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

Carbone, Alessandro

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5075/epfl-thesis-11552

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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.
We introduce a real-frequency framework for the quantum embedding of solids, based on sum-over-poles (SOP) representations. We partition the lattice into periodically repeated interacting fragments and derive an embedded Dyson equation for the fragment, to be solved self-consistently. The resulting scheme generalizes the algorithmic-inversion method of our previous work to interacting Hamiltonians and is closely related to dynamical mean-field theory (DMFT), especially when an extended impurity problem is introduced. By exploiting the compact pole expansion of the embedding potential, we carry out the self-consistent procedure close to the real axis, which avoids ill-posed analytic continuation. In addition, the SOP representation allows us to introduce a pole-and-residue dynamics, which we use to reach the self-consistent solution. After deriving the method, we benchmark it against exact-diagonalization implementations of DMFT on the one-dimensional Hubbard ring at half filling. The framework offers a natural route to real-space dynamical quantum embedding, for periodic systems as well as for strongly correlated defects in materials.
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 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.
We propose a method for estimating smooth real-frequency self-energy in the dynamical mean-field theory with the finite-temperature exact diagonalization (DMFT-ED). One of the benefits of DMFT-ED calculations is that one can obtain real-frequency spectra without a numerical analytic continuation. However, these spectra are spiky and strongly depend on the way to discretize a continuous bath (e.g., the number of the bath sites). The present scheme is based on a recently proposed compact representation of imaginary-time Green's functions, the intermediate representation (IR). The projection onto the IR basis acts as a noise filter for the discretization errors in the self-energy. This enables to extract the physically relevant part from the noisy self-energy. We demonstrate the method for single-site DMFT calculations of the single-band Hubbard model. We also show the results can be further improved by numerical analytic continuation.
Dynamical mean-field theory (DMFT) is a widely used, nonperturbative computational framework for strongly correlated electron systems. This thesis extends DMFT in two directions: treating disorder and higher-order correlators away from equilibrium, and using near-term quantum computers as impurity solvers. In the first part, we develop two methodological advances for the nonequilibrium DMFT treatment of disordered, interacting systems. The first overcomes the central computational bottleneck of the recently developed nonequilibrium DMFT plus coherent potential approximation (DMFT+CPA) by replacing its explicit numerical disorder averaging with an exact analytical integration, exploiting a normalization property of the Schwinger-Keldysh contour; applied to a disorder-and-interaction quench of the Anderson-Hubbard model, this reveals that weak random disorder promotes thermalization. The second demonstrates that the out-of-time-order correlator (OTOC) can be successfully computed within the existing DMFT+CPA framework, by generalizing the Schwinger-Keldysh contour to a double-folded structure; as a proof-of-principle demonstration, weak disorder is found to accelerate the OTOC's decay in the half-filled Anderson-Hubbard model in the intermediate-coupling regime. In the second part, we explore the use of near-term quantum computers as DMFT impurity solvers. A stable real-time DMFT iteration scheme, designed for compatibility with quantum hardware, is first developed and validated classically against the metal-to-insulator transition. A symmetry-adapted variational quantum eigensolver (VQE) is then benchmarked as the Anderson impurity solver for the converged DMFT Hamiltonian, and is found to reproduce ground-state energies accurately and to extract the single-particle Green's function reliably at intermediate to strong coupling, while accurate ground-state energies are shown not to guarantee accurate dynamics at weak coupling, a key challenge for extending such approaches to correlated metals. Together, we demonstrate that DMFT can be methodologically extended to study disordered and interacting systems away from equilibrium, and to incorporate near-term quantum hardware as a genuinely new computational resource to enhance its capabilities.
We demonstrate an algebraic construction of frequency-dependent bath orbitals which can be used in a robust and rigorously self-consistent DMFT-like embedding method, here called $\ensuremath{\omega}$-DMFT, suitable for use with Hamiltonian-based impurity solvers. These bath orbitals are designed to exactly reproduce the hybridization of the impurity to its environment, while allowing for a systematic expansion of this bath space as impurity interactions couple frequency points. In this way, the difficult nonlinear fit of bath parameters necessary for many Hamiltonian-formulation impurity solvers in DMFT is avoided, while the introduction of frequency dependence in this bath space is shown to allow for more compact bath sizes. This has significant potential use with a number of new, emerging Hamiltonian solvers which allow for the embedding of large impurity spaces within a DMFT framework. We present results of the $\ensuremath{\omega}$-DMFT approach for the Hubbard model on the Bethe lattice, a one-dimensional chain, and the two-dimensional square lattice, which show excellent agreement with standard DMFT results, with fewer bath orbitals and more compact support for the hybridization representation in the key impurity model of the method.