nLab quasicrystal






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.




  • 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

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:

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 *C^\ast-algebras and K-theory, 2005 (pdf)

  • Ian F. Putnam, Non-commutative methods for the K-theory of C *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 *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.


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)


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

Last revised on September 11, 2022 at 16:11:18. See the history of this page for a list of all contributions to it.