Band structure engineering and non-equilibrium dynamics in Floquet topological insulators

Abstract

Non-equilibrium topological phenomena can be induced in quantum many-body systems using time-periodic fields (for example, by laser or microwave illumination). This Review begins with the key principles underlying Floquet band engineering, wherein such fields are used to change the topological properties of a system’s single-particle spectrum. In contrast to equilibrium systems, non-trivial band structure topology in a driven many-body system does not guarantee that robust topological behaviour will be observed. In particular, periodically driven many-body systems tend to absorb energy from their driving fields and thereby tend to heat up. We survey various strategies for overcoming this challenge of heating and for obtaining new topological phenomena in this non-equilibrium setting. We describe how drive-induced topological edge states can be probed in the regime of mesoscopic transport, and three routes for observing topological phenomena beyond the mesoscopic regime: long-lived transient dynamics and prethermalization, disorder-induced many-body localization, and engineered couplings to external baths. We discuss the types of phenomena that can be explored in each of the regimes covered, and their experimental realizations in solid-state, cold atomic, and photonic systems.

Key points

  • Time-periodic fields provide a versatile platform for inducing non-equilibrium topological phenomena in quantum systems.

  • In contrast to equilibrium systems, non-trivial band structure topology does not guarantee that robust topological behaviour will be observed in a many-body system.

  • Various strategies can be employed to obtain novel topological phenomena in this non-equilibrium many-body setting.

  • Driving-induced topological edge states can be revealed through (non-quantized) mesoscopic transport.

  • Working in regimes where heating rates are strongly suppressed allows robust topological behaviour to be observed in long-lived transients.

  • Many-body localization due to strong disorder provides a mechanism for completely eliminating heating in certain types of systems, allowing a sharp delineation of intrinsic phases of many-body Floquet systems.

  • Bath engineering provides a powerful means for ensuring that the topological properties of a driven system coupled to its natural environment are reflected in its steady state.

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: Floquet band engineering for gapless and gapped systems.
Fig. 2: Topological features of Floquet bands without analogues in non-driven systems.
Fig. 3: Edge state transport in Floquet topological insulators.
Fig. 4: Heating via energy absorption from the drive.
Fig. 5: Steady states of Floquet topological insulators coupled to external baths.

References

  1. 1.

    Basov, D. N., Averitt, R. D. & Hsieh, D. Towards properties on demand in quantum materials. Nat. Mater. 16, 1077–1088 (2017).

    ADS  Google Scholar 

  2. 2.

    Floquet, G. Sur les equations differentielles lineaires a coefficients periodiques. Ann. Ecole Norm. Superieure 12, 47–88 (1883).

    MATH  Google Scholar 

  3. 3.

    Jaksch, D. & Zoller, P. Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. New J. Phys. 5, 56.1–56.11 (2003).

    Google Scholar 

  4. 4.

    Mueller, E. J. Artificial electromagnetism for neutral atoms: escher staircase and Laughlin liquids. Phys. Rev. A 70, 041603 (2004).

    ADS  Google Scholar 

  5. 5.

    Sørensen, A. S., Demler, E. & Lukin, M. D. Fractional quantum Hall states of atoms in optical lattices. Phys. Rev. Lett. 94, 086803 (2005).

    ADS  Google Scholar 

  6. 6.

    Yao, W., MacDonald, A. H. & Niu, Q. Optical control of topological quantum transport in semiconductors. Phys. Rev. Lett. 99, 047401 (2007).

    ADS  Google Scholar 

  7. 7.

    Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  Google Scholar 

  8. 8.

    Oka, T. & Aoki, H. Photovoltaic Hall effect in graphene. Phys. Rev. B 79, 081406 (2009).

    ADS  Google Scholar 

  9. 9.

    Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

    ADS  Google Scholar 

  10. 10.

    Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).

    Google Scholar 

  11. 11.

    D’Alessio, L. & Rigol, M. Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014).

    Google Scholar 

  12. 12.

    Lazarides, A., Das, A. & Moessner, R. Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014).

    ADS  Google Scholar 

  13. 13.

    Prosen, T. Time evolution of a quantum many-body system: transition from integrability to ergodicity in the thermodynamic limit. Phys. Rev. Lett. 80, 1808 (1998).

    ADS  Google Scholar 

  14. 14.

    Kukuljan, I. & Prosen, T. Corner transfer matrices for 2D strongly coupled many-body Floquet systems. J. Stat. Mech. 2016, 043305 (2016).

    MathSciNet  Google Scholar 

  15. 15.

    Citro, R. et al. Dynamical stability of a many-body Kapitza pendulum. Ann. Phys. 360, 694–710 (2015).

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Chandran, A. & Sondhi, S. L. Interaction-stabilized steady states in the driven O(N) model. Phys. Rev. B 93, 174305 (2016).

    ADS  Google Scholar 

  17. 17.

    Haldar, A., Moessner, R. & Das, A. Onset of Floquet thermalization. Phys. Rev. B 97, 245122 (2018).

