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arXiv:2505.21287

A mathematical analysis of the discretized IPT-DMFT equations

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

Published May 27, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2505.21287

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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.
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.
Pseudogap physics in strongly correlated systems is essentially scale dependent. We generalize the dynamical mean field theory (DMFT) by including into the DMFT equations dependence on correlation length of pseudogap fluctuations via additional (momentum dependent) self-energy Sigma_k. This self-energy describes non-local dynamical correlations induced by short-ranged collective SDW-like antiferromagnetic spin (or CDW-like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with finite correlation length. This generalized DMFT+Sigma_k approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Coulomb interaction on a square lattice with nearest and next nearest neighbour hopping. The effective single impurity problem is solved by numerical renormalization group (NRG). Both types of strongly correlated metals, namely (i) doped Mott insulator and (ii) the case of bandwidth W
We present a time-domain iteration scheme for solving the Dynamical Mean-Field Theory (DMFT) self-consistent equations using retarded Green's functions in real time. Unlike conventional DMFT approaches that operate in imaginary time or frequency space, our scheme operates directly with real-time quantities. This makes it particularly suitable for near-term quantum computing hardware with limited Hilbert spaces, where real-time propagation can be efficiently implemented via Trotterization or variational quantum algorithms. We map the effective impurity problem to a finite one-dimensional chain with a small number of bath sites, solved via exact diagonalization as a proof-of-concept. The hybridization function is iteratively updated through time-domain fitting until self-consistency. We demonstrate stable convergence across a wide range of interaction strengths for the half-filled Hubbard model on a Bethe lattice, successfully capturing the metal-to-insulator transition. Despite using limited time resolution and a minimal bath discretization, the spectral functions clearly exhibit the emergence of Hubbard bands and the suppression of spectral weight at the Fermi level as interaction strength increases. This overcomes major limitations of two-site DMFT approximations by delivering detailed spectral features while preserving efficiency and compatibility with quantum computing platforms through real-time dynamics.
The convergence to the self-consistency in the dynamical mean-field theory (DMFT) calculations for models of correlated electron systems can be significantly accelerated by using an appropriate mixing of hybridization functions which are used as the input to the impurity solver. It is shown that the techniques and the past experience with the mixing of input charge densities in the density-functional theory calculations are also effective in DMFT. As an example, the increase in the computational requirements near the Mott metal-insulator transition in the Hubbard model due to critical slowing down can be strongly reduced by using the modified Broyden's method to numerically solve the nonlinear self-consistency equation. Speed-up factors as high as three were observed in practical calculations even for this relatively well-behaved problem. Furthermore, the convergence can be achieved in difficult cases where simple linear mixing is either not effective or even leads to divergence. Unstable and metastable solutions can also be obtained. We also determine the linear response of the system with respect to the variations in the hybridization function, which is related to the propagation of the information between the different energy scales during the iteration.
We investigate the electronic structure of the strongly correlated electron systems with an orbital degeneracy within the framework of the iterated-perturbation theory (IPT) (1, 2) of the dynamical mean fleld method. We consider the multi-orbital Hubbard model in the limit of inflnite lattice spatial dimensions. Our motivation comes from recent studies of the orbital efiects in strongly correlated elctron systems such as tran- sition metal oxides, strontium ruthenates, manganities, and organic metals. Almost all of these materials consist of valence bands coming from d shells, where intra- and inter- orbital Coulomb interactions are equally important. The important part of the paper is a detailed discussion on the problem of the analytical continuation of the complex frequency Matsubara Grenn function from the Matsubara points to the real frequency axis. The analytical continuation is a nessessary step in the numerical calculations if we are interested in calculating the physical quantities such as the density of states or transport quantities.