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arXiv:2606.05822

Study on the validity of IPT+parquet method as an impurity solver in DMFT focusing on orbital fluctuations

Aira Yamada, Ryota Mizuno, Masayuki Ochi, Kazuhiko Kuroki, Takuma Ohashi

Published Jun 4, 2026• 0 citations
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Abstract

A breakdown of calculations with exact impurity solvers in the dynamical mean field theory in multiband systems easily occurs due to the expensive numerical cost. To overcome this practical difficulty, three of the present authors developed an inexpensive and reliable impurity solver by combining the iterative perturbation theory (IPT) and parquet equation, and named it IPT+parquet [R. Mizuno, et al., Phys. Rev. B 104, 035160 (2021).]. In this study, we validate IPT+parquet focusing on the orbital fluctuation by comparing the numerically exact impurity solvers. We confirm that IPT+parquet can capture competition between orbital fluctuation channels, which the conventional IPT cannot capture.

DOI
10.48550/arxiv.2606.05822

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A breakdown of calculations with exact impurity solvers in the dynamical mean field theory in multiband systems easily occurs due to the expensive numerical cost. To overcome this practical difficulty, three of the present authors developed an inexpensive and reliable impurity solver by combining the iterative perturbation theory (IPT) and parquet equation, and named it IPT+parquet [R. Mizuno, et al., Phys. Rev. B 104, 035160 (2021).]. In this study, we validate IPT+parquet focusing on the orbital fluctuation by comparing the numerically exact impurity solvers. We confirm that IPT+parquet can capture competition between orbital fluctuation channels, which the conventional IPT cannot capture.
Although several impurity solvers in the dynamical mean field theory (DMFT) have been proposed, especially in multiband systems, there are practical difficulties arising from a trade-off between numerical costs and reliability. In this study, we reinterpret the iterative perturbation theory (IPT) as an approximation which captures the strong correlation effects by mimicking the particular frequency structures of the exact full vertex, and extend it such that it can have efficiency and reliability simultaneously by modifying IPT vertex using the parquet equations. We apply this method to several models to evaluate their validity. We confirm that our method shows good agreements with the numerically exact continuous-time quantum Monte Carlo method in the single-site DMFT calculation.
Predicting the physical properties of strongly correlated electron systems (SCES) requires the combined Density Functional Theory and Dynamical Mean-Field Theory (DFT+DMFT) framework. However, the computational bottle-neck lies in the impurity solver. This work focuses on rigorously benchmarking the Multi-Orbital Iterated Perturbation Theory (MO-IPT) impurity solver against the numerically exact Continuous-Time Quantum Monte Carlo (CTQMC) method, specifically through a detailed study of the three-orbital Hubbard model. The two-orbital model serves as a canonical system to test performance under realistic multi-band conditions. Our systematically derived MO-IPT overcomes the critical limitations of CTQMC, including exponential scaling with orbital number, the need for analytic continuation, and the severe fermionic sign problem. By demonstrating high accuracy across varying interaction strengths and temperatures, we validate MO-IPT as an inexpensive, yet reliable, replacement solver essential for future material-specific investigations.
Although several impurity solvers in the dynamical mean field theory (DMFT) have been proposed, especially in multi-degree-of-freedom systems, there are practical difficulties arising from a trade-off between numerical costs and reliability. In this study, we re-interpret the iterative perturbation theory (IPT) from the perspective of the frequency dependence, and extend it such that costs and reliability can be compatible. From the same perspective, we also develop the methods in which the dual fermion calculation can be performed efficiently and can be combined with any impurity solver. We apply these methods to several models to evaluate their validity. We confirm that our methods show good agreements with the numerically exact continuous-time quantum Monte Carlo (CT-QMC) method in the local DMFT calculation, and can also capture important behaviors such as the pseudo gap in the DMFT + non-local calculation. In addition, the numerical costs are kept sufficiently small.
Abstract We provide a mathematical analysis of the Dynamical Mean-Field Theory (DMFT), a celebrated representative of a class of approximations in quantum mechanics known as embedding methods . Within the simplest setting of this approximation, which we introduce briefly at this point, we prove that, under certain assumptions, the DMFT equations admit a solution for any set of physical parameters. Moreover, we establish some properties of the solution(s). For the reader’s convenience, we provide in appendix a pedagogical and self-contained mathematical formulation of the DMFT equations for the finite Hubbard model. After recalling the definition and properties of one-body time-ordered Green’s functions and self-energies, and the mathematical structure of the Hubbard and Anderson impurity models, we describe the specific impurity solver we have studied in this article, namely the Iterated Perturbation Theory (IPT) solver, which can be conveniently formulated using Matsubara’s Green’s functions.
We propose a fast multi-orbital impurity solver for dynamical mean field theory (DMFT). Our DMFT solver is based on the equations of motion (EOMs) for local Green's functions and is constructed by generalizing from the single-orbital case to the multi-orbital case with the inclusion of the inter-orbital hybridizations and applying a mean field approximation to the inter-orbital Coulomb interactions. The two-orbital Hubbard model is studied using this impurity solver within a large range of parameters. The Mott metal-insulator transition and the quasiparticle peak are well described. A comparison of the EOM method with the quantum Monte Carlo method is made for the two-orbital Hubbard model and good agreement is obtained. The developed method hence holds promise as a fast DMFT impurity solver in studies of strongly correlated systems.
In a previous contribution (E. Cancès, A. Kirsch and S. Perrin--Roussel, arXiv:2406.03384), we have proven the existence of a solution to the Dynamical Mean-Field Theory (DMFT) equations under the Iterated Perturbation Theory (IPT-DMFT) approximation. In view of numerical simulations, these equations need to be discretized. In this article, we are interested in a discretization of the \acrshort{ipt}-\acrshort{dmft} functional equations, based on the restriction of the hybridization function and local self-energy to a finite number of points in the upper half-plane (iωn)n∈∣[0,Nω]∣\left(iω_n\right)_{n \in |[0,N_ω]|}(iωn​)n∈∣[0,Nω​]∣​, where ωn=(2n+1)π/βω_n=(2n+1)π/ βωn​=(2n+1)π/β is the nnn-th Matsubara frequency and Nω∈NN_ω\in \mathbb NNω​∈N. We first prove the existence of solutions to the discretized equations in some parameter range depending on NωN_ωNω​. We then prove uniqueness for a smaller range of parameters. We also study more in depth the case of bipartite systems exhibiting particle-hole symmetry. In this case, the discretized IPT-DMFT equations have purely imaginary solutions, which can be obtained by solving a real algebraic system of (Nω+1)(N_ω+1)(Nω​+1) equations with (Nω+1)(N_ω+1)(Nω​+1) variables. We provide a complete characterization of the solutions for Nω=0N_ω=0Nω​=0 and some results for Nω=1N_ω=1Nω​=1 in the simple case of the Hubbard dimer. We finally present some numerical simulations on the Hubbard dimer.