    ADS  Google Scholar 

  18. 18.

    Seetharam, K., Titum, P., Kolodrubetz, M. & Refael, G. Absence of thermalization in finite isolated interacting Floquet systems. Phys. Rev. B 97, 014311 (2018).

    ADS  Google Scholar 

  19. 19.

    Grushin, A. G., Gómez-León, Á. & Neupert, T. Floquet fractional Chern insulators. Phys. Rev. Lett. 112, 156801 (2014).

    ADS  Google Scholar 

  20. 20.

    Klinovaja, J., Stano, P. & Loss, D. Topological floquet phases in driven coupled Rashba nanowires. Phys. Rev. Lett. 116, 176401 (2016).

    ADS  Google Scholar 

  21. 21.

    Liu, J., Hejazi, K. & Balents, L. Floquet engineering of multiorbital Mott insulators: applications to orthorhombic titanates. Phys. Rev. Lett. 121, 107201 (2018).

    ADS  Google Scholar 

  22. 22.

    Görg, F. et al. Enhancement and sign change of magnetic correlations in a driven quantum many-body system. Nature 553, 481–485 (2018).

    ADS  Google Scholar 

  23. 23.

    Kennes, D. M., de la Torre, A., Ron, A., Hsieh, D. & Millis, A. J. Floquet engineering in quantum chains. Phys. Rev. Lett. 120, 127601 (2018).

    ADS  Google Scholar 

  24. 24.

    Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).

    Google Scholar 

  25. 25.

    Nathan, F. & Rudner, M. S. Topological singularities and the general classification of Floquet–Bloch systems. New J. Phys. 17, 125014 (2015).

    ADS  Google Scholar 

  26. 26.

    Roy, R. & Harper, F. Floquet topological phases with symmetry in all dimensions. Phys. Rev. B 95, 195128 (2017).

    ADS  Google Scholar 

  27. 27.

    Roy, R. & Harper, F. Periodic table for Floquet topological insulators. Phys. Rev. B 96, 155118 (2017).

    ADS  Google Scholar 

  28. 28.

    Yao, S., Yan, Z. & Wang, Z. Topological invariants of Floquet systems: general formulation, special properties, and Floquet topological defects. Phys. Rev. B 96, 195303 (2017).

    ADS  Google Scholar 

  29. 29.

    Gómez-León, A. & Platero, G. Floquet–Bloch theory and topology in periodically driven lattices. Phys. Rev. Lett. 110, 200403 (2013).

    ADS  Google Scholar 

  30. 30.

    Graf, G. M. & Tauber, C. Bulk-edge correspondence for two-dimensional Floquet topological insulators. Ann. Henri Poincare 19, 709–741 (2018).

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Shapiro, J. & Tauber, C. Strongly disordered Floquet topological systems. Ann. Henri Poincare 20, 1837–1875 (2019).

    ADS  MathSciNet  MATH  Google Scholar 

  32. 32.

    Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. Phase structure of driven quantum systems. Phys. Rev. Lett. 116, 250401 (2016).

    ADS  Google Scholar 

  33. 33.

    Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).

    ADS  Google Scholar 

  34. 34.

    Else, D. V., Bauer, B. & Nayak, C. Prethermal phases of matter protected by time-translation symmetry. Phys. Rev. X 7, 011026 (2017).

    Google Scholar 

  35. 35.

    Rudner, M. S. & Song, J. C. W. Self-induced Berry flux and spontaneous non-equilibrium magnetism. Nat. Phys. 15, 1017–1021 (2019).

    Google Scholar 

  36. 36.

    Nag, T., Slager, R.-J., Higuchi, T. & Oka, T. Dynamical synchronization transition in interacting electron systems. Phys. Rev. B 100, 134301 (2019).

    ADS  Google Scholar 

  37. 37.

    Kinoshita, S., Murata, K. & Oka, T. Holographic floquet states II: Floquet condensation of vector mesons in nonequilibrium phase diagram. J. High Energy Phys. 2018, 96 (2018).

    Google Scholar 

  38. 38.

    Harper, F., Roy, R., Rudner, M. S. & Sondhi, S. Topology and broken symmetry in Floquet systems. Annu. Rev. Condens. Matter Phys. 11, 345–368 (2020).

    Google Scholar 

  39. 39.

    von Keyserlingk, C. W. & Sondhi, S. L. Phase structure of one-dimensional interacting Floquet systems. I. Abelian symmetry-protected topological phases. Phys. Rev. B 93, 245145 (2016).

    ADS  Google Scholar 

  40. 40.

    Potter, A. C., Morimoto, T. & Vishwanath, A. Classification of interacting topological Floquet phases in one dimension. Phys. Rev. X 6, 041001 (2016).

    Google Scholar 

  41. 41.

    Else, D. V. & Nayak, C. Classification of topological phases in periodically driven interacting systems. Phys. Rev. B 93, 201103 (2016).

    ADS  Google Scholar 

  42. 42.

