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arXiv:1908.03198 · Physical review. B./Physical review. B

Improved treatment of fermion-boson vertices and Bethe-Salpeter equations in nonlocal extensions of dynamical mean field theory

A. A. Katanin

Published Jan 8, 2020• 16 citations
arXivHTMLPDF

Abstract

We reconsider the procedure of calculating fermion-boson vertices and the numerical solution of Bethe-Salpeter equations, used in nonlocal extensions of dynamical mean field theory. Because of the frequency dependence of the vertices, a finite-frequency box for matrix inversions is typically used, which often requires some treatment of the asymptotic behavior of vertices. Recently [J. Kune\ifmmode \check{s}\else \v{s}\fi{}, Phys. Rev. B 83, 085102 (2011); A. Tagliavini et al., Phys. Rev. B 97, 235140 (2018)] it was proposed to split the considered frequency box into smaller and larger ones; in the smaller-frequency box the numerically exact vertices are used, while beyond this box asymptotics of vertices are applied. Yet, this method requires numerical treatment of vertex asymptotics (including corresponding matrix manipulations) in the larger-frequency box and/or knowledge of fermion-boson vertices, which may not be convenient for numerical calculations. In the present paper we derive the formulas which treat analytically the contribution of vertices beyond the chosen frequency box, such that only numerical operations with vertices in the chosen small-frequency box are required. The method is tested on the Hubbard model and can be used in a broad range of applications of nonlocal extensions of dynamical mean field theory.

DOI
10.1103/physrevb.101.035110

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We reconsider the procedure of calculating fermion-boson vertices and the numerical solution of Bethe-Salpeter equations, used in nonlocal extensions of dynamical mean field theory. Because of the frequency dependence of the vertices, a finite-frequency box for matrix inversions is typically used, which often requires some treatment of the asymptotic behavior of vertices. Recently [J. Kune\ifmmode \check{s}\else \v{s}\fi{}, Phys. Rev. B 83, 085102 (2011); A. Tagliavini et al., Phys. Rev. B 97, 235140 (2018)] it was proposed to split the considered frequency box into smaller and larger ones; in the smaller-frequency box the numerically exact vertices are used, while beyond this box asymptotics of vertices are applied. Yet, this method requires numerical treatment of vertex asymptotics (including corresponding matrix manipulations) in the larger-frequency box and/or knowledge of fermion-boson vertices, which may not be convenient for numerical calculations. In the present paper we derive the formulas which treat analytically the contribution of vertices beyond the chosen frequency box, such that only numerical operations with vertices in the chosen small-frequency box are required. The method is tested on the Hubbard model and can be used in a broad range of applications of nonlocal extensions of dynamical mean field theory.
We present here two alternative schemes designed to correct the high-frequency truncation errors in the numerical treatment of the Bethe-Salpeter equations. The schemes are applicable to all Bethe-Salpeter calculations with a local two-particle irreducible vertex, which is relevant, e.g., for the dynamical mean-field theory (DMFT) and its diagrammatic extensions. In particular, within a purely diagrammatic framework, we could extend existing algorithms for treating the static case in the particle-hole sector to more general procedures applicable to all bosonic frequencies and all channels. After illustrating the derivation and the theoretical interrelation of the two proposed schemes, these have been applied to the Bethe-Salpeter equations for the auxiliary Anderson impurity models of selected DMFT calculations, where results can be compared against a numerically ``exact'' solution. The successful performance of the proposed schemes suggests that their implementation can significantly improve the accuracy of DMFT calculations at the two-particle level, in particular for more realistic multiorbital calculations where the large number of degrees of freedom substantially restricts the actual frequency range for numerical calculations, as well as---on a broader perspective---of the diagrammatic extensions of DMFT.
Dynamical mean-field theory describes the impact of strong local correlation effects in many-electron systems. While the single-particle spectral function is directly obtained within the formalism, two-particle susceptibilities can also be obtained by solving the Bethe-Salpeter equation. The solution requires handling infinite matrices in Matsubara frequency space. This is commonly treated using a finite frequency cutoff, resulting in slow linear convergence. A decomposition of the two-particle response in local and nonlocal contributions enables a reformulation of the Bethe-Salpeter equation inspired by the dual boson formalism. The reformulation has a drastically improved cubic convergence with respect to the frequency cutoff, considerably facilitating the calculation of susceptibilities in multi-orbital systems. This improved convergence arises from the fact that local contributions can be measured in the impurity solver. The dual Bethe-Salpeter equation uses the fully reducible vertex which is free from vertex divergences. We benchmark the approach on several systems including the spin susceptibility of strontium ruthenate Sr2RuO4 , a strongly correlated Hund's metal with three active orbitals. Published by the American Physical Society 2024
Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and nonlocal static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.
The contributions from the non-local fluctuations to the dynamical mean field theory (DMFT) were studied using the recently proposed dual fermion approach. Straight forward cluster extensions of DMFT need the solution of a small cluster, where all the short-range correlations are fully taken into account. All the correlations beyond the cluster scope are treated in the mean-field level. In the dual fermion method, only a single impurity problem needs to be solved. Both the short and long-range correlations could be considered on equal footing in this method. The weak-coupling nature of the dual fermion ensures the validity of the finite order diagram expansion. The one and two particle Green's functions calculated from the dual fermion approach agree well with the Quantum Monte Carlo solutions, and the computation time is considerably less than with the latter method. The access of the long-range order allows us to investigate the collective behavior of the electron system, e.g. spin wave excitations.
Self-consistent Hartree-Fock approximation combined with solutions of the Bethe-Salpeter equation offers a powerful tool for studies of strong correlation effects arising in condensed matter models, nuclear physics, quantum field theories, and real materials. The standard finite-temperature approach would be to first solve the problem in the Matsubara representation and then apply numerical analytic continuation to the real-frequency axis to link theoretical results with experimental probes, but this ill-conditioned procedure often distorts important spectral features even for very accurate imaginary-frequency data. We demonstrate that the ladder-type finite-temperature Bethe-Salpeter equation in the Hartree-Fock basis for the three-point vertex function and, ultimately, system's polarization can be accurately solved directly on the real frequency axis using the diagrammatic Monte Carlo technique and series resummation. We illustrate the method by applying it to the homogeneous electron gas model and demonstrate how multiple scattering events renormalize Landau damping.