Computation Kernel for Feynman Diagrams

TL;DR

The paper introduces a general, problem-agnostic computation kernel for finite-temperature Feynman diagrams by combining the discrete Lehmann representation (DLR) with Algorithmic Matsubara Integration (AMI). The Green's function is projected into a fixed auxiliary pole basis, yielding a kernel $\mathcal{K}^{\mathcal{D}}_{i\nu_x,\beta}(\{\ell\})$ that encodes all frequency/temperature dependence, while diagram-specific spatial information is captured by coefficients $C^{\mathcal{D}}_{\beta}(\{\ell\})$, so any diagram can be evaluated as a dot product over the discrete pole indices. The authors demonstrate the approach on two model Hamiltonians (a two-band impurity and the 2D Hubbard lattice) by constructing the second-order self-energy kernel and showing that the same kernel suffices to solve distinct physical problems, highlighting the kernel’s universality and reuse potential. The framework promises scalable, self-consistent perturbative calculations by precomputing kernels once and reusing them across problems, with practical storage costs and potential integration with tensor-network techniques and large quantitative models.

Abstract

We present a general representation for solving problems in many-body perturbation theory. By projecting the single-particle Green's function to an auxiliary space we show how one can convert an arbitrary Feynman graph to a universal kernel representation. Once constructed, the computation kernel contains no problem specific information yet contains all explicit temperature and frequency dependence of the diagram. This computation kernel is problem agnostic, and valid for any physical problem that would normally leverage the Matsubara formalism of many-body perturbation theory. The result of any diagram can be written as a linear combination of these computation kernel elements with coefficients given by a sum over products of known tensor elements that are themselves problem specific and represent spatial degrees of freedom. We probe the efficacy of this approach by generating the computation kernel for a low order self-energy diagram which we then use to construct solutions to distinct problems.

Figures (3)

• Figure 1: Schematic representation of applications of a discrete Lehmann representation to increasingly complex datasets. From top to bottom these represent: an impurity Green's function, a diagonal Green's function, a matrix-valued Green's function, and a collection of Green's functions (as appear in Feynman diagrams). On the left of each schematic is a physical space of states while on the right is a particular basis of DLR poles, $x_\ell$.
• Figure 2: Labelled second order self-energy diagram, $\Sigma^{(2)}$. The diagram has external frequency, $i\nu_x$, and external band-indices $a_x$ and $b_x$. Internal band indices are arbitrarily assigned to simplify the form of the interaction matrix elements.
• Figure 3: The imaginary part of the second order self energy for the first 10 Matsubara frequencies at $\beta=5$ utilizing a precomputed computation kernel for the diagram. (a) The 2D Hubbard result for external momentum $k_x=(\pi,\pi)$ at $U/t=1$ for the $L\times L$ system with $L=11,21$, and 31. Also included is a benchmark reference for the $L\to\infty$ limit. (b) The H$_2$ molecule for the sto-6G basis at equilibrium separation $r=0.74$Å which exhibits only diagonal elements of the self-energy.