Deep Learning the Functional Renormalization Group

TL;DR

This paper introduces a deep learning approach using Neural ODEs to efficiently model the complex fRG flow in the 2D Hubbard model, enabling compact representations of electron interactions.

Contribution

It presents a novel application of neural ODEs and dynamic mode decomposition to reduce the dimensionality of fRG data for correlated electron systems.

Findings

Deep learning accurately captures fRG dynamics.

A small number of modes suffices to describe the system.

AI methods can extract compact representations of vertex functions.

Abstract

We perform a data-driven dimensionality reduction of the scale-dependent 4-point vertex function characterizing the functional Renormalization Group (fRG) flow for the widely studied two-dimensional $t - t'$ Hubbard model on the square lattice. We demonstrate that a deep learning architecture based on a Neural Ordinary Differential Equation solver in a low-dimensional latent space efficiently learns the fRG dynamics that delineates the various magnetic and $d$-wave superconducting regimes of the Hubbard model. We further present a Dynamic Mode Decomposition analysis that confirms that a small number of modes are indeed sufficient to capture the fRG dynamics. Our work demonstrates the possibility of using artificial intelligence to extract compact representations of the 4-point vertex functions for correlated electrons, a goal of utmost importance for the success of cutting-edge quantum field theoretical methods for tackling the many-electron problem.