Real-space mutual information neural estimation algorithm for single-step extraction of renormalisation group-relevant degrees of freedom

Deriving the emergent macroscopic properties of matter from microscopic models of interacting constituents is a perpetual theoretical challenge. In statistical physics a powerful framework for addressing it is provided by the renormalization group (RG), which associates physical theories at different scales by recursively combining local degrees of freedom. It has been proposed that an optimal RG rule for a given system maximises the real-space mutual information (RSMI). Here we show that the RSMI-optimal coarse-graining is a remarkable object itself, extracting the relevant degrees of freedom in a single step, instead of relying on the iterative nature of the RG procedure. It can be used to characterise spatial correlations, locate phase transitions and construct order parameters. In this sense it allows to extract and interpret low-energy physics at the outset of the RG flow. We develop an efficient numerical algorithm based on recent rigorous results on mutual information estimation with neural networks. We validate its capabilities on an interacting lattice dimer model with a nontrivial RG flow and discuss further applications, including in non-equilibrium problems. Our findings introduce a new conceptual paradigm and a numerical tool in investigating statistical systems.