Calculating Feynman diagrams with matrix product states

TL;DR

This paper reviews a pedagogical approach to automating Feynman diagram calculations in quantum nanoelectronics using tensor network algorithms, offering an alternative to Monte Carlo methods and series resummation techniques.

Contribution

It introduces a novel application of the Tensor Cross Interpolation algorithm for calculating Feynman diagrams, improving efficiency over traditional Monte Carlo methods.

Findings

Tensor Cross Interpolation effectively computes Feynman diagrams.

The method handles diagram proliferation in quantum nanoelectronics.

Series resummation enhances the accuracy of calculations.

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.