  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.