    Harper, F. & Roy, R. Floquet topological order in interacting systems of bosons and fermions. Phys. Rev. Lett. 118, 115301 (2017).

    ADS  Google Scholar 

  43. 43.

    Moessner, R. & Sondhi, S. L. Equilibration and order in quantum Floquet matter. Nat. Phys. 13, 424–428 (2017).

    Google Scholar 

  44. 44.

    Eckardt, A. Colloquium: atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys. 89, 011004 (2017).

    ADS  MathSciNet  Google Scholar 

  45. 45.

    Cooper, N. R., Dalibard, J. & Spielman, I. B. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).

    ADS  MathSciNet  Google Scholar 

  46. 46.

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

    ADS  MathSciNet  Google Scholar 

  47. 47.

    Wang, Y. H., Steinberg, H., Jarillo-Herrero, P. & Gedik, N. Observation of Floquet–Bloch states on the surface of a topological insulator. Science 342, 453–457 (2013).

    ADS  Google Scholar 

  48. 48.

    McIver, J. W. et al. Light-induced anomalous Hall effect in graphene. Nat. Phys. 16, 38–41 (2019).

    Google Scholar 

  49. 49.

    Rudner, M. S. & Lindner, N. H. The Floquet engineer’s handbook. Preprint at https://arxiv.org/abs/2003.08252 (2020).

  50. 50.

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

    ADS  MathSciNet  Article  Google Scholar 

  51. 51.

    Kitagawa, T., Oka, T., Brataas, A., Fu, L. & Demler, E. Transport properties of nonequilibrium systems under the application of light: photoinduced quantum Hall insulators without Landau levels. Phys. Rev. B 84, 235108 (2011).

    ADS  Google Scholar 

  52. 52.

    Lindner, N. H., Bergman, D. L., Refael, G. & Galitski, V. Topological Floquet spectrum in three dimensions via a two-photon resonance. Phys. Rev. B 87, 235131 (2013).

    ADS  Google Scholar 

  53. 53.

    Sie, E. J. et al. Valley-selective optical Stark effect in monolayer WS2. Nat. Mater. 14, 290–294 (2015).

    ADS  Google Scholar 

  54. 54.

    Usaj, G., Perez-Piskunow, P. M., Foa Torres, L. E. F. & Balseiro, C. A. Irradiated graphene as a tunable Floquet topological insulator. Phys. Rev. B 90, 115423 (2014).

    ADS  Google Scholar 

  55. 55.

    Kundu, A., Fertig, H. A. & Seradjeh, B. Effective theory of Floquet topological transitions. Phys. Rev. Lett. 113, 236803 (2014).

    ADS  Google Scholar 

  56. 56.

    Quelle, A., Goerbig, M. O. & Smith, C. M. Bandwidth-resonant Floquet states in honeycomb optical lattices. New J. Phys. 18, 015006 (2016).

    ADS  Google Scholar 

  57. 57.

    Gu, Z., Fertig, H. A., Arovas, D. P. & Auerbach, A. Floquet spectrum and transport through an irradiated graphene ribbon. Phys. Rev. Lett. 107, 216601 (2011).

    ADS  Google Scholar 

  58. 58.

    Rodriguez-Vega, M. & Seradjeh, B. Universal fluctuations of Floquet topological invariants at low frequencies. Phys. Rev. Lett. 121, 036402 (2018).

    ADS  Google Scholar 

  59. 59.

    Delplace, P., Gómez-León, A. & Platero, G. Merging of Dirac points and Floquet topological transitions in ac-driven graphene. Phys. Rev. B 88, 245422 (2013).

    ADS  Google Scholar 

  60. 60.

    Sentef, M. A. et al. Theory of Floquet band formation and local pseudospin textures in pump–probe photoemission of graphene. Nat. Commun. 6, 7047 (2015).

    ADS  Google Scholar 

  61. 61.

    Iadecola, T. et al. Materials design from nonequilibrium steady states: driven graphene as a tunable semiconductor with topological properties. Phys. Rev. Lett. 110, 176603 (2013).

    ADS  Google Scholar 

  62. 62.

    Jiang, L. et al. Majorana fermions in equilibrium and in driven cold-atom quantum wires. Phys. Rev. Lett. 106, 220402 (2011).

    ADS  Google Scholar 

  63. 63.

    Thakurathi, M., Loss, D. & Klinovaja, J. Floquet Majorana fermions and parafermions in driven Rashba nanowires. Phys. Rev. B 95, 155407 (2017).

    ADS  Google Scholar 

  64. 64.

    Kennes, D. M. et al. Chiral one-dimensional Floquet topological insulators beyond the rotating wave approximation. Phys. Rev. B 100, 041103 (2019).

    ADS  Google Scholar 

  65. 65.

    Wang, R., Wang, B., Shen, R., Sheng, L. & Xing, D. Y. Floquet Weyl semimetal induced by off-resonant light. Europhys. Lett. 105, 17004 (2014).

    ADS  Google Scholar 

  66. 66.

