Deep Learning the Functional Renormalization Group Flow for Correlated Fermions

We perform a data-driven dimensionality reduction of the one-particle irreducible 4-point vertex function at varying next-nearest-neighbor hopping t' for the example of the two-dimensional t-t' Hubbard model on the square lattice for particle densities close to the Van Hove filling. On the one hand, a non-linear spectral embedding that implements a Laplacian eigenmap and a spectral decomposition of the graph Laplacian is used to find low-dimensional representations preserving the closeness of the trajectories inherent the temperature flow of one-loop functional renormalization group (fRG). On the other hand, a Dynamic Mode Decomposition shows that a small number of normal modes is sufficient to capture the fRG dynamics. We then propose a deep learning architecture based on convolutional neural networks and a neural ordinary differential equation solver in a low-dimensional latent space, to efficiently learn a reduced order model fRG dynamics in the various magnetic and d-wave superconducting regimes of the Hubbard model. Our work puts forward the possibility of compact representations of 4-point vertex functions that are likely useful also to other vertex-based numerical methods.