Pseudospin-1 Systems as a New Frontier for Research on Relativistic Quantum Chaos

  • Ying-Cheng LaiEmail author
Conference paper
Part of the Understanding Complex Systems book series (UCS)


Pseudospin-1 systems are characterized by the feature that their band structure consists of a pair of Dirac cones and a topologically flat band. Such systems can be realized in a variety of physical systems ranging from dielectric photonic crystals to electronic materials. Theoretically, massless pseudospin-1 systems are described by the generalized Dirac-Weyl equation governing the evolution of a three-component spinor. Recent works have demonstrated that such systems can exhibit unconventional physical phenomena such as revival resonant scattering, superpersistent scattering, super-Klein tunneling, perfect caustics, vanishing Berry phase, and isotropic low energy scattering. We argue that investigating the interplay between pseudospin-1 physics and classical chaos may constitute a new frontier area of research in relativistic quantum chaos with significant applications.



This Review is based on Refs. [52, 53, 54]. I thank my former student and current post-doctoral fellow Dr. H.-Y. Xu - the main contributor of these works. I would like to acknowledge support from the Pentagon Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through Grant No. N00014-16-1-2828.


  1. 1.
    K.S. Novoselov et al., Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004)CrossRefGoogle Scholar
  2. 2.
    C. Berger et al., Ultrathin epitaxial graphite: 2D electron gas properties and a route toward graphene-based nanoelectronics. J. Phys. Chem. B 108, 19912–19916 (2004)CrossRefGoogle Scholar
  3. 3.
    T. Wehling, A. Black-Schaffer, A. Balatsky, Dirac materials. Adv. Phys. 63, 1–76 (2014)CrossRefGoogle Scholar
  4. 4.
    J. Wang, S. Deng, Z. Liu, Z. Liu, The rare two-dimensional materials with Dirac cones. Natl. Sci. Rev. 2(1), 22–39 (2015)CrossRefGoogle Scholar
  5. 5.
    M.Z. Hasan, C.L. Kane, Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)CrossRefGoogle Scholar
  6. 6.
    X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011)CrossRefGoogle Scholar
  7. 7.
    X.-L. Qi, T.L. Hughes, S.-C. Zhang, Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008)CrossRefGoogle Scholar
  8. 8.
    A.M. Essin, J.E. Moore, D. Vanderbilt, Magnetoelectric polarizability and axion electrodynamics in crystalline insulators. Phys. Rev. Lett. 102, 146805 (2009)CrossRefGoogle Scholar
  9. 9.
    C.-Z. Chang et al., Zero-field dissipationless chiral edge transport and the nature of dissipation in the quantum anomalous hall state. Phys. Rev. Lett. 115, 057206 (2015)CrossRefGoogle Scholar
  10. 10.
    Y.H. Wang et al., Observation of chiral currents at the magnetic domain boundary of a topological insulator. Science 349, 948–952 (2015)CrossRefGoogle Scholar
  11. 11.
    M.C. Rechtsman et al., Topological creation and destruction of edge states in photonic graphene. Phys. Rev. Lett. 111, 103901 (2013)CrossRefGoogle Scholar
  12. 12.
    Y. Plotnik et al., Observation of unconventional edge states in photonic graphene. Nat. Mater. 13, 57–62, (2014) (Article)CrossRefGoogle Scholar
  13. 13.
    Z. Wang, Y.D. Chong, J.D. Joannopoulos, M. Soljačić, Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008)CrossRefGoogle Scholar
  14. 14.
    Z. Wang, Y. Chong, J.D. Joannopoulos, M. Soljacic, Observation of unidirectional backscattering-immune topological electromagnetic states. Nature (London) 461, 772–775 (2009)CrossRefGoogle Scholar
  15. 15.
    M. Hafezi, E.A. Demler, M.D. Lukin, J.M. Taylor, Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011)CrossRefGoogle Scholar
  16. 16.
    K. Fang, Z. Yu, S. Fan, Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photonics 6, 782–787 (2012)CrossRefGoogle Scholar
  17. 17.
    A.B. Khanikaev et al., Photonic topological insulators. Nat. Mater. 12, 233–239 (2013)CrossRefGoogle Scholar
  18. 18.
    L. Lu, J.D. Joannopoulos, M. Soljaclc, Topological photonics. Nat. Photonics 8, 821–829 (2014)CrossRefGoogle Scholar
  19. 19.
    X. Huang, Y. Lai, Z.H. Hang, H. Zheng, C.T. Chan, Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials. Nat. Mater. 10, 582–586 (2011)CrossRefGoogle Scholar
  20. 20.
    J. Mei, Y. Wu, C.T. Chan, Z.-Q. Zhang, First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals. Phys. Rev. B 86, 035141 (2012)CrossRefGoogle Scholar
  21. 21.
