Building Tensor Networks for Holographic States

TL;DR

This paper introduces a new class of continuous tensor network states derived from Euclidean path integrals in deformed holographic CFTs, which satisfy a Ryu-Takayanagi-like entanglement entropy bound and relate to the bulk Hartle-Hawking wavefunction.

Contribution

It proposes a novel interpretation of Euclidean path integral states as continuous tensor networks in holographic CFTs, connecting boundary states with bulk geometry and entanglement bounds.

Findings

CTN states satisfy a Ryu-Takayanagi-like entropy bound.

The CTN states correspond to the bulk time-reflection symmetric slice.

The original CFT state can be expressed as a superposition of CTN states.

Abstract

We discuss a one-parameter family of states in two-dimensional holographic conformal field theories which are constructed via the Euclidean path integral of an effective theory on a family of hyperbolic slices in the dual bulk geometry. The effective theory in question is the CFT flowed under a $T\overline{T}$ deformation, which "folds" the boundary CFT towards the bulk time-reflection symmetric slice. We propose that these novel Euclidean path integral states in the CFT can be interpreted as continuous tensor network (CTN) states. We argue that these CTN states satisfy a Ryu-Takayanagi-like minimal area upper bound on the entanglement entropies of boundary intervals, with the coefficient being equal to $\frac{1}{4G_N}$; the CTN corresponding to the bulk time-reflection symmetric slice saturates this bound. We also argue that the original state in the CFT can be written as a superposition of such CTN states, with the corresponding wavefunction being the bulk Hartle-Hawking wavefunction.