Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings

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 Poincaré 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.

Krylov complexity from integrability to chaos

“…4 Finally, since a particularly interesting application of K-complexity is in the context of holographic duality, it is desirable to incorporate K-complexity into the holographic dictionary. In this context it is intriguing to remark that [49] recently proposed a discrete holographic dual of the aperiodic XXZ chain. It may be of interest to generalize our results to this case, to allow a more direct comparison to a simple discrete holographic code.…”

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.…”