    P. Moitra et al., Realization of an all-dielectric zero-index optical metamaterial. Nat. Photonics 7, 791–795 (2013)CrossRefGoogle Scholar
  22. 22.
    Y. Li et al., On-chip zero-index metamaterials. Nat. Photonics 9, 738–742 (2015)CrossRefGoogle Scholar
  23. 23.
    A. Fang, Z.Q. Zhang, S.G. Louie, C.T. Chan, Klein tunneling and supercollimation of pseudospin-1 electromagnetic waves. Phys. Rev. B 93, 035422 (2016)CrossRefGoogle Scholar
  24. 24.
    D. Guzmán-Silva et al., Experimental observation of bulk and edge transport in photonic Lieb lattices. New J. Phys. 16, 063061 (2014)CrossRefGoogle Scholar
  25. 25.
    S. Mukherjee et al., Observation of a localized flat-band state in a photonic Lieb lattice. Phys. Rev. Lett. 114, 245504 (2015)CrossRefGoogle Scholar
  26. 26.
    R.A. Vicencio et al., Observation of localized states in Lieb photonic lattices. Phys. Rev. Lett. 114, 245503 (2015)CrossRefGoogle Scholar
  27. 27.
    F. Diebel, D. Leykam, S. Kroesen, C. Denz, A.S. Desyatnikov, Conical diffraction and composite Lieb bosons in photonic lattices. Phys. Rev. Lett. 116, 183902 (2016)CrossRefGoogle Scholar
  28. 28.
    S. Taie et al., Coherent driving and freezing of bosonic matter wave in an optical Lieb lattice. Sci. Adv. 1, e1500854 (2015)CrossRefGoogle Scholar
  29. 29.
    M. Rizzi, V. Cataudella, R. Fazio, Phase diagram of the Bose-Hubbard model with \(T\_3\) symmetry. Phys. Rev. B 73, 144511 (2006)Google Scholar
  30. 30.
    A.A. Burkov, E. Demler, Vortex-peierls states in optical lattices. Phys. Rev. Lett. 96, 180406 (2006)CrossRefGoogle Scholar
  31. 31.
    D. Bercioux, D.F. Urban, H. Grabert, W. Häusler, Massless Dirac-Weyl fermions in a \({T}_{3}\) optical lattice. Phys. Rev. A 80, 063603 (2009)CrossRefGoogle Scholar
  32. 32.
    B. Dóra, J. Kailasvuori, R. Moessner, Lattice generalization of the Dirac equation to general spin and the role of the flat band. Phys. Rev. B 84, 195422 (2011)CrossRefGoogle Scholar
  33. 33.
    A. Raoux, M. Morigi, J.-N. Fuchs, F. Piéchon, G. Montambaux, From dia- to paramagnetic orbital susceptibility of massless fermions. Phys. Rev. Lett. 112, 026402 (2014)CrossRefGoogle Scholar
  34. 34.
    T. Andrijauskas et al., Three-level Haldane-like model on a dice optical lattice. Phys. Rev. A 92, 033617 (2015)CrossRefGoogle Scholar
  35. 35.
    F. Wang, Y. Ran, Nearly flat band with Chern number \(c=2\) on the dice lattice. Phys. Rev. B 84, 241103 (2011)CrossRefGoogle Scholar
  36. 36.
    J. Wang, H. Huang, W. Duan, Z. Liu, Identifying Dirac cones in carbon allotropes with square symmetry. J. Chem. Phys. 139, 184701 (2013)CrossRefGoogle Scholar
  37. 37.
    W. Li, M. Guo, G. Zhang, Y.-W. Zhang, Gapless \({\text{ MoS }}_2\) allotrope possessing both massless Dirac and heavy fermions. Phys. Rev. B 89, 205402 (2014)CrossRefGoogle Scholar
  38. 38.
    J. Romhanyi, K. Penc, R. Ganesh, Hall effect of triplons in a dimerized quantum magnet. Nat. Commun. 6, 6805 (2015)Google Scholar
  39. 39.
    G. Giovannetti, M. Capone, J. van den Brink, C. Ortix, Kekulé textures, pseudospin-one Dirac cones, and quadratic band crossings in a graphene-hexagonal indium chalcogenide bilayer. Phys. Rev. B 91, 121417 (2015)CrossRefGoogle Scholar
  40. 40.
    G.-L. Wang, H.-Y. Xu, Y.-C. Lai, Mechanical topological semimetals with massless quasiparticles and a finite berry curvature. Phys. Rev. B 95, 235159 (2017)Google Scholar
  41. 41.
    R. Shen, L.B. Shao, B. Wang, D.Y. Xing, Single Dirac cone with a flat band touching on line-centered-square optical lattices. Phys. Rev. B 81, 041410 (2010)CrossRefGoogle Scholar
  42. 42.
