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arXiv:2609.26703

Real-Frequency Diagrammatics with Pole Representations

Rayan Farid, Lei Zhang, Aiman Al-Eryani, Agnieszka Jażdżewska, Emanuel Gull

Published Sep 22, 2026• 0 citations
arXivHTMLPDF

Abstract

Finite-temperature many-body theories are typically formulated in imaginary time, where they are amenable to efficient numerical treatment. However, the extraction of spectral properties from the imaginary axis is difficult and requires numerical analytic continuation, while a direct formulation on the real axis requires prohibitively expensive numerical quadrature. In this paper, we represent the objects of many-body perturbation theory using systematically improvable pole and moment expansions and develop the corresponding framework for evaluating diagrammatic contributions directly on the real axis. We introduce a controlled recompression procedure that prevents the proliferation of poles under algebraic and diagrammatic operations. We benchmark the approach on paradigmatic quantum impurity problems and apply it to self-consistent calculations of the uniform electron gas. The resulting spectral quantities converge systematically towards numerical accuracy. This work provides a practical framework for controlled and systematically improvable calculations of spectral quantities within diagrammatic many-body theories, including self-consistent conserving approaches.

DOI
10.48550/arxiv.2609.26703

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Finite-temperature many-body theories are typically formulated in imaginary time, where they are amenable to efficient numerical treatment. However, the extraction of spectral properties from the imaginary axis is difficult and requires numerical analytic continuation, while a direct formulation on the real axis requires prohibitively expensive numerical quadrature. In this paper, we represent the objects of many-body perturbation theory using systematically improvable pole and moment expansions and develop the corresponding framework for evaluating diagrammatic contributions directly on the real axis. We introduce a controlled recompression procedure that prevents the proliferation of poles under algebraic and diagrammatic operations. We benchmark the approach on paradigmatic quantum impurity problems and apply it to self-consistent calculations of the uniform electron gas. The resulting spectral quantities converge systematically towards numerical accuracy. This work provides a practical framework for controlled and systematically improvable calculations of spectral quantities within diagrammatic many-body theories, including self-consistent conserving approaches.
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This dissertation develops the complex pole representation as a unified framework for compactly representing Green's functions in quantum many-body simulations, with applications to numerical analytic continuation, hybridization fitting, compact representation and quantum impurity solvers, accompanied with software implementation. The central idea is to replace continuous spectral and correlation functions by a small number of complex poles and complex weights, while preserving the analytic structure needed for physical interpretation and numerical simulation. By connecting analytic continuation, hybridization fitting, compact representation, impurity solvers, and software implementation within a single pole-based perspective, this dissertation provides a systematically improvable approach to extracting, representing, and directly manipulating real-frequency information in quantum many-body simulations.
Analytic continuation is a central step in the simulation of finite-temperature field theories in which numerically obtained Matsubara data are continued to the real frequency axis for a physical interpretation. Numerical analytic continuation is considered to be an ill-posed problem where uncertainties on the Matsubara axis are amplified exponentially. Here, we present a systematic and controlled procedure that approximates any Matsubara function by a minimal pole representation to within a predefined precision. We then show a systematic convergence to the exact spectral function on the real axis as a function of our control parameter for a range of physically relevant setups. Our methodology is robust to noise and paves the way towards reliable analytic continuation in many-body theory and, by providing access to the analytic structure of the functions, a direct theoretical interpretation of physical properties.
Self-consistent Hartree-Fock approximation combined with solutions of the Bethe-Salpeter equation offers a powerful tool for studies of strong correlation effects arising in condensed matter models, nuclear physics, quantum field theories, and real materials. The standard finite-temperature approach would be to first solve the problem in the Matsubara representation and then apply numerical analytic continuation to the real-frequency axis to link theoretical results with experimental probes, but this ill-conditioned procedure often distorts important spectral features even for very accurate imaginary-frequency data. We demonstrate that the ladder-type finite-temperature Bethe-Salpeter equation in the Hartree-Fock basis for the three-point vertex function and, ultimately, system's polarization can be accurately solved directly on the real frequency axis using the diagrammatic Monte Carlo technique and series resummation. We illustrate the method by applying it to the homogeneous electron gas model and demonstrate how multiple scattering events renormalize Landau damping.
This thesis develops and applies numerical methods for computing correlated electrons at finite temperatures within perturbation theory, formulated in terms of Feynman diagrams. First we compute the fully renormalized effective interaction W in the 2D Hubbard model beyond the conventional low-order random phase approximation (RPA) and additionally resolve W into same-spin and opposite-spin channels. This analysis shows that starting from a purely local repulsive bare interaction U we get attractive effective nearest-neighbor interaction. We identify regimes of inverse temperature t and interaction U=t, where t denotes the nearest-neighbor hopping amplitude, in which this attraction appears. Such an emergent of attractive interaction is known to be able to promote superconductivity in the repulsive Hubbard model. Next, we introduce a method for incorporating higher-order diagrams within selfconsistent perturbative schemes. We merge algorithmic Matsubara integration (AMI) with discrete pole representations, including the discrete Lehmann representation and Prony (minimal pole) representation. AMI automates nested internal Matsubara frequency summations for arbitrary diagrams topologies through symbolic contour integration, which requires explicit access to the analytic pole structure of the input Green’s functions. By compressing the Green’s function into a small set of poles and corresponding weights, we regain access to its analytic pole structure, which allows us to iterate the calculation within self-consistent schemes. Benchmarks on the Bethe lattice demonstrate accurate single-shot and self-consistent self-energies on both Matsubara and real axes. For self-consistent calculations, we use two standard schemes, where AMI combined with the discrete pole representation works as a selfenergy and impurity solver. We further extend calculations to third and fourth order of interaction in self-energy diagrams. This approach yields the ability to perform self-consistent calculations at a fraction of the cost compared to other Matsubara-axis methods without moving to the real axis at every step of the iteration. Finally we demonstrate the universality of the AMI+DLR framework, by decomposing any diagram evaluation into a computational kernel that depends only on the diagram topology and DLR parameters, and a set of coefficients determined by the physical system. This separation enables precomputation and reuse of the computational kernel for a certain diagram topology and any physical problem, both in single-shot and iterative calculations. We illustrate the universality by reusing the same computational kernel for two simple systems: the 2D Hubbard square lattice and the H₂ molecule.
We develop a self-consistent spectral quadrature (sc-SQ) framework for the calculation of many-body Green functions. The method approximates the Källén--Lehmann spectral measure by Gauss--Christoffel (GC) quadrature, yielding a rational Green function representation with guaranteed spectral positivity that exactly reproduces the first 2N2N2N spectral moments at pole order NNN. A key component is an SVD-based rank-selection criterion on the Hankel matrix, which identifies the numerically resolvable pole rank N∗N^*N∗ from the singular-value gap and acts as a precision-guided diagnostic of correlation complexity. The scheme is made self-consistent by requiring that the spectral function used to evaluate expectation values coincides with the spectral function generated by the quadrature reconstruction. This defines a fixed-point hierarchy that connects systematically to established approximations, including Hartree--Fock and Hubbard-I, and incorporates non-perturbative features such as multi-peak spectral structure. We benchmark the approach for the Anderson impurity model against numerical renormalization group (NRG) results and apply it within dynamical mean-field theory for the Hubbard model on the Bethe lattice. The method captures the three-peak Anderson impurity spectrum and the suppression of quasiparticle weight in the half-filled Hubbard model on the Bethe lattice, including Mott-gap formation on the insulating branch for N⩾5N\geqslant 5N⩾5, in qualitative agreement with NRG references.