Central charges of aperiodic holographic tensor-network models

Central to the AdS/CFT correspondence is a precise relationship between the curvature of an anti–de Sitter (AdS) space-time and the central charge of the dual conformal field theory (CFT) on its boundary. Our work shows that such a relationship can also be established for tensor network models of AdS/CFT based on regular bulk geometries, leading to an analytical form of the maximal central charges exhibited by the boundary states. We identify a class of tensors based on Majorana dimer states that saturate these bounds in the large curvature limit, while also realizing perfect and block-perfect holographic quantum error correcting codes. Furthermore, the renormalization group description of the resulting model is shown to be analogous to the strong disorder renormalization group, thus giving an example of an exact quantum error correcting code that gives rise to a well-understood critical system. These systems exhibit a large range of fractional central charges, tunable by the choice of bulk tiling. Our approach thus provides a precise physical interpretation of tensor network models on regular hyperbolic geometries and establishes quantitative connections to a wide range of existing models.

Figure 1

(a) Continuous and (b) discretized geodesic ${\gamma }_A$ in the Poincaré disk with a deformation in the center and a boundary cutoff shown as a dashed curve. In the asymptotic region towards the boundary, the shape of ${\gamma }_A$ is independent of bulk deformations.

Figure 2

Vertex inflation of (a) the $\{3,7\}$ and (b) the $\{4,5\}$ tiling, with vertices labeled by type and each inflation layer color-coded.

Figure 3

Sketch of a $\{5,4\}$ tiling in the Poincaré disk with three reference points and one edge marked.

Figure 4

Central charge bounds and AdS radii for $\{n,k\}$ tilings, with the continuum Brown-Hennaux formula for $G=s/4ln\chi$ shown as a dashed line. The data series start at $k=7$ for $n=3, k=5$ for $n=4$, and $k=4$ for both $n=5$ and $n=6$ (first data point of each series in the upper-right corner).

Figure 5

Fibonacci singlets (top) and $\{5,4\}$ HyPeC Majorana dimers (bottom) shown in a disk projection along with their corresponding scaling of entanglement entropy $S_A$ with subsystem size $\ell =\left|A\right|$. The translation-invariant form [26] with effective central charge $c=c_{\text{Fib}}$ from (38) and $c=c_{\{5,4\}}^{\text{d}}$ from (B7) are shown as dashed curves.

Figure 6

Edge- and vertex-based inflation of the $\{5,4\}$ HyPeC in the form (42), with inflation layers color-coded. The full tiling in the original Poincaré disk projection is shown on the left, while the dimers at the first three inflation layers are shown on the right, unfolded onto a line (Poincaré half-space projection).

Figure 7

Central charges for the $\{5,k\}, \{9,k\}$, and $\{13,k\}$ Majorana dimer models (solid curves, bottom to top) and corresponding geodesic bounds (dashed curves). The continuum Brown-Hennaux formula for $G=s/4ln\chi$ is shown as a dashed line.

Figure 8

Local scale transformation of Poincaré disk angle $\varphi$ on the boundary under a translation in the bulk.

Figure 9

(a) Global scale transformation by growing the hyperbolic bulk tiling through vertex inflation. (b) Local scale transformation by a Möbius transformation composed of a bulk translation (first step) and a rotation (second step). (c) Successive reflections around a bulk edge and its tiling-symmetric rotation (green lines), with the same effect as a Möbius transformation.

Figure 10

(a) A contraction of a hyperbolic tensor network built from Majorana dimers (left) leads to a boundary Majorana dimer state (right). (b) An inflation step on the tiling (right) leads to a global scale transformation on the boundary state (right). (c) Certain combinations of Poincaré disk translations and rotations in the bulk (left) produce a local scaling transformation on the boundary state (right).