    D.F. Urban, D. Bercioux, M. Wimmer, W. Häusler, Barrier transmission of Dirac-like pseudospin-one particles. Phys. Rev. B 84, 115136 (2011)CrossRefGoogle Scholar
  43. 43.
    M. Vigh et al., Diverging dc conductivity due to a flat band in a disordered system of pseudospin-1 Dirac-Weyl fermions. Phys. Rev. B 88, 161413 (2013)CrossRefGoogle Scholar
  44. 44.
    J.T. Chalker, T.S. Pickles, P. Shukla, Anderson localization in tight-binding models with flat bands. Phys. Rev. B 82, 104209 (2010)CrossRefGoogle Scholar
  45. 45.
    J.D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, S. Flach, Flatbands under correlated perturbations. Phys. Rev. Lett. 113, 236403 (2014)CrossRefGoogle Scholar
  46. 46.
    E.H. Lieb, Two theorems on the Hubbard model. Phys. Rev. Lett. 62, 1201–1204 (1989)MathSciNetCrossRefGoogle Scholar
  47. 47.
    H. Tasaki, Ferromagnetism in the Hubbard models with degenerate single-electron ground states. Phys. Rev. Lett. 69, 1608–1611 (1992)MathSciNetCrossRefGoogle Scholar
  48. 48.
    H. Aoki, M. Ando, H. Matsumura, Hofstadter butterflies for flat bands. Phys. Rev. B 54, R17296–R17299 (1996)CrossRefGoogle Scholar
  49. 49.
    C. Weeks, M. Franz, Topological insulators on the Lieb and perovskite lattices. Phys. Rev. B 82, 085310 (2010)CrossRefGoogle Scholar
  50. 50.
    N. Goldman, D.F. Urban, D. Bercioux, Topological phases for fermionic cold atoms on the Lieb lattice. Phys. Rev. A 83, 063601 (2011)CrossRefGoogle Scholar
  51. 51.
    J. Vidal, R. Mosseri, B. Douçot, Aharonov-Bohm cages in two-dimensional structures. Phys. Rev. Lett. 81, 5888–5891 (1998)CrossRefGoogle Scholar
  52. 52.
    H.-Y. Xu, Y.-C. Lai, Revival resonant scattering, perfect caustics, and isotropic transport of pseudospin-1 particles. Phys. Rev. B 94, 165405 (2016)CrossRefGoogle Scholar
  53. 53.
    H.-Y. Xu, Y.-C. Lai, Superscattering of a pseudospin-1 wave in a photonic lattice. Phys. Rev. A 95, 012119 (2017)Google Scholar
  54. 54.
    H.-Y. Xu, L. Huang, D. Huang, Y.-C. Lai, Geometric valley Hall effect and valley filtering through a singular Berry flux. Phys. Rev. B 96, 045412 (2017)Google Scholar
  55. 55.
    M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Chiral tunnelling and the Klein paradox in graphene. Nat. Phys. 2, 620–625 (2006)CrossRefGoogle Scholar
  56. 56.
    D.S. Novikov, Elastic scattering theory and transport in graphene. Phys. Rev. B 76, 245435 (2007)CrossRefGoogle Scholar
  57. 57.
    M.I. Katsnelson, F. Guinea, A.K. Geim, Scattering of electrons in graphene by clusters of impurities. Phys. Rev. B 79, 195426 (2009)CrossRefGoogle Scholar
  58. 58.
    J.-S. Wu, M.M. Fogler, Scattering of two-dimensional massless Dirac electrons by a circular potential barrier. Phys. Rev. B 90, 235402 (2014)CrossRefGoogle Scholar
  59. 59.
    J. Cserti, A. Pályi, C. Péterfalvi, Caustics due to a negative refractive index in circular graphene \(p\rm \text{- }n\) junctions. Phys. Rev. Lett. 99, 246801 (2007)CrossRefGoogle Scholar
  60. 60.
    R.L. Heinisch, F.X. Bronold, H. Fehske, Mie scattering analog in graphene: Lensing, particle confinement, and depletion of Klein tunneling. Phys. Rev. B 87, 155409 (2013)CrossRefGoogle Scholar
  61. 61.
    M.M. Asmar, S.E. Ulloa, Rashba spin-orbit interaction and birefringent electron optics in graphene. Phys. Rev. B 87, 075420 (2013)CrossRefGoogle Scholar
  62. 62.
    B. Liao, M. Zebarjadi, K. Esfarjani, G. Chen, Isotropic and energy-selective electron cloaks on graphene. Phys. Rev. B 88, 155432 (2013)CrossRefGoogle Scholar
  63. 63.
