Tensor Network Models of Multiboundary Wormholes

TL;DR

The paper investigates how entanglement in CFT states dual to multiboundary wormholes is encoded in tensor-network models on hyperbolic tilings and their discrete quotients. It demonstrates a lattice version of the Ryu–Takayanagi bound, $S_A \le |\gamma_A| \ln \chi$, realized on networks built from perfect or random tensors, and constructs multiboundary states by quotienting tilings by discrete isometries. A key result is the existence of tilings where entanglement is purely bipartite across minimal cuts, even for generic moduli, while other tilings exhibit a residual multipartite component detectable via negative or entanglement measures; the structure aligns with expectations from high-temperature CFT limits and the presence or absence of a causal shadow. The work provides a computationally tractable framework for exploring holographic code subspaces, bulk reconstruction, and the role of the causal shadow in governing multipartite entanglement across multiboundary geometries, with potential extensions to higher bond dimensions and more complex topologies.

Abstract

We study the entanglement structure of states dual to multiboundary wormhole geometries using tensor network models. Perfect and random tensor networks tiling the hyperbolic plane have been shown to provide good models of the entanglement structure in holography. We extend this by quotienting the plane by discrete isometries to obtain models of the multiboundary states. We show that there are networks where the entanglement structure is purely bipartite, extending results obtained in the large temperature limit. We analyse the entanglement structure in a range of examples.

Figures (17)

• Figure 1: The surface $\Sigma$ as a quotient of the Poincaré disc for the pair of pants. The pairs of labeled geodesics (blue and red in colour version) are identified by the action of $\Gamma$. The region of the Poincaré disc bounded by these geodesics provides a fundamental domain for the quotient. $B_3$, $B_2$ and $B_1 = B_{1+} \cup B_{1-}$ become the desired three circular boundaries. There are corresponding minimal closed geodesics $H_3$, $H_2$ and $H_1 = H_{1+} \cup H_{1-}$. The lengths $L_a$ of these geodesics fully characterize the geometry of $\Sigma$.
• Figure 2: The geometry of the pair-of-pants in the high temperature limit and the resulting entanglement structure. a) the pair-of-pants geometry with the three coloured boundaries indicated and labelled 1-3, the black lines depict the horizons pertaining to each exterior region, and are labelled $H_a$, $a=1,2,3$. The interior of the horizons is the causal shadow region. b) Cartoon of a) in the high-temperature limit, with fixed ratios of the moduli. The exterior cylinders shrink (the strips should be thought of as being extremely thin, we've exaggerated them here) and the distance between the horizons across the causal shadow region is small almost everywhere. The black lines represent identifications between horizons, which is true to exponential accuracy away from the junctions. c) The resulting entanglement structure can be depicted with this "wheel" diagram: the path integral locally identifies the states in portions of the three boundaries. States localised in some boundary interval are purified by an interval of the same size on the opposite side of the seam, which may lie on any of the three boundaries, as the ratios of the moduli are varied. The resulting entanglement structure is almost entirely bipartite.
• Figure 3: The [2,4,6] tiling of the hyperbolic plane $H^2$. One of the seed triangles is indicated, the whole tiling is covered by the action of reflecting the seed along its edges. The black, primary tiling lines divide the network into a regular array of hexagons.
• Figure 4: An illustration of the quotient operation in the [2,4,6] tiling. We take $r_A$, $r_B$, $r_C$ to be the reflections in the geodesics labelled in the left-hand diagram. Quotienting by $\Gamma$ generated by $g_1 = r_A r_B$ gives the tiling of BTZ shown in the middle diagram. Quotienting by $\Gamma$ generated by $g_1 = r_A r_B$ and $g_2 = r_B r_C$ gives the tiling of the pair of pants shown in the right-hand diagram. The unshaded region is a fundamental region for the identification in both cases.
• Figure 5: Illustrations of three-boundary tilings obtained by quotienting the tilings of the hyperbolic plane indicated by each column, and with discrete moduli in the regimes indicated by each row. The unshaded region is a fundamental region for the identification. The minimal closed paths along tile boundaries homologous to each conformal boundary are indicated. The cases with two paths of the same colour, where one is dashed, represent degeneracies in the choice of closed path. The second group of examples show cases where there is no causal shadow; the minimal closed paths coincide, so the entanglement between different regions is entirely bipartite.