Dualities and non-Abelian mechanics

Abstract

Dualities are mathematical mappings that reveal links between apparently unrelated systems in virtually every branch of physics1,2,3,4,5,6,7,8. Systems mapped onto themselves by a duality transformation are called self-dual and exhibit remarkable properties, as exemplified by the scale invariance of an Ising magnet at the critical point. Here we show how dualities can enhance the symmetries of a dynamical matrix (or Hamiltonian), enabling the design of metamaterials with emergent properties that escape a standard group theory analysis. As an illustration, we consider twisted kagome lattices9,10,11,12,13,14,15, reconfigurable mechanical structures that change shape by means of a collapse mechanism9. We observe that pairs of distinct configurations along the mechanism exhibit the same vibrational spectrum and related elastic moduli. We show that these puzzling properties arise from a duality between pairs of configurations on either side of a mechanical critical point. The critical point corresponds to a self-dual structure with isotropic elasticity even in the absence of spatial symmetries and a twofold-degenerate spectrum over the entire Brillouin zone. The spectral degeneracy originates from a version of Kramers’ theorem16,17 in which fermionic time-reversal invariance is replaced by a hidden symmetry emerging at the self-dual point. The normal modes of the self-dual systems exhibit non-Abelian geometric phases18,19 that affect the semiclassical propagation of wavepackets20, leading to non-commuting mechanical responses. Our results hold promise for holonomic computation21 and mechanical spintronics by allowing on-the-fly manipulation of synthetic spins carried by phonons.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Symmetries and dualities.
Fig. 2: Twisted kagome lattices and their band structures.
Fig. 3: Schematic action of the duality operator.
Fig. 4: Mechanical spintronics via non-Abelian geometric phases.

Data availability

No external data set was used during the current study.

Code availability

The code used to compute the band structures and the holonomies, to perform the group-theoretical analysis, to integrate the semiclassical equations of motion and to verify the duality relations is available on Zenodo at https://doi.org/10.5281/zenodo.3417426 under the 2-clause BSD licence.

References

  1. 1.

    Kramers, H. A. & Wannier, G. H. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60, 252–262 (1941).

    ADS  MathSciNet  MATH  Google Scholar 

  2. 2.

    Savit, R. Duality in field theory and statistical systems. Rev. Mod. Phys. 52, 453–487 (1980).

    ADS  MathSciNet  Google Scholar 

  3. 3.

    Urade, Y., Nakata, Y., Nakanishi, T. & Kitano, M. Frequency-independent response of self-complementary checkerboard screens. Phys. Rev. Lett. 114, 237401 (2015).

    ADS  PubMed  Google Scholar 

  4. 4.

    Senthil, T., Vishwanath, A., Balents, L., Sachdev, S. & Fisher, M. P. A. Deconfined quantum critical points. Science 303, 1490–1494 (2004).

    ADS  CAS  PubMed  Google Scholar 

  5. 5.

    Louvet, T., Delplace, P., Fedorenko, A. A. & Carpentier, D. On the origin of minimal conductivity at a band crossing. Phys. Rev. B 92, 155116 (2015).

    ADS  Google Scholar 

  6. 6.

    Devetak, I. Triangle of dualities between quantum communication protocols. Phys. Rev. Lett. 97, 140503 (2006).

    ADS  MathSciNet  CAS  PubMed  Google Scholar 

  7. 7.

    Hull C. M. & Townsend, P. K. Unity of superstring dualities. Nucl. Phys. B 438, 109–137 (1995).

    ADS  MathSciNet  MATH  Google Scholar 

  8. 8.

    Maldacena, J. The large-N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Guest, S. & Hutchinson, J. W. On the determinacy of repetitive structures. J. Mech. Phys. Solids 51, 383–391 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  10. 10.

    Souslov, A., Liu, A. J. & Lubensky, T. C. Elasticity and response in nearly isostatic periodic lattices. Phys. Rev. Lett. 103, 205503 (2009).

    ADS  PubMed  Google Scholar 

  11. 11.

    Sun, K., Souslov, A., Mao, X. & Lubensky, T. C. Surface phonons, elastic response, and conformal invariance in twisted kagome lattices. Proc. Natl Acad. Sci. USA 109, 12369–12374 (2012).

    ADS  CAS  PubMed  Google Scholar 

  12. 12.

    Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2013).

    Google Scholar 

  13. 13.

    Paulose, J., Gin-ge Chen, B. & Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys. 11, 153–156 (2015).

    CAS  Google Scholar 

  14. 14.

    Rocklin, D. Z., Zhou, S., Sun, K. & Mao, X. Transformable topological mechanical metamaterials. Nat. Commun. 8, 14201 (2017).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  15. 15.

    Ma, J., Zhou, D., Sun, K., Mao, X. & Gonella, S. Edge modes and asymmetric wave transport in topological lattices: experimental characterization at finite frequencies. Phys. Rev. Lett. 121, 094301 (2018).

    ADS  CAS  PubMed  Google Scholar 

  16. 16.

    Kramers, H. A. Théorie générale de la rotation paramagnétique dans les cristaux. Proc. K. Akad. Wet. C 33, 959–972 (1930).

    CAS  MATH  Google Scholar 

  17. 17.

    Klein, M. J. On a degeneracy theorem of Kramers. Am. J. Phys. 20, 65–71 (1952).

    ADS  CAS  MATH  Google Scholar 

  18. 18.

    Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A 392, 45–57 (1984).

    ADS  MathSciNet  MATH  Google Scholar 

  19. 19.

    Wilczek, F. & Zee, A. Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984).

    ADS  MathSciNet  CAS  Google Scholar 

  20. 20.

    Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    ADS  MathSciNet  CAS  MATH  Google Scholar 

  21. 21.

    Zanardi, P. & Rasetti, M. Holonomic quantum computation. Phys. Lett. A 264, 94–99 (1999).

    ADS  MathSciNet  CAS  MATH  Google Scholar 

  22. 22.

    Coleman, S. Aspects of Symmetry (Cambridge Univ. Press, 1985).

  23. 23.

    Khanikaev, A. B., Fleury, R., Hossein Mousavi, S. & Alù, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nat. Commun. 6, 8260 (2015).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  24. 24.

    Süsstrunk, R. & Huber, S. D. Classification of topological phonons in linear mechanical metamaterials. Proc. Natl Acad. Sci. USA 113, E4767–E4775 (2016).

    ADS  PubMed  Google Scholar 

  25. 25.

    Huber, S. D. Topological mechanics. Nat. Phys. 12, 621–623 (2016).

    CAS  Google Scholar 

  26. 26.

    Matlack, K. H., Serra-Garcia, M., Palermo, A., Huber, S. D. & Daraio, C. Designing perturbative metamaterials from discrete models. Nat. Mater. 17, 323–328 (2018).

    ADS  CAS  PubMed  Google Scholar 

  27. 27.

    Fruchart, M. et al. Soft self-assembly of Weyl materials for light and sound. Proc. Natl Acad. Sci. USA 115, E3655–E3664 (2018).

    CAS  PubMed  Google Scholar 

  28. 28.

    Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    ADS  MathSciNet  CAS  Google Scholar 

  29. 29.

    Ningyuan, J., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time- and site-resolved dynamics in a topological circuit. Phys. Rev. X 50, 021031 (2015).

    Google Scholar 

  30. 30.

    Albert, V. V., Glazman, L. I. & Jiang, L. Topological properties of linear circuit lattices. Phys. Rev. Lett. 114, 173902 (2015).

    ADS  MathSciNet  PubMed  Google Scholar 

  31. 31.

    Lee, C. H. et al. Topolectrical circuits. Commun. Phys. 1, 39 (2018).

    Google Scholar 

  32. 32.

    Culcer, D., Yao, Y. & Niu, Q. Coherent wave-packet evolution in coupled bands. Phys. Rev. B 72, 085110 (2005).

    ADS  Google Scholar 

  33. 33.

