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  • Open Access

Conformal Quasicrystals and Holography

Latham Boyle1, Madeline Dickens2, and Felix Flicker2,3

  • 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
  • 2Department of Physics, University of California, Berkeley, California 94720, USA
  • 3Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Department of Physics, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom

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Phys. Rev. X 10, 011009 – Published 14 January, 2020

DOI: https://doi.org/10.1103/PhysRevX.10.011009

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.

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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.

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References (51)

  1. J. M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, Int. J. Theor. Phys. 38, 1113 (1999); [Adv. Theor. Math. Phys. 2, 231 (1998)].
  2. E. Witten, Anti-de Sitter Space and Holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  3. J. Kaplan, Lectures on Ads/CFT from the Bottom Up, Johns Hopkins University course notes (to be published).
  4. M. Van Raamsdonk, Building Up Spacetime with Quantum Entanglement, Gen. Relativ. Gravit. 42, 2323 (2010); [Int. J. Mod. Phys. D 19, 2429 (2010)].
  5. K. G. Wilson, Confinement of Quarks, Phys. Rev. D 10, 2445 (1974).
  6. G. Vidal, Entanglement Renormalization, Phys. Rev. Lett. 99, 220405 (2007).
  7. G. Vidal, Class of Quantum Many-Body States that Can Be Efficiently Simulated, Phys. Rev. Lett. 101, 110501 (2008).
  8. G. Vidal, Entanglement Renormalization: An Introduction, in Understanding Quantum Phase Transitions (CRC Press, New York, 2010).
  9. R. N. C. Pfeifer, G. Evenbly, and G. Vidal, Entanglement Renormalization, Scale Invariance, and Quantum Criticality, Phys. Rev. A 79, 040301(R) (2009).
  10. G. Evenbly and G. Vidal, Tensor Network States and Geometry, J. Stat. Phys. 145, 891 (2011).
  11. R. Orus, Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography, Eur. Phys. J. B 87, 280 (2014).
  12. B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86, 065007 (2012).
  13. B. Swingle, Constructing Holographic Spacetimes Using Entanglement Renormalization, arXiv:1209.3304.
  14. X. L. Qi, Exact Holographic Mapping and Emergent Space-Time Geometry, arXiv:1309.6282.
  15. J. Molina-Vilaplana, Holographic Geometries of One-Dimensional Gapped Quantum Systems from Tensor Network States, J. High Energy Phys. 05 (2013) 024.
  16. F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Holographic Quantum Error-Correcting Codes: Toy Models for the Bulk/Boundary Correspondence, J. High Energy Phys. 06 (2015) 149.
  17. P. Hayden, S. Nezami, X. L. Qi, N. Thomas, M. Walter, and Z. Yang, Holographic Duality from Random Tensor Networks, J. High Energy Phys. 11 (2016) 009.
  18. B. Czech, L. Lamprou, S. McCandlish, and J. Sully, Tensor Networks from Kinematic Space, J. High Energy Phys. 07 (2016) 100.
  19. M. Han and L. Y. Hung, Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy, Phys. Rev. D 95, 024011 (2017).
  20. X. L. Qi, Z. Yang, and Y. Z. You, Holographic Coherent States from Random Tensor Networks, J. High Energy Phys. 08 (2017) 060.
  21. G. Evenbly, Hyperinvariant Tensor Networks and Holography, Phys. Rev. Lett. 119, 141602 (2017).
  22. T. J. Osborne and D. E. Stiegemann, Dynamics for Holographic Codes, arXiv:1706.08823.
  23. N. Bao, C. Cao, S. M. Carroll, and A. Chatwin-Davies, De Sitter Space as a Tensor Network: Cosmic No-Hair, Complementarity, and Complexity, Phys. Rev. D 96, 123536 (2017).
  24. A. Jahn, M. Gluza, F. Pastawski, and J. Eisert, Holography and Criticality in Matchgate Tensor Networks, Sci. Adv. 5, eaaw0092 (2019).
  25. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2010).
  26. B. Czech, L. Lamprou, S. McCandlish, and J. Sully, Integral Geometry and Holography, J. High Energy Phys. 10 (2015) 175.
  27. A. Milsted and G. Vidal, arXiv:1812.00529.
  28. H. S. M. Coxeter, Regular Honeycombs in Hyperbolic Space, Proc. Intl. Congr. Math. 3, 155 (1954).

