Hyperinvariant Tensor Networks and Holography

We propose a new class of tensor network state as a model for the $\mathrm{AdS}/\mathrm{CFT}$ correspondence and holography. This class is demonstrated to retain key features of the multiscale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible. Yet, unlike the MERA, they are built according to a uniform tiling of hyperbolic space, without inherent directionality or preferred locations in the holographic bulk, and thus circumvent key arguments made against the MERA as a model for $\mathrm{AdS}/\mathrm{CFT}$. Novel holographic features of this tensor network class are examined, such as an equivalence between the causal cones $\mathcal{C}(\mathcal{R})$ and the entanglement wedges $\mathcal{E}(\mathcal{R})$ of connected boundary regions $\mathcal{R}$.

Figure 1

(a) A hyperinvariant tensor network based on a $\{7,3\}$ tiling of the hyperbolic disk, where a 3-index tensor $A$ is placed on each node of the tiling, with a matrix $B$ then placed on each edge adjoining two nodes. (b) Tensors $A$ are constrained to be rotationally invariant, while matrices $B$ are constrained to be symmetric. (c) The network can be unwrapped into concentric layers $V_z$ about any chosen center (where $z$ is a label over scale), with each layer as a connected string of alternating $A$ and $B$ tensors. (d) All layers (with the exception of that immediately following the top) are formed from a combination of two-site and three-site unit cells.

Figure 2

(a) The product of an $A$ and two $B$ tensors is constrained to form an isometry $w$ which, by definition, annihilates to identity with its conjugate $w^{†}$. (b) Tensor $u$, which is formed from three $A$ tensors together with five $B$ tensors, is constrained to be an isometric mapping from 3-to-2 indices. (c) The tensors in each layer $V$ can be grouped as a product of $w$ and $u$ isometries in many different ways. (d) Under the action of layer $V$, a one-site local operator $\sigma$ is mapped to a coarse-grained operator, ${\sigma }^{\prime }\equiv V^{†}\sigma V$, which remains local due to the cancellation of tensors in $V$ with their conjugates in $V^{†}$ (where tensors have been grouped into isometries $w$ and $u$ in such a way as to minimize the support of ${\sigma }^{\prime }$).

Figure 3

(a) A schematic representation of the hyperinvariant network, which has been organized into layers about a chosen bulk point $T$. The apparent causal cone ${\mathcal{C}}_T(\mathcal{R})$ of a boundary region $\mathcal{R}$ of $L$ sites shaded, where the true causal cone $\mathcal{C}(\mathcal{R})$ is a subset of ${\mathcal{C}}_T(\mathcal{R})$. (b) The entanglement wedge $\mathcal{E}(\mathcal{R})$ is the bulk region bounded by the minimal surface ${\gamma }_{\mathcal{R}}$ and $\mathcal{R}$. (c),(d) Part of a $\{7,3\}$ hyperinvariant network that has been organized into layers, with the apparent causal cone ${\mathcal{C}}_T(\mathcal{R})$ of a boundary region $\mathcal{R}$ shaded. In the evaluation of the reduced density matrix $\rho (\mathcal{R})$, many tensors in ${\mathcal{C}}_T(\mathcal{R})$ cancel, such that $\rho (\mathcal{R})$ depends only on tensors within a subset $\mathcal{C}(\mathcal{R})\subset {\mathcal{C}}_T(\mathcal{R})$, where it is observed that $\mathcal{C}(\mathcal{R})$ is exactly coincident with the entanglement wedge $\mathcal{E}(\mathcal{R})$. (e) Depictions of the causal cones $\mathcal{C}(\mathcal{R})$ and the corresponding reduced density matrices $\rho (\mathcal{R})$ for one-, two-, and three-site regions $\mathcal{R}$, where the causal cones also correspond to entanglement wedges $\mathcal{E}(\mathcal{R})$.

Figure 4

(a) A hyperinvariant network organised into layers $V$ about bulk point $T$, with the apparent causal cone ${\mathcal{C}}_T(\mathcal{R})$ and minimal surface ${\gamma }_{\mathcal{R}}$ of boundary region $\mathcal{R}$ shown. (b) Choosing point $T$ at the apex of ${\gamma }_{\mathcal{R}}$ gives ${\mathcal{C}}_T(\mathcal{R})=\mathcal{E}(\mathcal{R})$.