Space-time random tensor networks and holographic duality

TL;DR

The paper develops a covariant holographic framework using space-time random tensor networks in which bulk random projections define the boundary theory. In the large bond-dimension limit, the averaged network maps onto a discrete Z2 gauge theory, with boundary entanglement entropies governed by a covariant HRT-like minimal surface; bulk entanglement adds controlled corrections. It extends to higher Renyi entropies, establishes an operator correspondence and entanglement wedge reconstruction, and analyzes unitarity within code subspaces versus the full boundary. To control fluctuations and recover bulk geometry effects, the authors implement a gauge-fixing procedure based on spanning trees and discuss finite-D corrections. Overall, the work provides a covariant, microscopically articulated model of holographic duality with tractableentanglement and correlation structure, while highlighting open questions about dynamical gravity and background independence.

Abstract

In this paper we propose a space-time random tensor network approach for understanding holographic duality. Using tensor networks with random link projections, we define boundary theories with interesting holographic properties, such as the Renyi entropies satisfying the covariant Hubeny-Rangamani-Takayanagi formula, and operator correspondence with local reconstruction properties. We also investigate the unitarity of boundary theory in spacetime geometries with Lorenzian signature. Compared with the spatial random tensor networks, the space-time generalization does not require a particular time slicing, and provides a more covariant family of microscopic models that may help us to understand holographic duality.

Figures (10)

• Figure 1: (a) Illustration of a space-time tensor network which defines a statistical model, with the degrees of freedom defined on each link. With operator insertions $O_1,O_2,...,O_n$ (yellow boxes), evaluating this tensor network computes multipoint correlation functions in this system. (b) The definition of tensor that corresponds to the two-dimensional ferromagnetic Ising model.
• Figure 2: (a) The definition of multi-point functions of the boundary theory by a tensor network with one higher dimension. The red arrow along each link defines a random projector, as is illustrated in (b). (c) The random average over the direct product of two random projectors (denoted by the blue dashed line) results in the superposition of two operators, the identity operator and the swap operator, acting on the doubled Hilbert space. It should be noted that this figure has not taken into account the gauge redundancy subtlety discussed in Sec. \ref{['subsec:concept']} and Sec. \ref{['sec:fluctuations']}.
• Figure 3: (a) The network representation of the boundary density matrix $\rho$, with open legs at the $\tau=0$ cut representing the two indices of $\rho$. (b) For a boundary region $A$, ${\rm tr}\left[\rho_A^2\right]={\rm tr}\left[\left(\rho\otimes \rho\right)X_A\right]$ (see text), which is represented by two copies of the same tensor network with an insertion of boundary operator $X_A$ (defined in Eq. (\ref{['eqXA']})). (c) Carrying the random average over the random projectors at each link following Fig. \ref{['fig:TNofgrav']} (c), one obtains the partition function of a $Z_2$ gauge theory. The grey arrow is defined in (d) as the average of $P_{xy}\otimes P_{xy}$.
• Figure 4: An illustration of the gauge field configuration $\left\{\sigma_{xy}\right\}$. For a boundary region $A$ (the boundary of the organge region in the bulk), the boundary condition of $\sigma_{xy}$ is $\sigma_{xy}=-1$ for the red links across $A$, and $\sigma_{xy}=1$ elsewhere on the boundary. This boundary condition induces a flux co-dimension-$2$ surface $\gamma_A$ bounding the boundary of $A$, which is the purple dashed line. A gauge choice can be made by choosing $\sigma_{xy}=-1$ in the bulk for all links crossing a co-dimension $1$ surface $E_A$ which bounds $\gamma_A\cup A$, and $\sigma_{xy}=1$ elsewhere.
• Figure 5: (a) The tensor network that corresponds to the loop state we consider in Sec. \ref{['subsec:RT']}. The red arrows stand for random projectors, the same as in Fig. \ref{['fig:TNofgrav']}. (b) A more explicit definition of the vertex tensor, for a generic vertex that is adjacient to $n$ plaquettes. Each plaquette are adjacient to two links, which are labeled by $\mu_i,\nu_i$, $i=1,2,...,n$. $\mu_i,\nu_i=1,2,...,D$. (c) The generalization of loop state by introducing bulk entanglement, as is discussed in Sec. \ref{['subsec:RTwithbulk']}. Now each link is labeled by the loops passing through it, and an additional index $a=1,2,...,D_b$ labeling the remaining "quantum field theory" degrees of freedom in the bulk. (d) An example of a tensor network with loops and additional bulk degrees of freedom.