Calculating Feynman diagrams with matrix product states

TL;DR

This work presents a non‑Monte‑Carlo, tensor‑network–based route to computing out‑of‑equilibrium Feynman diagrams for quantum nanoelectronics. By expanding observables in the interaction strength $U$, it reduces the high‑dimensional integrals to structured forms that are efficiently handled by Tensor Cross Interpolation (TCI) and related matrix factorization techniques, followed by cross‑extrapolation to reconstruct the full $Q(U,t)$. The approach is demonstrated on the out‑of‑equilibrium single impurity Anderson model (SIAM), achieving accurate results for dot occupation and current, and exposing characteristic features such as Coulomb diamonds and the Kondo ridge with controlled error bars. The methodology leverages non‑interacting Green's functions, Wick determinants, and a scalable, modular implementation, opening routes to treat other correlated, out‑of‑equilibrium quantum impurity problems with reduced sign problems and improved convergence. Overall, the work advances a practical, open‑source framework that can transform how high‑order diagrammatics are computed in nonequilibrium many‑body physics.

Abstract

This text reviews, hopefully in a pedagogical manner, a series of work on the automatic calculations of Feynman diagrams in the context of quantum nanoelectronics (Keldysh formalism) with an application to the Kondo effect in the out-of-equilibrium single impurity Anderson model. It includes a discussion of (A) how to deal with the proliferation of diagrams, (B) how to calculate them using the Tensor Cross Interpolation algorithm instead of Monte-Carlo and (C) how to resum the obtained series. These notes correspond to a lecture given at the Autumn School on Correlated Electrons 2025 in Jullich, Germany. The book with all the lectures of the school (edited by Eva Pavarini, Erik Koch, Alexander Lichtenstein, and Dieter Vollhardt) is available in open access.

Figures (10)

• Figure 1: Schematic of the overall approach. Starting from non interacting Green's functions, a decomposition of the multidimensional integral is obtained using Tensor Cross-Interpolation to compute the perturbative expansion, from which the physical observables $Q(U,t)$ are reconstructed as a function of interaction $U$ and time $t$ using cross-extrapolation. Adapted from jeannin2025
• Figure 2: Top: electrical diagram of the quantum dot coupled to two electrodes. Bottom: schematic of the SIAM. Adapted from jeannin2025.
• Figure 3: Left: Colorplot of the integrand of $Q_2$ as a function of the two times $u_1$ and $u_2$ for SIAM with $\mu_L=\mu_R=0$, $\epsilon_d=0$, $T=0$ and $t=10$. The four panels correspond to the 4 possible values of the two Keldysh indices $a_1$ and $a_2$. The explicit form of the integrand is $f(u_1,u_2,a_1,a_2)= -\Im m (-1)^{\sum_i a_i} \det \mathrm{{\textbf{M}}} _{2}(u_1,u_2,a_1,a_2)$. Right: Same parameters as on the left but the integrand has now been summed over Keldysh indices. The colorplot represents $f(u_1,u_2)= i \sum_{a_1,a_2} (-1)^{\sum_i a_i} \det \mathrm{{\textbf{M}}} _{2}(u_1,u_2,a_1,a_2)$ ($f$ is real). Note that the integrand is now real, positive and concentrated around $u_1 = u_2 = t$. Adapted from profumo2015
• Figure 4: Illustration of the cross interpolation (CI) of a matrix. The large red triangles indicate real pivots and the smaller red triangles indicate automatically generated pivots. The right-hand side only contains small subparts of the matrix. Adapted from nunezfernandez2022
• Figure 5: Error $|A_{ij}-[A_{\mathrm{CI}}]_{ij}|$ versus $i$ and $j$ at different stages of the Cross-Interpolation for a $M\times M$ matrix with $M=20$. In this toy example, $A_{ij}=\left(\frac{i/M}{i/M+1}\right)^{4}(1+e^{-(j/M)^{2}})\left[1+(j/M)\cos(j/M)e^{-(j/M)\frac{i/M}{(i/M)+1}}\right]$. The red dots indicate the pivots. The $x$ and $y$ axis have been rescaled to be in $[0,10]$. Adapted from Jeannin et al jeannin2025.