Holographic duality from random tensor networks

TL;DR

This paper develops a random-tensor-network framework as a versatile toy model for holographic duality. By averaging over Haar-random tensors, the authors map boundary entanglement calculations to a classical Sym_n spin model, recovering the Ryu-Takayanagi formula in the large bond-dimension limit and incorporating bulk entanglement corrections through entanglement wedges. The work also demonstrates that such networks realize bidirectional holographic codes with bulk-to-boundary and boundary-to-bulk isometries and error-correction properties, and it extends to higher Rényi entropies, boundary two-point functions, and finite-D corrections. Moreover, the authors connect the formalism to random measurements and entanglement of assistance and discuss extensions to 2-designs and stabilizer states, outlining a general bulk-boundary dictionary that could apply beyond AdS-like spacetimes.

Abstract

Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface topologically in a way similar to creation of a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by AdS/CFT duality, our main results define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.

Figures (13)

• Figure 1: (a) A tensor network that defines a state in the Hilbert space of the dangling indices. (b) A tensor network that defines a mapping from bulk legs (red) to boundary legs (blue). An arbitrary bulk state (orange triangle) is mapped to a boundary state. (For simplicity, we have drawn a pure state in the bulk. For a mixed state the map needs to be applied to both indices of the bulk density operator.) (c) The internal lines of the tensor network can always be combined with the bulk state and viewed as a state in an enlarged Hilbert space (enclosed by the dashed hexegon). In this view, each tensor acts independently on this generalized bulk state and maps it to the boundary state.
• Figure 2: (a) Graphic representation of the single site density operator $\ket{V_x}\!\!\bra{V_x}$ for a vertex in the tensor network shown in Fig. \ref{['fig:tensor']}. (b) The average over the state $\ket{V_x}\!\!\bra{V_x}\otimes \ket{V_x}\!\!\bra{V_x}$ in the Hilbert space (see Eq. (\ref{['Haaraverage']})). On the right side of the equality, the dashed line connected to the black dot stands for a sum over an Ising variable $s_x=\pm 1$. When $s_x=1$ ($s_x=-1$), each green rectangle represents an operator $I_x$ ($\mathcal{F}_x$), respectively. (c) The state average of $Z_1$ in Eq. (\ref{['Z1average']}) for the simple tensor network shown in Fig. \ref{['fig:tensor']}. We consider a region $A$ consisting of a single site, and the green rectangle with X represents the swap operator $\mathcal{F}_A$. After contracting the doubled line loops one obtains the partition function of an Ising model, with the blue arrows representing the Ising variables. The dashed lines in the right of last equality represent three different terms in the Ising model contributed by the links, the bulk state (middle triangle) and the choice of boundary region $A$.
• Figure 3: (a) An example of Ising spin configuration with boundary fields down ($h_x=-1$) in $A$ region and up ($h_x=+1$) elsewhere. $\gamma_A$ is the boundary of minimal energy spin-down domain configuration. $\Sigma$ (black dashed line) is an example of other domain wall configurations with higher energy. The spin-down domain $E_A$ is called the entanglement wedge of $A$. (b) The minimal surfaces bounding two far-away regions $A$ and $B$, which are also the boundary of the entanglement wedge of the completement region $CD$. (c) The effect of bulk entanglement in the same configuration as panel (b). The entanglement wedges are deformed.
• Figure 4: (a) Illustration of the setup. The orange disk-shaped bulk region of radius $b$ is in a random pure state. We study the second Rényi entropy of a boundary region $\theta\in[-\varphi,\varphi]$ at radius $r=1-\epsilon$. (b) The phase diagram of the boundary state as parametrized by the bond dimensions $D$ and $D_b$, corresponding to in-plane and bulk degrees of freedom, respectively. The blue line, obtained numerically, describes the phase boundary that separates the perturbed AdS phase and the small black hole phase. The red line distinguishes the small black hole phase and the maximal black hole phase. The three phases are discussed in more detail in the main text.
• Figure 5: Configuration of the minimal surfaces calculated numerically in the bulk for different boundary regions in the three phases. The random pure state is supported at the orange region. The parameters are set to $l_g^{-1}\log D=10$, $b=\tanh(1/2)$. Depending on the value of $l_g^{-2}\log D_b$, the phases of the system are given by (a) $l_g^{-2}\log D_b =1$, perturbed AdS phase; (b) $l_g^{-2}\log D_b=5$, small black hole phase; (c) $l_g^{-2}\log D_b= 15$, maximal black hole phase.