    Chan, C.-K., Lee, P. A., Burch, K. S., Han, J. H. & Ran, Y. When chiral photons meet chiral fermions: photoinduced anomalous Hall effects in Weyl semimetals. Phys. Rev. Lett. 116, 026805 (2016).

    ADS  Google Scholar 

  67. 67.

    Chan, C.-K., Oh, Y.-T., Han, J. H. & Lee, P. A. Type-II Weyl cone transitions in driven semimetals. Phys. Rev. B 94, 121106 (2016).

    ADS  Google Scholar 

  68. 68.

    Hübener, H., Sentef, M. A., de Giovannini, U., Kemper, A. F. & Rubio, A. Creating stable Floquet–Weyl semimetals by laser-driving of 3D Dirac materials. Nat. Commun. 8, 13940 (2017).

    ADS  Google Scholar 

  69. 69.

    Fleury, R., Khanikev, A. B. & Alu, A. Floquet topological insulators for sound. Nat. Commun. 7, 11744 (2016).

    ADS  Google Scholar 

  70. 70.

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

    ADS  Google Scholar 

  71. 71.

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

    ADS  Google Scholar 

  72. 72.

    Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009).

    ADS  MATH  Google Scholar 

  73. 73.

    Ryu, S., Schnyder, A. P., Furusaki, A. & Ludwig, A. W. W. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010).

    ADS  Google Scholar 

  74. 74.

    Thouless, D. J. Quantization of particle transport. Phys. Rev. B 27, 6083 (1983).

    ADS  MathSciNet  Google Scholar 

  75. 75.

    Gross, D., Nesme, V., Vogts, H. & Werner, R. F. Index theory of one dimensional quantum walks and cellular automata. Commun. Math. Phys. 310, 419–454 (2012).

    ADS  MathSciNet  MATH  Google Scholar 

  76. 76.

    Higashikawa, S., Nakagawa, M. & Ueda, M. Floquet chiral magnetic effect. Phys. Rev. Lett. 123, 066403 (2019).

    ADS  MathSciNet  Google Scholar 

  77. 77.

    Sun, X.-Q., Xiao, M., Bzdušek, Tcv, Zhang, S.-C. & Fan, S. Three-dimensional chiral lattice fermion in Floquet systems. Phys. Rev. Lett. 121, 196401 (2018).

    ADS  Google Scholar 

  78. 78.

    Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    ADS  Google Scholar 

  79. 79.

    Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185–2190 (1982).

    ADS  Google Scholar 

  80. 80.

    Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697 (1993).

    ADS  MathSciNet  MATH  Google Scholar 

  81. 81.

    Carpentier, D., Delplace, P., Fruchart, M. & Gawedzki, K. Topological index for periodically driven time-reversal invariant 2D systems. Phys. Rev. Lett. 114, 106806 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  82. 82.

    Broome, M. A. et al. Discrete single-photon quantum walks with tunable decoherence. Phys. Rev. Lett. 104, 153602 (2010).

    ADS  Google Scholar 

  83. 83.

    Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

    ADS  Google Scholar 

  84. 84.

    Hu, W. et al. Measurement of a topological edge invariant in a microwave network. Phys. Rev. X 5, 011012 (2015).

    Google Scholar 

  85. 85.

    Cheng, Q. et al. Observation of anomalous π modes in photonic Floquet engineering. Phys. Rev. Lett. 122, 173901 (2019).

    ADS  Google Scholar 

  86. 86.

    Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly-driven photonic lattice. Nat. Commun. 8, 13918 (2017).

    ADS  Google Scholar 

  87. 87.

    Maczewsky, L. J., Zeuner, J. M., Nolte, S. & Szameit, A. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).

    ADS  Google Scholar 

  88. 88.

    Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296–300 (2016).

    Google Scholar 

  89. 89.

    Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350–354 (2016).

    Google Scholar 

  90. 90.

    Reichl, M. D. & Mueller, E. J. Floquet edge states with ultracold atoms. Phys. Rev. A 89, 063628 (2014).

    ADS  Google Scholar 

  91. 91.

    Quelle, A., Weitenberg, C., Sengstock, K. & Morais Smith, C. Driving protocol for a Floquet topological phase without static counterpart. New J. Phys. 19, 113010 (2017).

    ADS  Google Scholar 

  92. 92.

    Liu, D. T., Shabani, J. & Mitra, A. Floquet Majorana zero and π modes in planar Josephson junctions. Phys. Rev. B 99, 094303 (2019).

    ADS  Google Scholar 

  93. 93.

    Kundu, A. & Seradjeh, B. Transport signatures of Floquet Majorana fermions in driven topological superconductors. Phys. Rev. Lett. 111, 136402 (2013).

    ADS  Google Scholar 

  94. 94.

    Foa Torres, L. E. F., Perez-Piskunow, P. M., Balseiro, C. A. & Usaj, G. Multiterminal conductance of a Floquet topological insulator. Phys. Rev. Lett. 113, 266801 (2014).

    ADS  Google Scholar 

  95. 95.

    Farrell, A. & Pereg-Barnea, T. Photon-inhibited topological transport in quantum well heterostructures. Phys. Rev. Lett. 115, 106403 (2015).

    ADS  Google Scholar 

  96. 96.

    Farrell, A. & Pereg-Barnea, T. Edge-state transport in Floquet topological insulators. Phys. Rev. B 93, 045121 (2016).

    ADS  Google Scholar 

  97. 97.

    Kundu, A., Rudner, M. S., Berg, E. & Lindner, N. H. Quantized large-bias current in the anomalous Floquet–Anderson insulator. Phys. Rev. B 101, 041403(R) (2020).

    ADS  Google Scholar 

  98. 98.

    Salerno, G. et al. Quantized Hall conductance of a single atomic wire: a proposal based on synthetic dimensions. Phys. Rev. X 9, 041001 (2019).

    Google Scholar 

  99. 99.

    Kohler, S., Lehmann, J. & Hanggi, P. Driven quantum transport on the nanoscale. Phys. Rep. 406, 379–443 (2005).

    ADS  Google Scholar 

  100. 100.

    Perez-Piskunow, P. M., Foa Torres, L. E. F. & Usaj, G. Hierarchy of Floquet gaps and edge states for driven honeycomb lattices. Phys. Rev. A 91, 043625 (2015).

    ADS  Google Scholar 

  101. 101.

    Uhrig, G. S., Kalthoff, M. H. & Freericks, J. K. Positivity of the spectral densities of retarded Floquet Green functions. Phys. Rev. Lett. 122, 130604 (2019).

    ADS  Google Scholar 

  102. 102.

    Sengupta, K., Žutić, I., Kwon, H.-J., Yakovenko, V. M. & Das Sarma, S. Midgap edge states and pairing symmetry of quasi-one-dimensional organic superconductors. Phys. Rev. B 63, 144531 (2001).

    ADS  Google Scholar 

  103. 103.

    Law, K. T., Lee, P. A. & Ng, T. K. Majorana fermion induced resonant Andreev reflection. Phys. Rev. Lett. 103, 237001 (2009).

    ADS  Google Scholar 

  104. 104.

    Titum, P., Berg, E., Rudner, M. S., Refael, G. & Lindner, N. H. Anomalous Floquet–Anderson insulator as a nonadiabatic quantized charge pump. Phys. Rev. X 6, 021013 (2016).

    Google Scholar 

  105. 105.

    Dahlhaus, J. P., Fregoso, B. M. & Moore, J. E. Magnetization signatures of light-induced quantum Hall edge states. Phys. Rev. Lett. 114, 246802 (2015).

    ADS  Google Scholar 

  106. 106.

    Nathan, F., Rudner, M. S., Lindner, N. H., Berg, E. & Refael, G. Quantized magnetization density in periodically driven systems. Phys. Rev. Lett. 119, 186801 (2017).

    ADS  Google Scholar 

  107. 107.

    Mahmood, F. et al. Selective scattering between Floquet–Bloch and Volkov states in a topological insulator. Nat. Phys. 12, 306–310 (2016).

    Google Scholar 

  108. 108.

    Fregoso, B. M., Wang, Y. H., Gedik, N. & Galitski, V. Driven electronic states at the surface of a topological insulator. Phys. Rev. B 88, 155129 (2013).

    ADS  Google Scholar 

  109. 109.

    Farrell, A., Arsenault, A. & Pereg-Barnea, T. Dirac cones, Floquet side bands, and theory of time-resolved angle-resolved photoemission. Phys. Rev. B 94, 155304 (2016).

    ADS  Google Scholar 

  110. 110.

    Kandelaki, E. & Rudner, M. S. Many-body dynamics and gap opening in interacting periodically driven systems. Phys. Rev. Lett. 121, 036801 (2018).

    ADS  Google Scholar 

  111. 111.

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

    ADS  Google Scholar 

  112. 112.

    Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).

    Google Scholar 

  113. 113.

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

    ADS  Google Scholar 

  114. 114.

    Flaschner, N. et al. Observation of dynamical vortices after quenches in a system with topology. Nat. Phys. 14, 265–268 (2018).

    Google Scholar 

  115. 115.

    Tarnowski, M. et al. Measuring topology from dynamics by obtaining the Chern number from a linking number. Nat. Commun. 10, 1728 (2019).

    ADS  Google Scholar 

  116. 116.

    Asteria, L. et al. Measuring quantized circular dichroism in ultracold topological matter. Nat. Phys. 15, 449–454 (2019).

    Google Scholar 

  117. 117.

    Bukov, M., D’Alessio, L. & Polkovnikov, A. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139–226 (2015).

    ADS  Google Scholar 

  118. 118.

    Eckardt, A. & Anisimovas, E. High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective. New J. Phys. 17, 093039 (2015).

    ADS  Google Scholar 

  119. 119.

    Kuwahara, T., Mori, T. & Saito, K. Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems. Ann. Phys. 367, 96–124 (2016).

    ADS  MathSciNet  MATH  Google Scholar 

  120. 120.

