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.

  • Review Article
  • Published:

Thouless pumping and topology

Nature Reviews Physics volume 5, pages 87–101 (2023)Cite this article

Subjects

Abstract

Thouless pumping provides one of the simplest manifestations of topology in quantum systems and has attracted a lot of recent interest, both theoretically and experimentally. Since the seminal works by David Thouless and Qian Niu in 1983 and 1984, it has been argued that the quantization of the pumped charge is robust against weak disorder, but a clear characterization of the localization properties of the relevant states, and the breakdown of quantized transport in the presence of interaction or out of the adiabatic approximation, has long been debated. Thouless pumping is also the first example of a topological phase emerging in a periodically driven system. Driven systems can exhibit exotic topological phases without any static analogue and have been the subject of many recent proposals both in fermionic and in bosonic systems. Recent experimental studies have been performed in diverse platforms ranging from cold atoms to photonics and condensed-matter systems. This Review serves as a basis to understand the robustness of the topology of slowly driven systems and also highlights the rich properties of topological pumps and their diverse range of applications. Examples include systems with synthetic dimensions or work towards understanding higher-order topological phases, which underline the relevance of topological pumping for the fast-growing field of topological quantum matter.

Key points

  • A direct current is usually associated with a dissipative flow of electrons in response to an applied bias voltage. In quantum systems, however, coherent transport can be induced via adiabatic cyclic variation of at least two system parameters in the absence of any external bias.

  • A Thouless pump is a ‘quantum’ pump, in which the amount of ‘charge’ pumped during one cycle is quantized according to the Chern number — a topological invariant.

  • Quantized charge pumps are robust to perturbations, such as disorder or weak interactions, as long as they do not change the topology of the pump.

  • One-dimensional topological pumps can be seen as dynamical versions of the 2D integer quantum Hall effect in (1+1)-D, in which time plays the role of one spatial dimension and can be understood as a synthetic dimension.

  • The concept of synthetic dimensions opens the door towards realizations of exotic higher-dimensional systems, such as the 4D quantum Hall effect.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

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

Fig. 1: Classical and quantum pumping.
Fig. 2: Topological charge pumping in a superlattice.
Fig. 3: Rice–Mele (RM) model.
Fig. 4: Topological pumping with interactions.
Fig. 5: Synthetic dimensions and higher-dimensional systems.

Similar content being viewed by others

References

  1. Altshuler, B. & Glazman, L. Pumping electrons. Science 283, 1864–1865 (1999).

    Article  Google Scholar 

  2. Zhou, F., Spivak, B. & Altshuler, B. Mesoscopic mechanism of adiabatic charge transport. Phys. Rev. Lett. 82, 608 (1999).

    Article  ADS  Google Scholar 

  3. Niu, Q. Towards a quantum pump of electric charges. Phys. Rev. Lett. 64, 1812–1815 (1990).

    Article  ADS  Google Scholar 

  4. Pekola, J. P. et al. Single-electron current sources: toward a refined definition of the ampere. Rev. Mod. Phys. 85, 1421–1472 (2013).

    Article  ADS  Google Scholar 

  5. Das, K. K., Kim, S. & Mizel, A. Controlled flow of spin-entangled electrons via adiabatic quantum pumping. Phys. Rev. Lett. 97, 096602 (2006).

    Article  ADS  Google Scholar 

  6. Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).

    Article  ADS  Google Scholar 

  7. Verbin, M., Zilberberg, O., Lahini, Y., Kraus, Y. E. & Silberberg, Y. Topological pumping over a photonic fibonacci quasicrystal. Phys. Rev. B 91, 064201 (2015).

    Article  ADS  Google Scholar 

  8. Ke, Y. et al. Topological phase transitions and Thouless pumping of light in photonic waveguide arrays. Laser Photonics Rev. 10, 995–1001 (2016).

    Article  ADS  Google Scholar 

  9. Cerjan, A., Wang, M., Huang, S., Chen, K. P. & Rechtsman, M. C. Thouless pumping in disordered photonic systems. Light Sci. Appl. 9, 178 (2020).

    Article  ADS  Google Scholar 

  10. Grinberg, I. H. et al. Robust temporal pumping in a magneto-mechanical topological insulator. Nat. Commun. 11, 974 (2020).

    Article  ADS  Google Scholar 

  11. Xia, Y. et al. Experimental observation of temporal pumping in electromechanical waveguides. Phys. Rev. Lett. 126, 095501 (2021).

    Article  ADS  Google Scholar 

  12. 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).

    Article  Google Scholar 

  13. Nakajima, S. et al. Topological thouless pumping of ultracold fermions. Nat. Phys. 12, 296 (2016).

    Article  Google Scholar 

  14. Lu, H.-I. et al. Geometrical pumping with a Bose–Einstein condensate. Phys. Rev. Lett. 116, 200402 (2016).

    Article  ADS  Google Scholar 

  15. Tangpanitanon, J. et al. Topological pumping of photons in nonlinear resonator arrays. Phys. Rev. Lett. 117, 213603 (2016).

    Article  ADS  Google Scholar 

  16. Jürgensen, M., Mukherjee, S. & Rechtsman, M. C. Quantized nonlinear thouless pumping. Nature 596, 63–67 (2021).

    Article  ADS  Google Scholar 

  17. Walter, A.-S. et al. Breakdown of quantisation in a Hubbard–Thouless pump. Preprint at https://arXiv.org/2204.06561 (2022).

  18. Fedorova, Z., Qiu, H., Linden, S. & Kroha, J. Observation of topological transport quantization by dissipation in fast Thouless pumps. Nat. Commun. 11, 3758 (2020).

