Holographic tensor networks from hyperbolic buildings

TL;DR

This paper develops a unified framework for holographic tensor networks using hyperbolic buildings, enabling the modeling of complex boundary spaces like fractals and generalizing holographic entanglement entropy beyond integer dimensions.

Contribution

It introduces a novel construction of holographic tensor networks based on hyperbolic buildings, extending holographic principles to fractal and non-integer dimensional boundaries.

Findings

Constructs bulk regions satisfying complementary recovery.

Derives a Ryu--Takayanagi formula for non-integer dimensions.

Recovers and generalizes HaPPY-like codes and Bruhat--Tits trees.

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.