    Abanin, D. A., De Roeck, W., Ho, W. W. & Huveneers, F. Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems. Phys. Rev. B 95, 014112 (2017).

    ADS  Google Scholar 

  121. 121.

    Lindner, N. H., Berg, E. & Rudner, M. S. Universal Chiral quasisteady states in periodically driven many-body systems. Phys. Rev. X 7, 011018 (2017).

    Google Scholar 

  122. 122.

    Mori, T., Ikeda, T. N., Kamanishi, E. & Ueda, M. Thermalization and prethermalization in isolated quantum systems: a theoretical overview. J. Phys. B 51, 112001 (2018).

    ADS  Google Scholar 

  123. 123.

    Bukov, M., Heyl, M., Huse, D. A. & Polkovnikov, A. Heating and many-body resonances in a periodically driven two-band system. Phys. Rev. B 93, 155132 (2016).

    ADS  Google Scholar 

  124. 124.

    Abanin, D. A., De Roeck, W. & Huveneers, F. Exponentially slow heating in periodically driven many-body systems. Phys. Rev. Lett. 115, 256803 (2015).

    ADS  Google Scholar 

  125. 125.

    Bilitewski, T. & Cooper, N. R. Scattering theory for Floquet–Bloch states. Phys. Rev. A 91, 033601 (2015).

    ADS  MathSciNet  Google Scholar 

  126. 126.

    Reitter, M. et al. Interaction dependent heating and atom loss in a periodically driven optical lattice. Phys. Rev. Lett. 119, 200402 (2017).

    ADS  Google Scholar 

  127. 127.

    Abanin, D., De Roeck, W., Ho, W. W. & Huveneers, F. A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems. Commun. Math. Phys. 354, 809–827 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  128. 128.

    Mori, T. Floquet prethermalization in periodically driven classical spin systems. Phys. Rev. B 98, 104303 (2018).

    ADS  Google Scholar 

  129. 129.

    Howell, O., Weinberg, P., Sels, D., Polkovnikov, A. & Bukov, M. Asymptotic prethermalization in periodically driven classical spin chains. Phys. Rev. Lett. 122, 010602 (2019).

    ADS  Google Scholar 

  130. 130.

    Vogl, M., Laurell, P., Barr, A. D. & Fiete, G. A. Flow equation approach to periodically driven quantum systems. Phys. Rev. X 9, 021037 (2019).

    Google Scholar 

  131. 131.

    Regnault, N. & Bernevig, B. A. Fractional Chern insulator. Phys. Rev. X 1, 021014 (2011).

    Google Scholar 

  132. 132.

    Claassen, M., Jiang, H.-C., Mortiz, B. & Devereaux, T. P. Dynamical time-reversal symmetry breaking and photo-induced chiral spin liquids in frustrated mott insulators. Nat. Commun. 8, 1192 (2017).

    ADS  Google Scholar 

  133. 133.

    Liu, J., Hejazi, K. & Balents, L. Floquet engineering of multiorbital Mott insulators: applications to orthorhombic titanates. Phys. Rev. Lett. 121, 107201 (2018).

    ADS  Google Scholar 

  134. 134.

    Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126–1205 (2006).

    ADS  MATH  Google Scholar 

  135. 135.

    Oganesyan, V. & Huse, D. A. Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007).

    ADS  Google Scholar 

  136. 136.

    Pal, A. & Huse, D. A. Many-body localization phase transition. Phys. Rev. B 82, 174411 (2010).

    ADS  Google Scholar 

  137. 137.

    Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

    ADS  Google Scholar 

  138. 138.

    Lazarides, A., Das, A. & Moessner, R. Fate of many-body localization under periodic driving. Phys. Rev. Lett. 115, 030402 (2015).

    ADS  Google Scholar 

  139. 139.

    Ponte, P., Papić, Z., Huveneers, F. & Abanin, D. A. Many-body localization in periodically driven systems. Phys. Rev. Lett. 114, 140401 (2015).

    ADS  Google Scholar 

  140. 140.

    Bordia, P., Luschen, H., Schneider, U., Knap, M. & Bloch, I. Periodically driving a many-body localized quantum system. Nat. Phys. 13, 460–464 (2017).

    Google Scholar 

  141. 141.

    Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012).

    ADS  Google Scholar 

  142. 142.

    Zeng, T.-S. & Sheng, D. N. Prethermal time crystals in a one-dimensional periodically driven Floquet system. Phys. Rev. B 96, 094202 (2017).

    ADS  Google Scholar 

  143. 143.

    Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).

    ADS  Google Scholar 

  144. 144.

    Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017).

    ADS  Google Scholar 

  145. 145.

    Dykman, M. I., Bruder, C., Lörch, N. & Zhang, Y. Interaction-induced time-symmetry breaking in driven quantum oscillators. Phys. Rev. B 98, 195444 (2018).

    ADS  Google Scholar 

  146. 146.

    Nandkishore, R. & Potter, A. C. Marginal Anderson localization and many-body delocalization. Phys. Rev. B 90, 195115 (2014).

