Emergent Einstein Equation in p-adic Conformal Field Theory Tensor Networks

Lin Chen; Xirong Liu; Ling-Yan Hung

doi:10.1103/PhysRevLett.127.221602

Emergent Einstein Equation in p-adic Conformal Field Theory Tensor Networks

Phys Rev Lett. 2021 Nov 24;127(22):221602. doi: 10.1103/PhysRevLett.127.221602.

Authors

Lin Chen^{1

2

3}, Xirong Liu^{1

2

3}, Ling-Yan Hung^{1

2

3

4}

Affiliations

¹ State Key Laboratory of Surface Physics, Fudan University, 200433 Shanghai, China.
² Shanghai Qi Zhi Institute, 41st Floor, AI Tower, No. 701 Yunjin Road, Xuhui District, Shanghai 200232, China.
³ Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China.
⁴ Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, 200433 Shanghai, China.

PMID: 34889642
DOI: 10.1103/PhysRevLett.127.221602

Abstract

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) 170JHEPFG1029-847910.1007/JHEP04(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)TOMJAM0040-873510.2748/tmj/1325886283; Y. Ollivier, J. Funct. Anal. 256, 810 (2009)JFUAAW0022-123610.1016/j.jfa.2008.11.001] and physics [S. S. Gubser et al., J. High Energy Phys. 06 (2017) 157JHEPFG1029-847910.1007/JHEP06(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.