Conformal Quasicrystals and Holography

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.

Popular Summary

The two pillars of modern physics—general relativity and quantum mechanics—lead to paradoxes when combined. Important progress on this puzzle has come from the AdS/CFT correspondence, an example of holography that suggests that all the information about some region is encoded on its boundary. In recent years, researchers have developed a version of this idea in which the bulk of the region can be described as discrete rather than smooth—a step that has led to further exciting progress by revealing new links to concepts from condensed-matter physics and quantum computation. Now, we have made a fundamental new discovery: A discrete bulk naturally leads to a discrete boundary, described by a fascinating new mathematical structure we call a conformal quasicrystal.

In the AdS/CFT correspondence, a higher-dimensional negatively curved “bulk” spacetime with gravity can be equivalent to another theory that lives in a lower-dimensional “boundary” spacetime with no gravity. Previous studies have focused on discretizing the higher-dimensional bulk spacetime but have not thought through the corresponding implications for the boundary spacetime, as we now have. Such studies have suggested a new picture in which spacetime and gravity emerge from a type of quantum entanglement.

One of the most interesting things about these conformal quasicrystals is their close relation to the famous Penrose tiling, a mosaic of shapes renowned for its aesthetic appeal but now helping to understand the microscopic structure of spacetime. These quasicrystals also have additional remarkable properties that might not have been discovered without the holographic perspective.

The appearance of quasicrystals in holography is unprecedented and foretells a highly interdisciplinary new connection between quantum gravity, quantum information, condensed-matter physics, and mathematical crystallography.

Figure 1

The Penrose tiling [35]. The two different tiles (red and blue) cover the entire Euclidean plane in an aperiodic manner without gaps. Centers of approximate fivefold symmetry can be seen. The tiling can be created through a cut-and-project method from a five-dimensional hypercubic lattice or a four-dimensional $A_4$ lattice, or by applying “inflation rules” to a finite-size starting configuration [33, 34, 35].

Figure 2

Illustration of unit-cell and letter assignment on the boundary of a simply connected tile set $S$, in the $\{7,3\}$ tiling. (a) Unit-cell assignment. (b) Letter assignment. (c) Cyclically ordered strings of unit cells and letters. (d) Correspondence between unit cells and letters in the $\{7,3\}$ case.

Figure 3

Demonstration of how vertex completion induces inflation rules on boundary letters and unit cells, for the $\{7,3\}$ tiling. The induced letter inflation rule $(A,B)\mapsto (A^{-1}{B}^{-5},B^{4}A)$ is the same for every tile set $S$, convex or not. When $S$ is convex, only type-2 and type-3 unit cells appear, and there is an induced unit-cell inflation rule $(2,3)\mapsto (223,23)$. Note that vertex completion preserves convexity and that every tile set $S$ becomes convex after finitely many vertex completions.

Figure 4

A “quasi-MERA” tensor network based on the 1D Fibonacci quasicrystal. Vertices denote tensors, and edges denote contractions between tensors. Three- and four-legged tensors represent isometries and disentanglers, respectively. Dotted lines separate layers related by the Fibonacci deflation rule.