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arXiv:2610.28472

Sum-over-poles framework for quantum embedding to approach the real-frequency axis

Alessandro Carbone, Massimo Capone, Nicola Marzari, Tommaso Chiarotti

Published Oct 20, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2610.28472

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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.
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.
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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.
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