Machine Learning Holographic Mapping by Neural Network Renormalization Group

Mar 2, 2019
12 pages
Published in:
  • Phys.Rev.Res. 2 (2020) 2, 023369
  • Published: Jun 20, 2020
e-Print:

Citations per year

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Abstract: (APS)
Exact holographic mapping (EHM) provides an explicit duality map between a conformal field theory (CFT) configuration and a massive field propagating on an emergent classical geometry. However, designing the optimal holographic mapping is challenging. Here we introduce the neural network renormalization group as a universal approach to design generic EHM for interacting field theories. Given a field theory action, we train a flow-based hierarchical deep generative neural network to reproduce the boundary field ensemble from uncorrelated bulk field fluctuations. In this way, the neural network develops the optimal renormalization-group transformations. Using the machine-designed EHM to map the CFT back to a bulk effective action, we determine the bulk geodesic distance from the residual mutual information. We have shown that the geometry measured in this way is the classical saddle-point geometry. We apply this approach to the complex ϕ4 theory in two-dimensional Euclidian space-time in its critical phase, and show that the emergent bulk geometry matches the three-dimensional hyperbolic geometry when geometric fluctuation is neglected.
Note:
  • 9 pages, 7 figures + appendix
  • 05.10.Cc
  • 11.25.Hf
  • 04.62.+v
  • field theory: conformal
  • renormalization group: transformation
  • field theory: interaction
  • dimension: 3
  • dimension: 2
  • neural network
  • holography