nLab quasicrystal

Idea

(…)

Properties

Noncommutative Brillouin zone

The Brillouin zone of quasicrystals may be understood as having noncommutative topology/noncommutative geometry (Bellissard 03, Ypma 05, p. 8, Putnam 10).

For more see at noncommutative topology of quasiperiodicity.

References

General

Survey:

• Christian Janot, The Properties and Applications of Quasicrystals, Europhysics News 27 (1996) (pdf)

• Christian Janot, Quasicrystals – A Primer, Oxford Classic Texts in the Physical Sciences, Oxford University Press, 2012 (ISBN:978-0-19-965740-7)

See also

• Wikipedia, Quasicrystal

As topological phases

As or in relation to topological phases of matter:

• Yao Wang et al., Quantum Topological Boundary States in Quasi-crystal, Adv Mater 2019 Dec;31(49):e1905624 (doi:10.1002/adma.201905624)

• Dominic V. Else, Sheng-Jie Huang, Abhinav Prem, Andrey Gromov, Quantum many-body topology of quasicrystals (arXiv:2103.13393)

• Oded Zilberberg, Topology in quasicrystals, Optical Materials Express Vol. 11, Issue 4, pp. 1143-1157 (2021) (doi:10.1364/OME.416552)

As a substrate for topological quantum computation:

• Marcelo Amaral, David Chester, Fang Fang, Klee Irwin, Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing, Symmetry 14 9 (2022) 1780 [arXiv:2207.08928, doi:10.3390/sym14091780]

Mathematical description

Mathematical disucssion:

• A. Hof, On diffraction by aperiodic structures, Commun. Math. Phys., 169 (1995), p. 25-43.

• J-B. Gouere, Quasicrystals and almost periodicity, Commun. Math. Phys., 255 (2005), p. 655-681.

Discussion of the noncommutative topology/KK-theory of the Brillouin zone in the spirit of the K-theory classification of topological phases of matter:

• Jean Bellissard, The Noncommutative Geometry of Aperiodic Solids, in: Geometric and Topological Methods for Quantum Field Theory, pp. 86-156 (2003) (pdf, doi:10.1142/9789812705068_0002)

• Fonger Ypma, Quasicrystals, $C^\ast$-algebras and K-theory, 2005 (pdf)

• Ian F. Putnam, Non-commutative methods for the K-theory of $C^\ast$-algebras of aperiodic patterns from cut-and-project systems, Commun. Math. Phys. 294, 703–729 (2010) (pdf, doi:10.1007/s00220-009-0968-0)

• Hervé Oyono-Oyonoa, Samuel Petite, $C^\ast$-algebras of Penrose hyperbolic tilings, Journal of Geometry and Physics Volume 61, Issue 2, February 2011, Pages 400-424 (doi:10.1016/j.geomphys.2010.09.019)

For more see at noncommutative topology of quasiperiodicity.

Phenomenology

Analysis of possible quasicrystal configurations:

via machine learning:

• Roman Eremin, Pavel Zolotarev, Tilmann Leisegang, Pavlo Solokha, A machine learning approach for predicting formation enthalpy: A case study of Mackay-type approximants of icosahedral quasicrystals, AIP Conference Proceedings 2163, 020003 (2019) (doi:10.1063/1.5130082)

via topological data analysis:

• Pavlo Solokha et al., New Quasicrystal Approximant in the Sc–Pd System: From Topological Data Mining to the Bench, Chem. Mater. 2020, 32, 3, 1064–1079 (doi:10.1021/acs.chemmater.9b03767)

• Søren S. Sørensen, Revealing hidden medium-range order in amorphous materials using topological data analysis, Science Advances 09 Sep 2020: Vol. 6, no. 37, eabc2320 (doi:10.1126/sciadv.abc2320)

In AdS/CFT

Discussion of asymptotic boundaries of hyperbolic tensor networks as conformal quasicrystals (see also at AdS/CFT in solid state physics):

• Latham Boyle, Madeline Dickens, Felix Flicker, Conformal Quasicrystals and Holography, Phys. Rev. X 10, 011009 (2020) (arXiv:1805.02665)

• Alexander Jahn, Zoltán Zimborás, Jens Eisert, Central charges of aperiodic holographic tensor network models, Phys. Rev. A 102, 042407 (arXiv:1911.03485)

  (via Majorana dimer codes)