Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quasiperiodicity and topology transcend dimensions

Nature Physics volume 12, pages 624–626 (2016)Cite this article

Subjects

The topological state of matter depends on its dimension. Remarkably, topological properties of quasiperiodic systems are found to emerge from higher dimensions.

Many of us marvel at the beauty of nature, be it the vivid blossom of the lily, the fractal structures of snowflakes or other intriguing wonders. Their magic lies in the subtle combination of their symmetry and complexity, which come together in ordered patterns. Pythagoras of Samos was the first to express the harmony of nature in terms of mathematics. Since then, physicists have looked for beauty and symmetry not only in the outside world, but also in its mathematical description.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Tile arrangement in quasicrystals.
Figure 2: Quasiperiodic sequences.

References

  1. Janot, C. Quasicrystals, a Primer (Oxford Univ. Press, 1994).

    MATH  Google Scholar 

  2. Senechal, M. Quasicrystals and Geometry (Cambridge Univ. Press, 1996).

    MATH  Google Scholar 

  3. Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Phys. Rev. Lett. 53, 1951–1953 (1984).

    Article  ADS  Google Scholar 

  4. Bindi, L., Steinhardt, P. J., Yao, N. & Lu, P. J. Science 24, 1306–1309 (2009).

    Article  ADS  Google Scholar 

  5. Macia, E. Rep. Prog. Phys. 69, 397 (2006).

    Article  ADS  Google Scholar 

  6. Dubois, J. M Chem. Soc. Rev. 41, 6760–6777 (2012).

    Article  Google Scholar 

  7. Dal Negro, L. et al. Phys. Rev. Lett. 90, 055501 (2003).

    Article  ADS  Google Scholar 

  8. Man, W. et al. Nature 436, 993–996 (2005).

    Article  ADS  Google Scholar 

  9. Vardeny, Z. V., Nahata, A. & Agrawal, A. Nature Photon. 7, 177–187 (2013).

    Article  ADS  Google Scholar 

  10. Ryu, S. et al. New J. Phys. 12, 065010 (2010).

    Article  ADS  Google Scholar 

  11. Fu, L. Phys. Rev. Lett. 106, 106802 (2011).

    Article  ADS  Google Scholar 

  12. Slager, R. J. et al. Nature Phys. 9, 98–102 (2013).

    Article  ADS  Google Scholar 

  13. Asorey, M. Nature Phys. 12, 616–618 (2016).

    Article  ADS  Google Scholar 

  14. Bohr, H. Acta Math. 46, 101–214 (1925).

    Article  MathSciNet  Google Scholar 

  15. Bohr, H. Acta Math. 47, 237–281 (1926).

    Article  MathSciNet  Google Scholar 

  16. Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Phys. Rev. Lett. 109, 106402 (2012).

