Holographic tensor networks from hyperbolic buildings

Abstract

We introduce a unifying framework for the construction of holographic tensor networks, based on the theory of hyperbolic buildings. The underlying dualities relate a bulk space to a boundary which can be homeomorphic to a sphere, but also to more general spaces like a Menger sponge type fractal. In this general setting, we give a precise construction of a large family of bulk regions that satisfy complementary recovery. For these regions, our networks obey a Ryu-Takayanagi formula. The areas of Ryu-Takayanagi surfaces are controlled by the Hausdorff dimension of the boundary, and consistently generalize the behavior of holographic entanglement entropy in integer dimensions to the non-integer case. Our construction recovers HaPPY-like codes in all dimensions, and generalizes the geometry of Bruhat-Tits trees. It also provides examples of infinite-dimensional nets of holographic conditional expectations, and opens a path towards the study of conformal field theory and holography on fractal spaces.

Entanglement entropy and non-local duality: quantum channels and quantum algebras

“…As they currently stand, these tools are most applicable to discrete systems such as the TFIM. For example, the pursuit of a discrete holographic model (see for instance [32][33][34] and references therein for recent work) could be a natural candidate for the methods detailed here. Beyond such discrete models, a generalization to continuous degrees of freedom would be necessary.…”

“…In fact for all our boundary conditions we used the same duality map, and since entanglement is a statement about the states and not their energies, the same result holds for these states with periodic or self-dual boundary conditions as well. The state (32) requires an additional flip of the order of the spins be included in the action of duality to match the self-dual boundary condition ground states (24), so in this case we compare the dual ground states…”

Breitenlohner-Freedman bound on hyperbolic tilings

“…It remains an open question, though, how to establish a duality in the sense of a map between bulk and boundary theories for hyperbolic tessellations. Steps in this direction were taken in [18,19] using modular discretizations and in [20] via tensor networks on hyperbolic buildings. In this work we focus on hyperbolic tilings as a discretization scheme instead.…”