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arXiv:2306.05157 · Physical review. B./Physical review. B

Dual Bethe-Salpeter equation for the multiorbital lattice susceptibility within dynamical mean-field theory

Erik G. C. P. van Loon, Hugo U. R. Strand

Published Apr 23, 2024• 7 citations
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Abstract

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

DOI
10.1103/physrevb.109.155157

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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
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.
The dynamical susceptibility of strongly correlated electronic systems can be calculated within the framework of dynamical mean-field theory (DMFT). The required measurement of the four-point vertex of the auxiliary impurity model is however costly and restricted to a finite grid of Matsubara frequencies, leading to a cutoff error. It is shown that the propagation of this error to the lattice response function can be minimized by virtue of an exact decomposition of the DMFT polarization function into local and nonlocal parts. The former is measured directly by the impurity solver, while the latter is given in terms of a ladder equation for the Hedin vertex that features an unprecedentedly fast decay of frequency summations compared to previous calculation schemes, such as the one of the dual boson approach. At strong coupling, the local approximation of the TRILEX approach is viable, but vertex corrections to the polarization should be dropped on equal footing to recover the correct prefactor of the effective exchange. In finite dimensions, the DMFT susceptibility exhibits spurious mean-field criticality, therefore, a two-particle self-consistent and frequency-dependent correction term is introduced, similar to the Moriya $\ensuremath{\lambda}$ correction of the dynamical vertex approximation. Applications to the two- and three-dimensional Hubbard models on the square and cubic lattices show that the expected critical behavior near an antiferromagnetic instability is recovered.
Computing momentum-dependent susceptibilities in the dynamical mean-field theory (DMFT) requires solving the Bethe-Salpeter equation, which demands large computational cost. Exploiting the strong-coupling feature of local fluctuations, we derive a simplified formula that can be solved at a considerably lower cost. The validity and the physical meaning of the formula are confirmed by deriving the effective intersite interactions in the strong-coupling limit, such as the kinetic exchange and RKKY interactions. Furthermore, numerical calculations for single-orbital and multiorbital models demonstrate surprisingly wider applicability including a weak-coupling region. Based on this formula, we propose three levels of practical approximations that can be chosen depending on complexity of problems. Simpler evaluations of spin and orbital susceptibilities in multiorbital systems thus become possible within DMFT.
In this paper, we show how the two-particle Green function can be obtained within the framework of the dual fermion approach. This facilitates the calculation of the susceptibility in strongly correlated systems where long-ranged nonlocal correlations cannot be neglected. We formulate the Bethe--Salpeter equations for the full vertex in the particle-particle and particle-hole channels and introduce an approximation for practical calculations. The scheme is applied to the two-dimensional Hubbard model at half-filling. The spin-spin susceptibility is found to strongly increase for the wave vector $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$, which indicates the antiferromagnetic instability. We find a suppression of the critical temperature compared to the mean-field result due to the incorporation of the nonlocal spin fluctuations.
In this paper, we present the details of the dual fermion (DF) method to study the nonlocal corrections to the single-site dynamical mean-field theory (DMFT). The DMFT two-particle Green's function is calculated using continuous-time quantum Monte Carlo method. The momentum dependence of the vertex function is analyzed and its renormalization based on the Bethe-Salpeter equation is performed in the particle-hole channel. We found magnetic instability for both the dual and the lattice fermions. Furthermore, the lattice fermion susceptibility is calculated in this method and also in another recently proposed method, namely, dynamical vertex approximation $(\text{D}\ensuremath{\Gamma}\text{A})$. The comparisons between these two methods are presented in both weak- and strong-coupling regions. Compared to the results from the quantum Monte Carlo calculation, the susceptibility obtained from the DF and $\text{D}\ensuremath{\Gamma}\text{A}$ methods are satisfactory.
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.