    Article  ADS  Google Scholar 

  17. Verbin, M., Zilberberg, O., Lahini, Y., Kraus, Y. E. & Silberberg, Y. Phys. Rev. B 91, 064201 (2015).

    Article  ADS  Google Scholar 

  18. Lohse, M. et al. Nature Phys. 12, 350–354 (2015).

    Article  ADS  Google Scholar 

  19. Nakajima, S. et al. Nature Phys. 12, 296–300 (2016).

    Article  ADS  Google Scholar 

  20. Lu, H. I. et al. Phys. Rev. Lett. 116, 200402 (2016).

    Article  ADS  Google Scholar 

  21. Lu, L., Joannopoulos, J. D. & Soljačić, M. Nature Phys. 12, 626–629 (2016).

    Article  ADS  Google Scholar 

  22. Goldman, N., Budich, J. C. & Zoller, P. Nature Phys. 12, 639–645 (2016).

    Article  ADS  Google Scholar 

  23. Prodan, E. Phys. Rev. B 91, 245104 (2015).

    Article  ADS  Google Scholar 

  24. Kraus, Y. E. & Zilberberg, O. Phys. Rev. Lett. 109, 116404 (2012).

    Article  ADS  Google Scholar 

  25. Dana, I. Phys. Rev. B 89, 205111 (2014).

    Article  ADS  Google Scholar 

  26. El Hassouani, Y. et al. Phys. Rev. B 74, 035314 (2006).

    Article  ADS  Google Scholar 

  27. Pang, X. N., Dong, J. W. & Wang, H. Z. J. Opt. Soc. Am. B 27, 2009–2013 (2010).

    Article  ADS  Google Scholar 

  28. Verbin, M. et al. Phys. Rev. Lett. 110, 076403 (2013).

    Article  ADS  Google Scholar 

  29. Bellissard, J. From Number Theory to Physics 538–630 (Springer, 1992).

    Book  Google Scholar 

  30. Levy, E. et al. Preprint at http://arxiv.org/abs/1509.04028 (2015).

  31. Yang, C. N. J. Math. Phys. 19, 320–328 (1978).

    Article  ADS  Google Scholar 

  32. Zhang, S.-C. & Hu, J. Science 294, 823–828 (2001).

    Article  ADS  Google Scholar 

  33. Kraus, Y. E., Ringel, Z. & Zilberberg, O. Phys. Rev. Lett. 111, 226401 (2013).

    Article  ADS  Google Scholar 

  34. Ben-Abraham, S. I. & Quandt, A. Phil. Mag. 91, 2718–2727 (2011).

    Article  ADS  Google Scholar 

  35. Bandres, M. A., Rechtsman, M. C. & Segev, M. Phys. Rev. X 6, 011016 (2016).

    Google Scholar 

  36. McGrath, R. et al. J. Phys. Cond. Matter 22, 084022 (2010).

    Article  ADS  Google Scholar 

  37. Tran, D. T. et al. Phys. Rev. B 91, 085125 (2015).

    Article  ADS  Google Scholar 

  38. Fulga, I., Pikulin, D. & Loring, T. Preprint at http://arxiv.org/abs/1510.06035 (2015).

  39. Mourik, V. et al. Science 336, 1003–1007 (2012).

    Article  ADS  Google Scholar 

  40. Das, A. et al. Nature Phys. 8, 887–895 (2012).

    Article  ADS  Google Scholar 

  41. Deng, M. et al. Nano Lett. 12, 6414–6419 (2012).

    Article  ADS  Google Scholar 

  42. Lang, L. J. & Chen, S. Phys. Rev. B 86, 205135 (2012).

    Article  ADS  Google Scholar 

  43. Tezuka, M. & Kawakami, N. Phys. Rev. B 88, 155428 (2013).

    Article  ADS  Google Scholar 

  44. Cai, X., Lang, L. J., Chen, S. & Wang, Y. Phys. Rev. Lett. 110, 176403 (2013).

    Article  ADS  Google Scholar 

  45. DeGottardi, W., Sen, D. & Vishveshwara, S. Phys. Rev. Lett. 110, 146404 (2013).

    Article  ADS  Google Scholar 

  46. Wang, J. et al. Phys. Rev. B 93, 104504 (2015).

    Article  ADS  Google Scholar 

  47. Beenakker, C. & Kouwenhoven, L. Nature Phys. 12, 618–621 (2016).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank H. M. Price for feedback on the article.

Author information

Authors and Affiliations

  1. Yaacov E. Kraus is in the Department of Physics, Holon Institute of Technology, Holon 5810201, Israel,

    Yaacov E. Kraus

  2. Oded Zilberberg is in the Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland,

    Oded Zilberberg

Authors
  1. Yaacov E. Kraus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oded Zilberberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oded Zilberberg.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kraus, Y., Zilberberg, O. Quasiperiodicity and topology transcend dimensions. Nature Phys 12, 624–626 (2016). https://doi.org/10.1038/nphys3784

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys3784

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing