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

Abstract: Characterizing complex many-body phases of matter has been a central question in quantum physics for decades. Numerical methods built around approximations of the renormalization group (RG) flow equations have offered reliable and systematically improvable answers to the initial question -- what simple physics drives quantum order and disorder? The flow equations are a very high dimensional set of coupled nonlinear equations whose solution is the two particle vertex function, a function of three continuous momenta that describes particle-particle scattering and encodes much of the low energy physics including whether the system exhibits various forms of long ranged order. In this work, we take a simple and interpretable data-driven approach to the open question of compressing the two-particle vertex. We use PCA and an autoencoder neural network to derive compact, low-dimensional representations of underlying physics for the case of interacting fermions on a lattice. We quantify errors in the representations by multiple metrics and show that a simple linear PCA offers more physical insight and better out-of-distribution (zero-shot) generalization than the nominally more expressive nonlinear models. Even with a modest number of principal components (10 - 20), we find excellent reconstruction of vertex functions across the phase diagram. This result suggests that many other many-body functions may be similarly compressible, potentially allowing for efficient computation of observables. Finally, we identify principal component subspaces that are shared between known phases, offering new physical insight.