Machine learning-based compression of quantum many body physics: PCA and autoencoder representation of the vertex function

The vertex function, a continuous function of three momenta describing particle-particle scattering that is typically obtained by sophisticated calculations, plays a central role in the Feynman diagram approach to quantum many-body physics. Here, we use Principal Component Analysis (PCA) and a deep convolutional autoencoder to derive compact, low-dimensional representations of the vertex functions derived using the functional renormalization group for the two dimensional Hubbard model, a paradigmatic theoretical model of interacting electrons on a lattice. Both methodologies successfully reduced the dimensionality to a mere few dimensions while preserving accuracy. PCA demonstrated superior performance in dimensionality reduction compared to the autoencoder. The results suggest the presence of a fundamental underlying structure in the vertex function and suggest paths to dramatically reducing the computational complexity of quantum many-body calculations.