    Shindou, R. & Imura, K.-I. Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons. Nucl. Phys. B 720, 399–435 (2005).

    ADS  MathSciNet  MATH  Google Scholar 

  34. 34.

    Onoda, M., Murakami, S. & Nagaosa, N. Hall effect of light. Phys. Rev. Lett. 93, 083901 (2004).

    ADS  PubMed  Google Scholar 

  35. 35.

    Stern, A. & Lindner, N. H. Topological quantum computation—from basic concepts to first experiments. Science 339, 1179–1184 (2013).

    ADS  CAS  PubMed  Google Scholar 

  36. 36.

    Iadecola, T., Schuster, T. & Chamon, C. Non-Abelian braiding of light. Phys. Rev. Lett. 117, 073901 (2016).

    ADS  PubMed  Google Scholar 

  37. 37.

    Barlas, Y. & Prodan, E. Topological braiding of Majorana-like modes in classical metamaterials. Preprint at https://arxiv.org/abs/1903.00463 (2019).

  38. 38.

    Liu, Y., Liu, Y. & Prodan, E. Braiding flux-tubes in topological quantum and classical lattice models from class-D. Preprint at https://arxiv.org/abs/1905.02457 (2019).

  39. 39.

    Li, N.et al. Phononics: manipulating heat flow with electronic analogs and beyond. Rev. Mod. Phys. 84, 1045–1066 (2012).

    ADS  Google Scholar 

  40. 40.

    Wilczek, F. & Shapere, A. Geometric Phases in Physics (World Scientific, 1989).

  41. 41.

    Chruściński, D. & Jamiołkowski, A. Geometric Phases in Classical and Quantum Mechanics (Birkhäuser Boston, 2004).

  42. 42.

    Cohen, E. et al. Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond. Nat. Rev. Phys. 1, 437–449 (2019).

    Google Scholar 

  43. 43.

    Wu, T. T. & Yang, C. N. Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D 12, 3845–3857 (1975).

    ADS  MathSciNet  Google Scholar 

  44. 44.

    Wilson, K. G. Confinement of quarks. Phys. Rev. D 10, 2445–2459 (1974).

    ADS  CAS  Google Scholar 

  45. 45.

    Bliokh, K. Y. & Bliokh, Y. P. Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, Berry phase, and the optical Magnus effect. Phys. Rev. E 70, 026605 (2004).

    ADS  Google Scholar 

  46. 46.

    Onoda, M., Murakami, S. & Nagaosa, N. Geometrical aspects in optical wave-packet dynamics. Phys. Rev. E 74, 066610 (2006).

    ADS  MathSciNet  Google Scholar 

  47. 47.

    Bliokh, K. Y., Frolov D. Y. & Kravtsov Y. A. Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium. Phys. Rev. A 75, 053821 (2007).

    ADS  Google Scholar 

  48. 48.

    Bliokh, K. Y. & Freilikher, V. D. Polarization transport of transverse acoustic waves: Berry phase and spin Hall effect of phonons. Phys. Rev. B 74, 174302 (2006).

    ADS  Google Scholar 

  49. 49.

    Mehrafarin, M. &Torabi, R. Geometric aspects of phonon polarization transport. Phys. Lett. A 373, 2114–2116 (2009).

    ADS  CAS  MATH  Google Scholar 

  50. 50.

    Torabi, R. & Mehrafarin M. Berry effect in acoustical polarization transport in phononic crystals. JETP Lett. 88, 590–594 (2009).

    ADS  Google Scholar 

  51. 51.

    Alden Mead, C. Molecular Kramers degeneracy and non-Abelian adiabatic phase factors. Phys. Rev. Lett. 59, 161–164 (1987).

    ADS  MathSciNet  Google Scholar 

  52. 52.

    Zee, A. Non-Abelian gauge structure in nuclear quadrupole resonance. Phys. Rev. A 38, 1–6 (1988).

    ADS  MathSciNet  CAS  MATH  Google Scholar 

  53. 53.

