Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2406.03384 · Communications in Mathematical Physics

A Mathematical Analysis of IPT-DMFT

Eric Cancès B, Alfred Kirsch, Solal Perrin-Roussel

Published Jun 6, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s00220-026-05647-9

Related papers

A mathematical analysis of the discretized IPT-DMFT equations

Eric Cancès B, Kirsch, A., Solal Perrin-Roussel

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 $\left(iω_n\right)_{n \in |[0,N_ω]|}$, where $ω_n=(2n+1)π/ β$ is the $n$-th Matsubara frequency and $N_ω\in \mathbb N$. We first prove the existence of solutions to the discretized equations in some parameter range depending on $N_ω$. 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 $(N_ω+1)$ equations with $(N_ω+1)$ variables. We provide a complete characterization of the solutions for $N_ω=0$ and some results for $N_ω=1$ in the simple case of the Hubbard dimer. We finally present some numerical simulations on the Hubbard dimer.

arXiv:2505.21287•May 27, 2025•0 citations

Development of an efficient impurity solver in dynamical mean field theory for multiband systems: Iterative perturbation theory combined with parquet equations

Ryota Mizuno, Masayuki Ochi, Kazuhiko Kuroki

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.

arXiv:2101.04466•Physical review. B./Physical review. B•Jul 30, 2021•6 citations

Sum-over-poles framework for quantum embedding in dynamical mean-field theory

Carbone, Alessandro

This thesis develops a theoretical framework for quantum embedding based on compact sum-over-poles (SOP) representations of Green's functions, with a particular focus on dynamical mean-field theory (DMFT) formulated close to the real-frequency axis. The work starts from the algorithmic inversion method (AIM-SOP) and generalizes it to many-body lattice systems, proving that any interacting fragment embedded in an arbitrary bath can be represented by an auxiliary Hamiltonian with a finite set of non-interacting bath sites. The author embeds this formalism into a DMFT self-consistency loop formulated on complex contours close to the real axis, complemented by a SOP-based fitting strategy and exact-diagonalization (ED) impurity solvers, constructing a DMFT scheme that does not rely on approximate analytic continuation techniques. In particular, the framework enables us to work on compact (SOP) representations of Green's functions and self-energies alone, and leads to a frequency-independent formulation of DMFT, grounded in effective free energy (Baym-Kadanoff and Klein) functionals. Applications to a $1000$-site one-dimensional Hubbard ring demonstrate that compact SOP forms reproduce state-of-the-art ED-DMFT results on the Matsubara frequencies and retain key spectral features as the contour approaches the real axis, while also revealing convergence trends and multiple-solution behavior near challenging correlation regimes. Beyond its numerical validation, this approach paves the way for novel variational implementations of DMFT and realistic applications to materials, harnessing its promising ability to resolve fine spectral features.

•Jan 1, 2026•0 citations

Two-particle correlations and the metal-insulator transition: Iterated perturbation theory revisited

Erik G. C. P. van Loon

Recent advances in many-body physics have made it possible to study correlated electron systems at the two-particle level. In dynamical mean-field theory (DMFT), it has been shown that the metal-insulator phase diagram is closely related to the eigenstructure of the susceptibility. So far, this situation has been studied using accurate but numerically expensive solvers. Here, the iterated perturbation theory (IPT) approximation is used instead. Its simplicity makes it possible to obtain analytical results for the two-particle vertex and the DMFT Jacobian. The limited computational cost also enables a detailed comparison of analytical expressions for the response functions to results obtained using finite differences. At the same time, the approximate nature of IPT precludes an interpretation of the metal-insulator transition in terms of a Landau free energy functional.

