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This study applies machine learning to reconstruct the emergent bulk geometry of the 3D Ising model, revealing that a single classical metric fails to simultaneously describe correlation functions for different boundary operators. The research demonstrates a fundamental tension between these operator channels, suggesting that theories with low central charge may require more complex geometric descriptions than standard AdS/CFT duality implies. By training differentiable solvers on Monte Carlo data, the authors provide numerical evidence that the bulk structure is not captured by a unified metric, advancing our understanding of how AI tools can probe deep theoretical physics constraints.
Learning the Emergent Bulk Geometry of the Three Dimensional Ising Model
arXiv:2609.20210v1 Announce Type: cross Abstract: We use Monte Carlo data for the three-dimensional Ising model at the critical temperature to predict an emergent radial geometry of the bulk using machine learning. Specifically, we employ a differentiable solver for a probe field on an asymptotically $\mathrm{AdS}_4$ background, trained to reproduce the measured momentum-space two-point functions of the boundary operators ($\sigma$ and $\epsilon$). We find that a single metric cannot simultaneously serve both operators. Fitting each channel separately improves its own residual by a factor of three to five, but degrades the other by up to two orders of magnitude. We quantify this tension between the two operators and show that it survives changes to the momentum window and the lattice size. Our results numerically demonstrate that the bulk geometry of a theory with an $\mathcal{O}(1)$ central charge, such as the 3D Ising model, is not captured by a single classical metric.
Abstract:We use Monte Carlo data for the three-dimensional Ising model at the critical temperature to predict an emergent radial geometry of the bulk using machine learning. Specifically, we employ a differentiable solver for a probe field on an asymptotically $\mathrm{AdS}_4$ background, trained to reproduce the measured momentum-space two-point functions of the boundary operators ($\sigma$ and $\epsilon$). We find that a single metric cannot simultaneously serve both operators. Fitting each channel separately improves its own residual by a factor of three to five, but degrades the other by up to two orders of magnitude. We quantify this tension between the two operators and show that it survives changes to the momentum window and the lattice size. Our results numerically demonstrate that the bulk geometry of a theory with an $\mathcal{O}(1)$ central charge, such as the 3D Ising model, is not captured by a single classical metric.
High Energy Physics - Theory (hep-th); Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech); Computational Physics (physics.comp-ph); Quantum Physics (quant-ph)
arXiv:2609.20210 [hep-th]
(or arXiv:2609.20210v1 [hep-th] for this version)
https://doi.org/10.48550/arXiv.2609.20210 arXiv-issued DOI via DataCite (pending registration)
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