    Alden Mead, C. The geometric phase in molecular systems. Rev. Mod. Phys. 64, 51–85 (1992).

    ADS  MathSciNet  Google Scholar 

  54. 54.

    Sugawa, S., Salces-Carcoba, F., Perry, A. R., Yue, Y. & Spielman, I. B. Second Chern number of a quantum-simulated non-Abelian Yang monopole. Science 360, 1429–1434 (2018).

    ADS  MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  55. 55.

    Bliokh, K. Y. Rodriguez-Fortuño, F. J. Nori, F. & Zayats, A. V. Spin–orbit interactions of light. Nat. Photonics 9, 796–808 (2015).

    ADS  CAS  Google Scholar 

  56. 56.

    Ma, L. B. et al. Spin–orbit coupling of light in asymmetric microcavities. Nat. Commun. 7, 10983 (2016).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  57. 57.

    Dalibard, J., Gerbier, F., Juzeliūnas, G. & Öhberg, P. Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011).

    ADS  CAS  Google Scholar 

  58. 58.

    Goldman, N., Juzeliūnas, G., Öhberg, P. & Spielman, I. B. Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys. 77, 126401 (2014).

    ADS  CAS  PubMed  Google Scholar 

  59. 59.

    Wu, Z. et al. Realization of two-dimensional spin–orbit coupling for Bose–Einstein condensates. Science 354, 83–88 (2016).

    ADS  CAS  PubMed  Google Scholar 

  60. 60.

    Huang, L. et al. Experimental realization of two-dimensional synthetic spin–orbit coupling in ultracold Fermi gases. Nat. Phys. 12, 540–544 (2016).

    CAS  Google Scholar 

  61. 61.

    Aidelsburger, M., Nascimbene, S. & Goldman, N. Artificial gauge fields in materials and engineered systems. C. R. Phys. 19, 394–432 (2018).

    ADS  CAS  Google Scholar 

  62. 62.

    Chen, Y. et al. Non-Abelian gauge field optics. Nat. Commun. 10, 3125 (2019).

    ADS  PubMed  PubMed Central  Google Scholar 

  63. 63.

    Yang, Y. et al. Synthesis and observation of non-Abelian gauge fields in real space. Science 365, 1021–1025 (2019).

    ADS  MathSciNet  CAS  PubMed  Google Scholar 

  64. 64.

    Leinaas, J. M. & Myrheim, J. On the theory of identical particles. Nuovo Cimento B 37, 1–23 (1977).

    ADS  Google Scholar 

  65. 65.

    Wilczek, F. Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982).

    ADS  MathSciNet  CAS  Google Scholar 

  66. 66.

    Fröhlich, J. in Nonperturbative Quantum Field Theory (eds ’t Hooft, G. et al.) 71–100 (Springer, 1988).

  67. 67.

    Wen, X. G. Non-Abelian statistics in the fractional quantum Hall states. Phys. Rev. Lett. 66, 802–805 (1991).

    ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  68. 68.

    Moore, G. & Read, N. Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1991).

    ADS  MathSciNet  Google Scholar 

  69. 69.

    Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    ADS  MathSciNet  CAS  MATH  Google Scholar 

  70. 70.

    Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722–723 (1984).

    ADS  Google Scholar 

  71. 71.

    Lahtinen, V. & Pachos, J. K. Non-Abelian statistics as a Berry phase in exactly solvable models. New J. Phys. 11, 093027 (2009).

    ADS  Google Scholar 

  72. 72.

    Noh, J. et al. Braiding photonic topological zero modes. Preprint at https://arxiv.org/abs/1907.03208 (2019).

  73. 73.

    Maldovan, M. Sound and heat revolutions in phononics. Nature 503, 209–217 (2013).

    ADS  CAS  PubMed  Google Scholar 

  74. 74.

    Born, M. et al. Principles of Optics (Cambridge Univ. Press, 1999).

  75. 75.

    Karplus, R. & Luttinger, J. M. Hall effect in ferromagnetics. Phys. Rev. 95, 1154–1160 (1954).

