Machine Learning Holography in Neural Network Renormalization Group

Previously, people have shown the close relations between renormalization group(RG) with both deep learning and holographic duality, and how holographic geometry can emerge from deep learning the entanglement feature of a quantum many-body state. Inspired by those, we propose any boundary conformal field theory can be mapped into its holographic bulk to an operator level. In our framework, renormalization group is constructed as a hierarchical unsupervised generative model. Coarse graining direction can be viewed as an emergent direction, and it pushes boundary field theory configurations to bulk field configurations. The inverse coarse graining direction generates boundary field configurations from bulk noises. The goal is to construct optimal RG that makes bulk variables as uncorrelated as possible. The leftover of correlations between bulk variables can be used to define measure of distance in the bulk. We studied two dimensional interacting bosonic system as a boundary field theory. RG network is trained to find the effective bulk field theory and we observed the emergence of hyperbolic geometry(AdS3 spatial geometry) as we tuned system towards critical point.