Data mining the many-body problem

Many-body systems are typically characterised via low-order correlation functions, that are directly related to response functions. In this talk, I will show how it is possible to provide a characterisation of many-body systems via a direct and assumption-free data mining of one of  the pillars of both classical and quantum statistical mechanics - the partition function. The core idea of this programme is the fact that, once sampled stochastically (such as in experiments or Monte Carlo simulations), partitions functions can be construed as a very high dimensional manifold. The topology of such manifold can be characterised via basic concepts, in particular, by their intrinsic dimension. I will discuss theoretical results for both classical and quantum many-body spin systems that illustrate how data structures undergo structural transitions whenever the underlying physical system does, and display universal (critical) behavior in both classical and quantum mechanical cases. I will conclude with remarks on the applicability of our theoretical framework to synthetic quantum systems, quantum computing architectures, and lattice gauge theories.