    ADS  MATH  Google Scholar 

  76. 76.

    Chang, M.-C. & Niu, Q. Berry phase, hyperorbits, and the Hofstadter spectrum: semiclassical dynamics in magnetic Bloch bands. Phys. Rev. B 53, 7010–7023 (1996).

    ADS  CAS  Google Scholar 

  77. 77.

    Sundaram, G. & Niu, Q. Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects. Phys. Rev. B 59, 14915–14925 (1999).

    ADS  CAS  Google Scholar 

  78. 78.

    Panati, G., Spohn, H. & Teufel, S. Effective dynamics for Bloch electrons: Peierls substitution and beyond. Commun. Math. Phys. 242, 547–578 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  79. 79.

    Chang, M.-C. & Niu, Q. Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields. J. Phys. Condens. Matter 20, 193202 (2008).

    ADS  Google Scholar 

  80. 80.

    Zener, C. A theory of the electrical breakdown of solid dielectrics. Proc. R. Soc. A 145, 523–529 (1934).

    ADS  CAS  MATH  Google Scholar 

  81. 81.

    Mendez, E. E. & Bastard, G. Wannier–Stark ladders and Bloch oscillations in superlattices. Phys. Today 46, 34–42 (1993).

    CAS  Google Scholar 

  82. 82.

    Raizen, M., Salomon, C. & Niu, Q. New light on quantum transport. Phys. Today 50, 30–34 (1997).

    CAS  Google Scholar 

  83. 83.

    Price, H. M. & Cooper, N. R. Mapping the Berry curvature from semiclassical dynamics in optical lattices. Phys. Rev. A 85, 033620 (2012).

    ADS  Google Scholar 

  84. 84.

    Atala, M. et al. Direct measurement of the Zak phase in topological Bloch bands. Nat. Phys. 9, 795–800 (2013).

    CAS  Google Scholar 

  85. 85.

    Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

    ADS  CAS  PubMed  Google Scholar 

  86. 86.

    Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2014).

    Google Scholar 

  87. 87.

    Flaschner, N. et al. Experimental reconstruction of the Berry curvature in a Floquet Bloch band. Science 352, 1091–1094 (2016).

    ADS  CAS  PubMed  Google Scholar 

  88. 88.

    Li, T. et al. Bloch state tomography using Wilson lines. Science 352, 1094–1097 (2016).

    ADS  CAS  PubMed  Google Scholar 

Download references

Acknowledgements

We thank B. Bradlyn, V. Cheianov, S. Huber, W. Irvine, P. Lidon, N. Mitchell, S. Ryu, C. Scheibner, D. Son, A. Souslov, P. Wiegmann and B. van Zuiden for discussions. V.V. was supported by the Complex Dynamics and Systems Program of the Army Research Office under grant no. W911NF-19-1-0268. M.F. was primarily supported by the Chicago MRSEC (US NSF grant DMR 1420709) through a Kadanoff–Rice postdoctoral fellowship and acknowledges partial support by the University of Chicago through a Big Ideas Generator (BIG) grant and the Netherlands Organization for Scientific Research (NWO/OCW) as part of the Frontiers of Nanoscience program. LEGO is a trademark of the LEGO Group of companies which does not sponsor, license or endorse its use in this work.

Author information

Affiliations

Authors

Contributions

M.F. and V.V. designed the research, performed the research, and wrote the paper. Y.Z. and M.F. fabricated the mechanical kagome lattices. All authors contributed to discussions and manuscript revision.

Corresponding authors

Correspondence to Michel Fruchart or Vincenzo Vitelli.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Muamer Kadic, Ronny Thomale and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Notes: contains supplementary information about the system, the derivation, and the numerical simulations.

Video 1

Video demonstrating the zero-energy mechanism and its relation with the duality.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fruchart, M., Zhou, Y. & Vitelli, V. Dualities and non-Abelian mechanics. Nature 577, 636–640 (2020). https://doi.org/10.1038/s41586-020-1932-6

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

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