Building Tensor Networks for Holographic States

TL;DR

This work links TTbar-deformed boundary theories to continuous tensor networks by folding Euclidean path integrals on hyperbolic slices into the bulk, proposing CTN states that encode holographic geometry. It derives a Ryu-Takayanagi-like entanglement bound with coefficient 1/(4G_N) and shows the bound saturates on the time-reflection slice, while the CFT vacuum appears as a Hartle-Hawking superposition of CTN states. The framework clarifies how TTbar coupling, regulator geometry, and bulk minimal surfaces interrelate, and discusses extensions to sub-AdS locality, bit threads, and real-time holography along with open issues on UV completion. Overall, it provides a principled path to realize holographic tensor networks from TTbar deformations, with implications for bulk locality, complexity, and quantum-error-correcting structure in AdS/CFT.

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.

Figures (6)

• Figure 1: Summary of our construction. We consider the Euclidean path integral of a CFT on the lower half plane (black dotted line). We deform this path integral with a specific $T\bar{T}$ deformation, which is holographically dual to folding the slices (parametrized by $w = w_c$, blue solid line) into the bulk. The metric on the $w=w_c$ slices is that of $\mathbb{H}_2$ and the $T\bar{T}$ coupling $\lambda$ is related to the bulk coordinate $w_c$ as $\lambda = 2\pi G_N w_c^2$. Euclidean path integrals of the deformed theory on these slices are interpreted as continuous tensor network (CTN) states. The endpoint of the flow is at $w_c = 1$, where the "folded" slice coincides with the time-reflection symmetric slice in the bulk at $\tau = 0$ (red dashed line).
• Figure 2: Left: Folded slices parametrized by $w$. The red dashed line is the time-reflection symmetric slice. Right: Same blue slice as in the left panel, but regulated with a UV cutoff at $z=\epsilon$; this leads to a flat space regulator strip in the boundary of width $2b$. $Z_{UV}^{(\pm)}$ are the gravity path integrals over the upper and lower orange regions, whereas $Z_{IR}$ is the gravity path integral over the wedge (grey).
• Figure 3: An alternative picture of the folding deformation. The CFT lives on flat space, but at each time step the $T\bar{T}$ coupling is decreased slightly, holographically this will result in a folded slice inside the bulk, since we go deeper in the bulk for earlier times $u$.
• Figure 4: Left: A cartoon for a tensor network preparation of a state. The entanglement entropy of some interval (shown in green) is upper bounded by the minimal cut (shown in red) through the network. Right: For the continuous case, the entanglement entropy of some subregion $R$ is upper bounded by $\log\,\text{dim}$ of the Hilbert space on any cut through the Euclidean path integral of the $T\overline{T}$ deformed theory on the half place at $w_c$.
• Figure 5: Left: Folded slices $\gamma^{(-)}(w_c)$ and $\gamma^{(*)}$. $\gamma^{(*)}$ is the metric on the time-reflection symmetric slice in the bulk, which is shaded gray. The blue slice is a particular folded slice at some $w_c$. The green region is the boundary region $R$ of which we want to compute the entanglement entropy. The dashed red curve is the minimal-rank cut on the constant $w_c$ slice and the solid red curve is the actual RT surface on the bulk time-reflection symmetric slices $(w_c = 1)$. Right: The path-integral computing the density matrix on $R$. The regulator strip of size $2b$ is shown in grey. The replica manifold $\mathcal{M}_n$ is obtained by taking the $n$-fold branched cover over $R$.