Automated evaluation of imaginary time strong coupling diagrams by sum-of-exponentials hybridization fitting

TL;DR

This work presents a deterministic, low-cost solver for high-order imaginary-time bold strong-coupling diagrams in DMFT by representing the hybridization function Δ with a tailored sum-of-exponentials (SOE) via the AAA rational approximation and a bilevel pole refinement. The method enables automatic evaluation of arbitrary-order diagrams for multi-orbital systems, significantly reducing computational cost compared to prior approaches and stabilizing self-consistent solutions at low temperatures. Demonstrations on a fermionic dimer, a two-band e_g model, and a Ca$_2$RuO$_4$ DMFT study show major speedups and accurate, order-by-order convergence, including SOC-induced anisotropy and AFM order; the approach enables low-temperature, multi-orbital impurity problems that are challenging for stochastic solvers. The work also provides detailed algorithmic steps for diagram decomposition, backbone evaluation, and complexity analyses, highlighting potential extensions to real-time diagrammatics and ab-initio DMFT workflows.

Abstract

We present an efficient separation of variables algorithm for the evaluation of imaginary time Feynman diagrams appearing in the bold pseudo-particle strong coupling expansion of the Anderson impurity model. The algorithm uses a fitting method based on AAA rational approximation and numerical optimization to obtain a sum-of-exponentials expansion of the hybridization function, which is then used to decompose the diagrams. A diagrammatic formulation of the algorithm leads to an automated procedure for diagrams of arbitrary order and topology. We also present methods of stabilizing the self-consistent solution of the pseudo-particle Dyson equation. The result is a low-cost and high-order accurate impurity solver for quantum embedding methods using general multi-orbital hybridization functions at low temperatures, appropriate for low-to-intermediate expansion orders. In addition to other benchmark examples, we use our solver to perform a dynamical mean-field theory study of a minimal model of the strongly correlated compound Ca$_2$RuO$_4$, describing the anti-ferromagnetic transition and the in- and out-of-plane anisotropy induced by spin-orbit coupling.

Figures (7)

• Figure 1: Fitting of hybridization functions (second row) generated from three spectral densities (first row)---sum of $\delta$-functions (left), semicircle (middle), and sum of Gaussians (right)--- for several inverse temperatures. Fitting errors \ref{['eq:l2err']} versus the number of poles $p$ are shown in the third row. We observe essentially exact fitting for discrete spectra, and systematic convergence for continuous spectra.
• Figure 2: (Left) Number of poles $p$ required by our hybridization fitting algorithm with tolerance $10^{-6}$ vs. inverse temperature $\beta$, for semicircle and sum of Gaussians spectra, a many-orbital random pole model ($p$ saturates to the plotted value as the number of random orbitals is increased; see bottom-right panel), as well as the number of DLR basis functions for equivalent parameters. (Right) Number of poles $p$ required to achieve accuracy $10^{-6}$ vs. actual number of poles $N$ in a random pole model, for single-orbital (top) and multi-orbital (bottom) cases with $\beta = 100$ (data for 1200 random experiments each).
• Figure 3: Single-particle Green's function for the fermionic spinless dimer model \ref{['eq:dimer']}, at several temperatures and strong coupling expansion orders, along with pointwise errors.
• Figure 4: (a) Wall clock timings for a single evaluation of all pseudo-particle self-energy ($\Sigma$) and single-particle Green's function ($G$) diagrams at a given order, for inverse temperature $\beta = 16$. The expected scaling is $\mathcal{O}(C(m) 2^{m-1}(np)^{m-1} (m r^2 N^3 + n r N^3))$, where $C(m)$ is the number of diagram topologies at expansion order $m$, and we choose the prefactor to match the $m=1$ data point. (b) $L^2$ error \ref{['eq:l2err']} of $G$ versus $m$ at various temperatures.
• Figure 5: Single particle Green's function for the two-band $e_g$ model with a discrete bath at several temperatures and strong coupling expansion orders, along with pointwise errors.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3