number of form factors in different regimes. While our numerical treatment of the parquet equations can be restricted to only a few Matsubara frequencies, reminiscent of Astretsov et al. [Phys. Rev. B 101, 075109 (2020)], the one- and two-particle spectral information is fully retained. In applications to the two-dimensional Hubbard model the method agrees quantitatively with a stochastic summation of diagrams over a wide range of parameters.