Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations

TL;DR

This paper demonstrates how tensor networks encode geometric data satisfying an emergent Einstein equation in the p-adic AdS/CFT correspondence, linking tensor network geometry to a Fisher metric between quantum states.

Contribution

It introduces a method to assign distances in tensor networks that satisfy a graph Einstein equation, connecting tensor network geometry with emergent gravitational dynamics.

Findings

Tensor network distances satisfy a graph Einstein equation.

The geometric data matches a Fisher metric between quantum states.

Perturbative analysis recovers known mathematical proposals.

Abstract

As an extended companion paper to [1], we elaborate in detail how the tensor network construction of a p-adic CFT encodes geometric information of a dual geometry even as we deform the CFT away from the fixed point by finding a way to assign distances to the tensor network. In fact we demonstrate that a unique (up to normalizations) emergent graph Einstein equation is satisfied by the geometric data encoded in the tensor network, and the graph Einstein tensor automatically recovers the known proposal in the mathematics literature, at least perturbatively order by order in the deformation away from the pure Bruhat-Tits Tree geometry dual to pure CFTs. Once the dust settles, it becomes apparent that the assigned distance indeed corresponds to some Fisher metric between quantum states encoding expectation values of bulk fields in one higher dimension. This is perhaps a first quantitative demonstration that a concrete Einstein equation can be extracted directly from the tensor network, albeit in the simplified setting of the p-adic AdS/CFT.