We take the tensor network describing explicit $p$-adic conformal field theory partition functions proposed in [L.-Y. Hung et al., J. High Energy Phys. 04 (2019) 170], and consider boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geometry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation function as the tensor network, at least in the perturbative limit order by order away from the pure BT tree. Moreover, the (perturbative) definition of the graph curvature in the mathematics [Y. Lin and S.-T. Yau, Tohoku Math. J. 63, 605 (2011); Y. Ollivier, J. Funct. Anal. 256, 810 (2009)] and physics [S. S. Gubser et al., J. High Energy Phys. 06 (2017) 157] literature naturally emerges from the consistency requirements of the emergent Einstein equation. This could provide new insights into the understanding of gravitational dynamics potentially encoded in more general tensor networks.

• Received 9 March 2021
• Accepted 5 October 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.221602

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society