Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs

Abstract

Discrete geometries in hyperbolic space are of longstanding interest in pure mathematics and have come to recent attention in holography, quantum information, and condensed matter physics. Working at a purely geometric level, we describe how any regular tessellation of ($d+1$)-dimensional hyperbolic space naturally admits a $d$-dimensional boundary geometry with self-similar ''quasicrystalline'' properties. In particular, the boundary geometry is described by a local, invertible, self-similar substitution tiling, that discretizes conformal geometry. We greatly refine an earlier description of these local substitution rules that appear in the 1D/2D example and use the refinement to give the first extension to higher dimensional bulks; including a detailed account for all regular 3D hyperbolic tessellations. We comment on global issues, including the reconstruction of bulk geometries from boundary data, and introduce the notion of a ''holographic foliation'': a foliation by a stack of self-similar quasicrystals, where the full geometry of the bulk (and of the foliation itself) is encoded in any single leaf in a local invertible way. In the $\{3,5,3\}$ tessellation of 3D hyperbolic space by regular icosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold symmetry which is not the Penrose tiling, and record and comment on a related conjecture of William Thurston. We end with a large list of open questions for future analytic and numerical studies.

26 pages bulk + 8 pages appendix, 14 large figures + 7 inline figures. v2 ref added, published version

References in corpus (17)

• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Crystallography of Hyperbolic Lattices
• Automorphic Bloch theorems for hyperbolic lattices
• Circuit Quantum Electrodynamics in Hyperbolic Space: From Photon Bound States to Frustrated Spin Models
• Holography on tessellations of hyperbolic space
• Non-Abelian hyperbolic band theory from supercells
• Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings
• Simulating Holographic Conformal Field Theories on Hyperbolic Lattices
• Breitenlohner-Freedman bound on hyperbolic tilings
• Noncrystalline topological superconductors
• Aperiodic spin chains at the boundary of hyperbolic tilings
• Hyperbolic Lattice for Scalar Field Theory in AdS${}_3$
• Topological linear response of hyperbolic Chern insulators
• Holographic tensor networks from hyperbolic buildings
• Anderson transition and mobility edges on hyperbolic lattices with randomly connected boundaries
• Hyperbolic lattices and two-dimensional Yang-Mills theory
• Holographic analysis of boundary correlation functions for the hyperbolic-lattice Ising model

Cited by in corpus (4)

• A Superalgebra Within: representations of lightest standard model particles form a ${\mathbb{Z}}_2^5$-graded algebra
• Tessellation codes: encoded quantum gates by geometric rotation
• Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices
• Topological states and flat bands in exactly solvable decorated Cayley trees