    M.M. Asmar, S.E. Ulloa, Spin-orbit interaction and isotropic electronic transport in graphene. Phys. Rev. Lett. 112, 136602 (2014)CrossRefGoogle Scholar
  64. 64.
    A. Ferreira, T.G. Rappoport, M.A. Cazalilla, A.H. Castro Neto, Extrinsic spin Hall effect induced by resonant skew scattering in graphene. Phys. Rev. Lett. 112, 066601 (2014)Google Scholar
  65. 65.
    Y. Zhao et al., Creating and probing electron whispering-gallery modes in graphene. Science 348, 672–675 (2015)CrossRefGoogle Scholar
  66. 66.
    W.S. Bakr, J.I. Gillen, A. Peng, S. Folling, M. Greiner, A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 74–77 (2009)CrossRefGoogle Scholar
  67. 67.
    Jin, D., et al., Topological magnetoplasmon (2016). arXiv:1602.00553
  68. 68.
    L.I. Schiff, Quantum Mechanics, 3rd edn. (McGraw-Hill, New York, 1968)Google Scholar
  69. 69.
    R. Newton, Scattering Theory of Waves and Particles. Dover Books on Physics (Dover Publications, New York, 1982)Google Scholar
  70. 70.
    M. Lewkowicz, B. Rosenstein, Dynamics of particle-hole pair creation in graphene. Phys. Rev. Lett. 102, 106802 (2009)CrossRefGoogle Scholar
  71. 71.
    B. Rosenstein, M. Lewkowicz, H.-C. Kao, Y. Korniyenko, Ballistic transport in graphene beyond linear response. Phys. Rev. B 81, 041416 (2010)CrossRefGoogle Scholar
  72. 72.
    B. Dóra, R. Moessner, Nonlinear electric transport in graphene: quantum quench dynamics and the Schwinger mechanism. Phys. Rev. B 81, 165431 (2010)Google Scholar
  73. 73.
    B. Dóra, R. Moessner, Dynamics of the spin Hall effect in topological insulators and graphene. Phys. Rev. B 83, 073403 (2011)Google Scholar
  74. 74.
    S. Vajna, B. Dóra, R. Moessner, Nonequilibrium transport and statistics of Schwinger pair production in Weyl semimetals. Phys. Rev. B 92, 085122 (2015)CrossRefGoogle Scholar
  75. 75.
    C.-Z. Wang, H.-Y. Xu, L. Huang, Y.-C. Lai, Nonequilibrium transport in the pseudospin-1 Dirac-Weyl system. Phys. Rev. B 96, 115440 (2017)CrossRefGoogle Scholar
  76. 76.
    W. Häusler, Flat-band conductivity properties at long-range Coulomb interactions. Phys. Rev. B 91, 041102 (2015)CrossRefGoogle Scholar
  77. 77.
    T. Louvet, P. Delplace, A.A. Fedorenko, D. Carpentier, On the origin of minimal conductivity at a band crossing. Phys. Rev. B 92, 155116 (2015)CrossRefGoogle Scholar
  78. 78.
    H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, New York, 1999)CrossRefGoogle Scholar
  79. 79.
    Haake, F. Quantum Signatures of Chaos, 3rd edn.. Springer Series in Synergetics (Springer, Berlin, 2010)CrossRefGoogle Scholar
  80. 80.
    A.H.C. Neto, K. Novoselov, Two-dimensional crystals: beyond graphene. Mater. Exp. 1, 10–17 (2011)CrossRefGoogle Scholar
  81. 81.
    P. Ajayan, P. Kim, K. Banerjee, Two-dimensional van der Waals materials. Phys. Today 69, 38–44 (2016)CrossRefGoogle Scholar
  82. 82.
    Y.-C. Lai, L. Huang, H.-Y. Xu, C. Grebogi, Relativistic quantum chaos - an emergent interdisciplinary field. Chaos 28, 052101 (2018)MathSciNetCrossRefGoogle Scholar
  83. 83.
    L. Huang, H.-Y. Xu, C. Grebogi, Y.-C. Lai, Relativistic quantum chaos. Phys. Rep. 753, 1–128 (2018)MathSciNetCrossRefGoogle Scholar
  84. 84.
    A. Mekis, J.U. Nöckel, G. Chen, A.D. Stone, R.K. Chang, Ray chaos and Q spoiling in lasing droplets. Phys. Rev. Lett. 75, 2682–2685 (1995)CrossRefGoogle Scholar
  85. 85.
    J.U. Nöckel, A.D. Stone, Ray and wave chaos in asymmetric resonant optical cavities. Nature 385, 45–47 (1997)CrossRefGoogle Scholar
  86. 86.
    C. Gmachl et al., High-power directional emission from microlasers with chaotic resonators. Science 280, 1556–1564 (1998)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Electrical, Computer and Energy Engineering, Arizona State UniversityTempeUSA

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