  29. Note that, in this paper, as in Refs. [16, 21, 24], we focus on ordinary regular {p,q} tessellations (i.e., tessellations where each tile is a regular p-gon, q such tiles meet at each vertex, and p and q are both finite) since these represent the natural discretizations of theories that are local in the bulk. Another fascinating possibility is to relax this locality restriction and consider {p,∞} tilings of the hyperbolic plane in which infinitely many tiles meet at each vertex; in this case, tile edges are infinitely long, and the tile vertices live at infinity, on the boundary of the hyperbolic plane, where they can then induce a periodic structure at infinity. Reference [22] contains a detailed exploration of this latter possibility and, in particular, in this context, presents an interesting toy model in which the discrete symmetries in the bulk may be related to a discretization of the conformal group of the boundary. The dual cases {∞,q} consist of regular polygons with an infinite number of sides (apeirogons), with q meeting at each vertex. When q is a prime p, the boundary theory is then naturally described by the p-adic numbers [30].

  30. L. Brekke and P. G. O. Freund, p-Adic Numbers in Physics, Phys. Rep. 233, 1 (1993).

  31. Yet another possibility is to replace the regular tiling in the bulk by a random tensor network as in Ref. [17]. In AdS/CFT, a key role is played by the isometry group of the bulk AdS space, and a regular tiling has the advantage that it preserves a large discrete subgroup of this symmetry. However, for some purposes, a random tensor network may also have certain advantages, and it would be interesting to investigate the mathematical structure that it induces on the boundary. If the random tensor network in the bulk respects the properties of the regular tiling in a statistical sense, then the random structure induced on the boundary should also respect the properties of the corresponding conformal quasicrystal in a statistical sense, but we leave the investigation of this topic to future work.

  32. J. M. Luck, A Classification of Critical Phenomena on Quasicrystals and Other Aperiodic Structures, Europhys. Lett. 24, 359 (1993).
  33. M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Cambridge, England, 1995).
  34. C. Janot, Quasicrystals: A Primer (Clarendon Press, Oxford, 1994).
  35. M. Baake and U. Grimm, Aperiodic Order: Volume 1: A Mathematical Invitation (Cambridge University Press, Cambridge, England, 2013).
  36. L. Boyle and P. J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv:1608.08220.
  37. L. Boyle and P. J. Steinhardt, Coxeter Pairs, Ammann Patterns and Penrose-like Tilings, arXiv:1608.08215.
  38. M. Gardner, Extraordinary Nonperiodic Tiling that Enriches the Theory of Tiles, Scientific American 236, 110 (1977).

  39. Strictly speaking, this is a τ quasicrystal of degree two since τ acts on two interval types; if τ acts on n interval types, we would have a τ quasicrystal of degree n.

  40. E. Bombieri and J. E. Taylor, Which Distributions of Matter Diffract? An Initial Investigation, J. Phys. (Paris), Colloq. 47, C3 (1986); Quasicrystals, Tilings, and Algebraic Number Theory: Some Preliminary Connections, Contemp. Math. 64, 241 (1987).
  41. H. S. M. Coxeter, Regular Polytopes (Methuen, London, 1948).
  42. R. Rietman, B. Nienhuis, and J. Oitmaa, The Ising Model on Hyperlattices, J. Phys. A 25, 6577 (1992).
  43. P. H. Ginsparg, Applied Conformal Field Theory, in Fields, Strings and Critical Phenomena (Les Houches, Session XLIX, 1988), edited by E. Brezin and J. Zinn-Justin (North-Holland Publishing Company, Amsterdam, 1988).
  44. P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory (Springer-Verlag, New York, 1997).
  45. W. Magnus, Noneuclidean Tesselations and Their Groups (Academic Press, New York and London, 1974).

  46. Note that, besides AdS/CFT, examples of holography include the Banks-Fischler-Shenker-Susskind matrix model [47] and the Klebanov-Strassler construction [48]. The ideas described in this paper relate to the AdS/CFT context, but we hope that progress in this context may lead to progress in the other holographic settings as well.

  47. T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, M Theory as a Matrix Model: A Conjecture, Phys. Rev. D 55, 5112 (1997).
  48. I. R. Klebanov and M. J. Strassler, Supergravity and a Confining Gauge Theory: Duality Cascades and χSB-Resolution of Naked Singularities, J. High Energy Phys. 08 (2000) 052.
  49. D. S. Rokhsar, D. C. Wright, and N. D. Mermin, The Two-Dimensional Quasicrystallographic Space Groups with Rotational Symmetries Less Than 23-fold, Acta Crystallogr. Sect. A 44, 197 (1988).
  50. L. Nemeth, The Growing Ratios of Hyperbolic Regular Mosaics with Bounded Cells, Armenian J. Math. 9, 1 (2017).
  51. J. Weeks, KaleidoTile, a computer program for creating spherical, Euclidean, and hyperbolic tilings.

Outline

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Phys. Rev. X 10, 011009– Published 14 January, 2020
  • Received 18 May 2018
  • Revised 22 August 2019
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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