    ADS  Google Scholar 

  147. 147.

    Po, H. C., Fidkowski, L., Morimoto, T., Potter, A. C. & Vishwanath, A. Chiral Floquet phases of many-body localized bosons. Phys. Rev. X 6, 041070 (2016).

    Google Scholar 

  148. 148.

    Po, H. C., Fidkowski, L., Vishwanath, A. & Potter, A. C. Radical chiral Floquet phases in a periodically driven Kitaev model and beyond. Phys. Rev. B 96, 245116 (2017).

    ADS  Google Scholar 

  149. 149.

    Nathan, F., Abanin, D., Berg, E., Lindner, N. H. & Rudner, M. S. Anomalous Floquet insulators. Phys. Rev. B 99, 195133 (2019).

    ADS  Google Scholar 

  150. 150.

    Liu, D. E. Classification of the Floquet statistical distribution for time-periodic open systems. Phys. Rev. B 91, 144301 (2015).

    ADS  Google Scholar 

  151. 151.

    Shirai, T., Mori, T. & Miyashita, S. Condition for emergence of the Floquet–Gibbs state in periodically driven open systems. Phys. Rev. E 91, 030101(R) (2015).

    ADS  MathSciNet  Google Scholar 

  152. 152.

    Shirai, T. et al. Effective Floquet–Gibbs states for dissipative quantum systems. New J. Phys. 18, 053008 (2016).

    ADS  Google Scholar 

  153. 153.

    Torres, M. & Kunold, A. Kubo formula for floquet states and photoconductivity oscillations in a two-dimensional electron gas. Phys. Rev. B 71, 115313 (2005).

    ADS  Google Scholar 

  154. 154.

    Mahan, G. D. Many-Particle Physics (Springer, 2000).

  155. 155.

    Kohn, W. Periodic thermodynamics. J. Stat. Phys. 103, 417–423 (2001).

    MathSciNet  MATH  Google Scholar 

  156. 156.

    Hone, D. W., Ketzmerick, R. & Kohn, W. Time-dependent Floquet theory and absence of an adiabatic limit. Phys. Rev. A 56, 4045 (1997).

    ADS  Google Scholar 

  157. 157.

    Hone, D. W., Ketzmerick, R. & Kohn, W. Statistical mechanics of Floquet systems: the pervasive problem of near degeneracies. Phys. Rev. E 79, 051129 (2009).

    ADS  MathSciNet  Google Scholar 

  158. 158.

    Dehghani, H., Oka, T. & Mitra, A. Out-of-equilibrium electrons and the Hall conductance of a Floquet topological insulator. Phys. Rev. B 91, 155422 (2015).

    ADS  Google Scholar 

  159. 159.

    Genske, M. & Rosch, A. Floquet–Boltzmann equation for periodically driven Fermi systems. Phys. Rev. A 92, 062108 (2015).

    ADS  Google Scholar 

  160. 160.

    Esin, I., Rudner, M. S., Refael, G. & Lindner, N. H. Quantized transport and steady states of Floquet topological insulators. Phys. Rev. B 97, 245401 (2018).

    ADS  Google Scholar 

  161. 161.

    Goebel, E. O. & Hildebrand, O. Thermalization of the electron–hole plasma in GaAs. Phys. Stat. Sol. 88, 645–652 (1978).

    ADS  Google Scholar 

  162. 162.

    Glazman, L. I. Resonant excitation of carriers in a seminconductor by a high-power laser pulse. Sov. Phys. JETP 53, 178–181 (1981).

    Google Scholar 

  163. 163.

    Glazman, L. I. Kinetics of electrons and holes in direct gap seminconductors photoexcited by high intensity pulses. Sov. Phys. Semi 17, 494–498 (1983).

    Google Scholar 

  164. 164.

    Chow, W. W. & Koch, S. W. Semiconductor-Laser Fundamentals (Springer, 1999).

  165. 165.

    Huang, M. H. et al. Room-temperature ultraviolet nanowire nanolasers. Science 292, 1897–1899 (2001).

    ADS  Google Scholar 

  166. 166.

    Röder, R. et al. Continuous wave nanowire lasing. Nano Lett. 13, 3602–3606 (2013).

    ADS  Google Scholar 

  167. 167.

    Dehghani, H., Oka, T. & Mitra, A. Dissipative Floquet topological systems. Phys. Rev. B 90, 195429 (2014).

    ADS  Google Scholar 

  168. 168.

    Seetharam, K. I., Bardyn, C.-E., Lindner, N. H., Rudner, M. S. & Refael, G. Controlled population of Floquet–Bloch states via coupling to Bose and Fermi baths. Phys. Rev. X 5, 041050 (2015).

    Google Scholar 

  169. 169.

    Iadecola, T., Neupert, T. & Chamon, C. Occupation of topological Floquet bands in open systems. Phys. Rev. B 91, 235133 (2015).

    ADS  Google Scholar 

  170. 170.