arXiv:2110.11116•Physical review. B./Physical review. B•Jun 6, 2022•16 citations

Exact diagonalization as an impurity solver in dynamical mean field theory

Yi Lu, M. W. Haverkort

The dynamical mean-field theory (DMFT) maps a correlated lattice problem onto an impurity problem of a single correlated site coupled to an uncorrelated bath. Most implementations solve the DMFT equations using quantum Monte-Carlo sampling on the imaginary time and frequency (Matsubara) axis. We will here review alternative methods using exact diagonalization, i.e., representing the many-body ground state of the impurity as a sum over Slater determinants and calculating Green’s functions using iterative Lanczos procedures. The advantage being that these methods have no sign problem, can handle involved multi-orbital Hamiltonians (low crystal symmetry, spin-orbit coupling) and – when working completely on the real axis – do not need a mathematically ill-posed analytical continuation. The disadvantage of traditional implementations of exact diagonalization has been the exponential scaling of the calculation problem as a function of number of bath discretization points. In the last part we will review how recent advances in exact diagonalization can evade the exponential barrier thereby increasing the number of bath discretization points to reach the thermodynamic limit.

The European Physical Journal Special Topics•Jul 1, 2017•17 citations

Dynamical Mean-Field Theory Studies of Correlated Electrons

Yang Yu

Correlated electron systems exhibit a variety of intriguing emergent phenomena, including Mott physics, pseudogap behavior, and unconventional superconductivity. One of the most widely used numerical approaches for investigating these systems is dynamical mean-field theory (DMFT). This dissertation addresses two related problems: how to develop better quantum impurity solvers for DMFT and how to use DMFT results to gain a better understanding of correlated electron systems. The first part of this dissertation develops numerical methods for solving quantum impurity problems, which constitute the central computational task in DMFT calculations. In Chap. 4, we develop a method that combines a tensor-train numerical integration scheme with the inchworm hybridization expansion for quantum impurity problems. Benchmarks on spinless, single-orbital, and multi-orbital impurity models demonstrate the convergence of the Green’s functions to exact or reference results and identify the regimes in which tensor-train compression of Feynman diagrams and the inchworm expansion itself are effective. In Chap. 5, we introduce the concept of counterterms within the hybridization expansion and apply it to a bare hybridization expansion solver that relies on a sum-of-exponentials integration technique. Tests on spinless, multi-orbital, and realistic embedding impurity problems show that counterterms can significantly reduce the required perturbative order and improve the efficiency of hybridization-expansion calculations. The second part of this dissertation applies the cluster extension of DMFT to the single-band Hubbard model. In Chap. 6, we introduce a fluctuation decomposition for the self-energy that eliminates the Fierz ambiguity encountered in conventional fluctuation diagnostics of the Hubbard model. The resulting analysis shows that spin fluctuations make an important contribution to the pseudogap self-energy, while multi-boson contributions are also necessary for a complete description. In Chap. 7, we analyze the spin-fermion coupling and its role in d-wave pairing. Our calculations show that, for antinodal fermions in the pseudogap regime, their coupling to antiferromagnetic spin fluctuations is strongly enhanced relative to the non-interacting limit. A modified Bethe–Salpeter equation analysis further shows that this enhanced spin–fermion coupling boosts the spin-fluctuation-induced contribution to the d-wave pairing susceptibility.

