Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings

TL;DR

This paper introduces a novel discrete holography model using hyperbolic tilings to construct aperiodic spin chains, exploring their properties and entanglement entropy growth to advance the understanding of AdS/CFT duality in discrete spaces.

Contribution

It presents a new class of aperiodic spin chains derived from hyperbolic tilings, providing a framework for discrete bulk reconstruction and entanglement analysis in holography.

Findings

Logarithmic entanglement entropy growth with subsystem size.

Effective central charges depend on bulk discretization parameters.

Tensor network construction for ground state from strong disorder renormalization.

Abstract

We propose a new example of discrete holography that provides a new step towards establishing the AdS/CFT duality for discrete spaces. A class of boundary Hamiltonians is obtained in a natural way from regular tilings of the hyperbolic Poincar\'e disk, via an inflation rule that allows to construct the tiling using concentric layers of tiles. The models in this class are aperiodic spin chains, whose sequences of couplings are obtained from the bulk inflation rule. We explicitly choose the aperiodic XXZ spin chain with spin 1/2 degrees of freedom as an example. The properties of this model are studied by using strong disorder renormalization group techniques, which provide a tensor network construction for the ground state of this spin chain. This can be regarded as discrete bulk reconstruction. Moreover we compute the entanglement entropy in this setup in two different ways: a discretization of the Ryu-Takayanagi formula and a generalization of the standard computation for the boundary aperiodic Hamiltonian. For both approaches, a logarithmic growth of the entanglement entropy in the subsystem size is identified. The coefficients, i.e. the effective central charges, depend on the bulk discretization parameters in both cases, albeit in a different way.