    Dykman, M. I., Marthaler, M. & Peano, V. Quantum heating of a parametrically modulated oscillator: spectral signatures. Phys. Rev. A 83, 052115 (2011).

    ADS  Google Scholar 

  171. 171.

    Galitskii, V. M., Goreslavskii, S. P. & Elesin, V. F. Electric and magnetic properties of a semiconductor in the field of a strong electromagnetic wave. Sov. Phys. JETP 30, 117–122 (1970).

    ADS  Google Scholar 

  172. 172.

    Sato, S. A. et al. Microscopic theory for the light-induced anomalous Hall effect in graphene. Phys. Rev. B 99, 214302 (2019).

    ADS  Google Scholar 

  173. 173.

    Singh, K. et al. Quantifying and controlling prethermal nonergodicity in interacting Floquet matter. Phys. Rev. X 9, 041021 (2019).

    Google Scholar 

  174. 174.

    Holthaus, M. Floquet engineering with quasienergy bands of periodically driven optical lattices. J. Phys. B 49, 013001 (2016).

    ADS  Google Scholar 

  175. 175.

    Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142 (1997).

    ADS  Google Scholar 

  176. 176.

    Asbóth, J. K. & Obuse, H. Bulk-boundary correspondence for chiral symmetric quantum walks. Phys. Rev. B 88, 121406 (2013).

    ADS  Google Scholar 

  177. 177.

    Asbóth, J. K., Tarasinski, B. & Delplace, P. Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems. Phys. Rev. B 90, 125143 (2014).

    ADS  Google Scholar 

  178. 178.

    Lababidi, M., Satija, I. I. & Zhao, E. Counter-propagating edge modes and topological phases of a kicked quantum Hall system. Phys. Rev. Lett. 112, 026805 (2014).

    ADS  Google Scholar 

  179. 179.

    Zhou, Z., Satija, I. I. & Zhao, E. Floquet edge states in a harmonically driven integer quantum Hall system. Phys. Rev. B 90, 205108 (2014).

    ADS  Google Scholar 

  180. 180.

    Yao, S., Yan, Z. & Wang, Z. Topological invariants of Floquet systems: general formulation, special properties, and Floquet topological defects. Phys. Rev. B 96, 195303 (2017).

    ADS  Google Scholar 

  181. 181.

    de Gennes, P. G. Superconductivity of Metals and Alloys (Springer, 2000).

  182. 182.

    Morimoto, T., Po, H. C. & Vishwanath, A. Floquet topological phases protected by time glide symmetry. Phys. Rev. B 95, 195155 (2017).

    ADS  Google Scholar 

  183. 183.

    Xu, S. & Wu, C. Space-time crystal and space-time group. Phys. Rev. Lett. 120, 096401 (2018).

    ADS  MathSciNet  Google Scholar 

  184. 184.

    Peng, Y. & Refael, G. Floquet second-order topological insulators from nonsymmorphic space-time symmetries. Phys. Rev. Lett. 123, 016806 (2019).

    ADS  Google Scholar 

  185. 185.

    Liu, D. E., Levchenko, A. & Baranger, H. U. Floquet Majorana fermions for topological qubits in superconducting devices and cold-atom systems. Phys. Rev. Lett. 111, 047002 (2013).

    ADS  Google Scholar 

  186. 186.

    Dal Lago, V., Atala, M. & Foa Torres, L. E. F. Floquet topological transitions in a driven one-dimensional topological insulator. Phys. Rev. A 92, 023624 (2015).

    ADS  Google Scholar 

  187. 187.

    Bauer, B. et al. Topologically protected braiding in a single wire using Floquet Majorana modes. Phys. Rev. B 100, 041102 (2019).

    ADS  Google Scholar 

Download references

Acknowledgements

The authors thank all of their collaborators on FTI-related work, with whom they have had many stimulating interactions. In particular, they acknowledge E. Berg, E. Demler, V. Galitski, T. Kitagawa, M. Levin and G. Refael, with whom they began their journey in this field. The authors also thank I. Esin for help with figures and helpful discussions. N.H.L. acknowledges support from the European Research Council (ERC) under the European Union Horizon 2020 Research and Innovation Programme (grant agreement number 639172), and from the Israeli Center of Research Excellence (I-CORE) ‘Circle of Light’. M.S.R. gratefully acknowledges the support of the European Research Council (ERC) under the European Union Horizon 2020 Research and Innovation Programme (grant agreement number 678862), the Villum Foundation, and CRC 183 of the Deutsche Forschungsgemeinschaft.

Author information

Affiliations

Authors

Contributions

M.S.R. and N.H.L. contributed equally to the organization and writing of this Review.

Corresponding author

Correspondence to Mark S. Rudner.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information

Nature Reviews Physics thanks the 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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rudner, M.S., Lindner, N.H. Band structure engineering and non-equilibrium dynamics in Floquet topological insulators. Nat Rev Phys 2, 229–244 (2020). https://doi.org/10.1038/s42254-020-0170-z

Download citation