Conformal Quasicrystals and Holography

Abstract

Recent studies of holographic tensor network models defined on regular tessellations of hyperbolic space have not yet addressed the underlying discrete geometry of the boundary. We show that the boundary degrees of freedom naturally live on a novel structure, a conformal quasicrystal, that provides a discrete model of conformal geometry. We introduce and construct a class of one-dimensional conformal quasicrystals, and discuss a higher-dimensional example (related to the Penrose tiling). Our construction permits discretizations of conformal field theories that preserve an infinite discrete subgroup of the global conformal group at the cost of lattice periodicity.

Black rubber and the non-linear elastic response of scale invariant solids

“…See e.g. [13] for a recent discussion of the possible realization of conformal symmetry in quasi-crystals and [14] for a possible relation between quasi-crystals and our holographic models. 2 Let us stress that the elastic response exhibited by our solutions differs from other notions of elasticity black brane horizons discussed previously [16,17].…”

Conformal Properties of Hyperinvariant Tensor Networks

“…Subsequent layers are tiled around the central polygon by applying the necessary inflation rules, as depicted in the green-and magenta-lettered layers of the resulting bulk tessellation. This procedure is described in more detail by [23,26,[34][35][36]. We add tensors to the vertices and edges to the entire bullk of the resultant hyMERA network.…”

“…As such, a tessellated Poincaré-disk model exhibits several differences from the non-tessellated version; perhaps the most interesting of which is that the boundary of such a tessellated space does not exhibit periodic boundary conditions [26] , provided that the tilings themselves are regular. The Poincaré disk model also features the Möbius transformations, which are formed from the Möbius group P SL(2, C) and leave the manifold invariant.…”

“…Just as in the original Poincaré disk model, one can tessellate a timeslice of AdS 3 spacetime; the resulting fractal pattern in the bulk of AdS 3 breaks the continuous symmetries of the global conformal group [23,[34][35][36] on the boundary. In [45], it was shown that the holographic dictionary survives the truncation associated with approximating a continuous Poincaré disk model of AdS 3 with a regular tessellation; moreover, as described in [26], a tensor-network realization that imitates a tessellated Poincaré disk can be described using tensors at every vertex of the bulk. Conversely, edge tensors can be added to a tensor-network manifold that mimics the tessellated Poincaré disk, as in hyMERA.…”

“…Preferred directionality is generally not a feature of uniform bulk AdS space. Additionally, other research works have suggested that tensor-network analogs that exhibit a holographic duality can be better studied in systems with impurities or disorder [23,[25][26][27].…”