•Jan 1, 2026•0 citations
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.
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.
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.
This thesis develops a theoretical framework for quantum embedding based on compact sum-over-poles (SOP) representations of Green's functions, with a particular focus on dynamical mean-field theory (DMFT) formulated close to the real-frequency axis. The work starts from the algorithmic inversion method (AIM-SOP) and generalizes it to many-body lattice systems, proving that any interacting fragment embedded in an arbitrary bath can be represented by an auxiliary Hamiltonian with a finite set of non-interacting bath sites. The author embeds this formalism into a DMFT self-consistency loop formulated on complex contours close to the real axis, complemented by a SOP-based fitting strategy and exact-diagonalization (ED) impurity solvers, constructing a DMFT scheme that does not rely on approximate analytic continuation techniques. In particular, the framework enables us to work on compact (SOP) representations of Green's functions and self-energies alone, and leads to a frequency-independent formulation of DMFT, grounded in effective free energy (Baym-Kadanoff and Klein) functionals. Applications to a 100010001000-site one-dimensional Hubbard ring demonstrate that compact SOP forms reproduce state-of-the-art ED-DMFT results on the Matsubara frequencies and retain key spectral features as the contour approaches the real axis, while also revealing convergence trends and multiple-solution behavior near challenging correlation regimes. Beyond its numerical validation, this approach paves the way for novel variational implementations of DMFT and realistic applications to materials, harnessing its promising ability to resolve fine spectral features.
Recent advances in many-body physics have made it possible to study correlated electron systems at the two-particle level. In dynamical mean-field theory (DMFT), it has been shown that the metal-insulator phase diagram is closely related to the eigenstructure of the susceptibility. So far, this situation has been studied using accurate but numerically expensive solvers. Here, the iterated perturbation theory (IPT) approximation is used instead. Its simplicity makes it possible to obtain analytical results for the two-particle vertex and the DMFT Jacobian. The limited computational cost also enables a detailed comparison of analytical expressions for the response functions to results obtained using finite differences. At the same time, the approximate nature of IPT precludes an interpretation of the metal-insulator transition in terms of a Landau free energy functional.
The dynamical mean-field theory (DMFT) maps a correlated lattice problem onto an impurity problem of a single correlated site coupled to an uncorrelated bath. Most implementations solve the DMFT equations using quantum Monte-Carlo sampling on the imaginary time and frequency (Matsubara) axis. We will here review alternative methods using exact diagonalization, i.e., representing the many-body ground state of the impurity as a sum over Slater determinants and calculating Green’s functions using iterative Lanczos procedures. The advantage being that these methods have no sign problem, can handle involved multi-orbital Hamiltonians (low crystal symmetry, spin-orbit coupling) and – when working completely on the real axis – do not need a mathematically ill-posed analytical continuation. The disadvantage of traditional implementations of exact diagonalization has been the exponential scaling of the calculation problem as a function of number of bath discretization points. In the last part we will review how recent advances in exact diagonalization can evade the exponential barrier thereby increasing the number of bath discretization points to reach the thermodynamic limit.
Correlated electron systems exhibit a variety of intriguing emergent phenomena, including Mott physics, pseudogap behavior, and unconventional superconductivity. One of the most widely used numerical approaches for investigating these systems is dynamical mean-field theory (DMFT). This dissertation addresses two related problems: how to develop better quantum impurity solvers for DMFT and how to use DMFT results to gain a better understanding of correlated electron systems. The first part of this dissertation develops numerical methods for solving quantum impurity problems, which constitute the central computational task in DMFT calculations. In Chap. 4, we develop a method that combines a tensor-train numerical integration scheme with the inchworm hybridization expansion for quantum impurity problems. Benchmarks on spinless, single-orbital, and multi-orbital impurity models demonstrate the convergence of the Green’s functions to exact or reference results and identify the regimes in which tensor-train compression of Feynman diagrams and the inchworm expansion itself are effective. In Chap. 5, we introduce the concept of counterterms within the hybridization expansion and apply it to a bare hybridization expansion solver that relies on a sum-of-exponentials integration technique. Tests on spinless, multi-orbital, and realistic embedding impurity problems show that counterterms can significantly reduce the required perturbative order and improve the efficiency of hybridization-expansion calculations. The second part of this dissertation applies the cluster extension of DMFT to the single-band Hubbard model. In Chap. 6, we introduce a fluctuation decomposition for the self-energy that eliminates the Fierz ambiguity encountered in conventional fluctuation diagnostics of the Hubbard model. The resulting analysis shows that spin fluctuations make an important contribution to the pseudogap self-energy, while multi-boson contributions are also necessary for a complete description. In Chap. 7, we analyze the spin-fermion coupling and its role in d-wave pairing. Our calculations show that, for antinodal fermions in the pseudogap regime, their coupling to antiferromagnetic spin fluctuations is strongly enhanced relative to the non-interacting limit. A modified Bethe–Salpeter equation analysis further shows that this enhanced spin–fermion coupling boosts the spin-fluctuation-induced contribution to the d-wave pairing susceptibility.