    Article  ADS  Google Scholar 

  19. Dreon, D. et al. Self-oscillating pump in a topological dissipative atom–cavity system. Nature 608, 494–498 (2022).

    Article  ADS  Google Scholar 

  20. Thouless, D. J. Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  21. Niu, Q. & Thouless, D. J. Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction. J. Phys. A Math. Gen. 17, 2453–2462 (1984).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Schouten, K. J. M. & Cheianov, V. Rapid-cycle Thouless pumping in a one-dimensional optical lattice. Phys. Rev. A 104, 063315 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  23. Minguzzi, J. et al. Topological pumping in a Floquet–Bloch band. Phys. Rev. Lett. 129, 053201 (2022).

    Article  ADS  Google Scholar 

  24. Ke, Y. et al. Topological pumping assisted by Bloch oscillations. Phys. Rev. Res. 2, 033143 (2020).

    Article  Google Scholar 

  25. Rice, M. J. & Mele, E. J. Elementary excitations of a linearly conjugated diatomic polymer. Phys. Rev. Lett. 49, 1455–1459 (1982).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Wang, L., Troyer, M. & Dai, X. Topological charge pumping in a one-dimensional optical lattice. Phys. Rev. Lett. 111, 026802 (2013).

    Article  ADS  Google Scholar 

  28. Switkes, M., Marcus, C. M., Campman, K. & Gossard, A. C. An adiabatic quantum electron pump. Science 283, 1905–1908 (1999).

    Article  ADS  Google Scholar 

  29. Talyanskii, V. I. et al. Single-electron transport in a one-dimensional channel by high-frequency surface acoustic waves. Phys. Rev. B 56, 15180–15184 (1997).

    Article  ADS  Google Scholar 

  30. Brouwer, P. W. Scattering approach to parametric pumping. Phys. Rev. B 58, R10135–R10138 (1998).

    Article  ADS  Google Scholar 

  31. Zhou, H.-Q., Cho, S. Y. & McKenzie, R. H. Gauge fields, geometric phases, and quantum adiabatic pumps. Phys. Rev. Lett. 91, 186803 (2003).

    Article  ADS  Google Scholar 

  32. Schweizer, C., Lohse, M., Citro, R. & Bloch, I. Spin pumping and measurement of spin currents in optical superlattices. Phys. Rev. Lett. 117, 170405 (2016).

    Article  ADS  Google Scholar 

  33. Sheng, D. N., Weng, Z. Y., Sheng, L. & Haldane, F. D. M. Quantum spin-Hall effect and topologically invariant Chern numbers. Phys. Rev. Lett. 97, 036808 (2006).

    Article  ADS  Google Scholar 

  34. Bernevig, B. A. Topological insulators and topological superconductors. In Topological Insulators and Topological Superconductors Ch. 10, 123–138 (Princeton Univ. Press, 2013).

  35. Shindou, R. Quantum spin pump in s=1/2 antiferromagnetic chains ‘holonomy of phase operators in sine-Gordon theory’. J. Phys. Soc. Jpn. 74, 1214–1223 (2005).

    Article  ADS  MATH  Google Scholar 

  36. Zhou, C. Q. et al. Proposal for a topological spin Chern pump. Phys. Rev. B 90, 085133 (2014).

    Article  ADS  Google Scholar 

  37. Chen, Q., Cai, J. & Zhang, S. Topological quantum pumping in spin-dependent superlattices with glide symmetry. Phys. Rev. A 101, 043614 (2020).

    Article  ADS  Google Scholar 

  38. Meidan, D., Micklitz, T. & Brouwer, P. W. Topological classification of interaction-driven spin pumps. Phys. Rev. B 84, 075325 (2011).

    Article  ADS  Google Scholar 

  39. Cazalilla, M. A., Citro, R., Giamarchi, T., Orignac, E. & Rigol, M. One dimensional bosons: from condensed matter systems to ultracold gases. Rev. Mod. Phys. 83, 1405–1466 (2011).

    Article  ADS  Google Scholar 

  40. Shih, W.-K. & Niu, Q. Nonadiabatic particle transport in a one-dimensional electron system. Phys. Rev. B 50, 11902–11910 (1994).

    Article  ADS  Google Scholar 

  41. von Klitzing, K. The quantized Hall effect. Rev. Mod. Phys. 58, 519–531 (1986).

    Article  ADS  Google Scholar 

  42. Avron, J. E. & Kons, Z. Quantum response at finite fields and breakdown of Chern numbers. J. Phys. A Math. Gen. 32, 6097–6113 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Privitera, L., Russomanno, A., Citro, R. & Santoro, G. E. Nonadiabatic breaking of topological pumping. Phys. Rev. Lett. 120, 106601 (2018).

    Article  ADS  Google Scholar 

  44. Sambe, H. Steady states and quasienergies of a quantum-mechanical system in an oscillating field. Phys. Rev. A 7, 2203–2213 (1973).

    Article  ADS  Google Scholar 

  45. Shirley, J. H. Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev. 138, B979–B987 (1965).

    Article  ADS  Google Scholar 

  46. Ferrari, R. Floquet energies and quantum hall effect in a periodic potential. Int. J. Mod. Phys. B 12, 1105–1123 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  48. Russomanno, A., Pugnetti, S., Brosco, V. & Fazio, R. Floquet theory of Cooper pair pumping. Phys. Rev. B 83, 214508 (2011).

    Article  ADS  Google Scholar 

  49. 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 

  50. Russomanno, A., Silva, A. & Santoro, G. E. Periodic steady regime and interference in a periodically driven quantum system. Phys. Rev. Lett. 109, 257201 (2012).

    Article  ADS  Google Scholar 

  51. Lazarides, A., Das, A. & Moessner, R. Periodic thermodynamics of isolated quantum systems. Phys. Rev. Lett. 112, 150401 (2014).

    Article  ADS  Google Scholar 

  52. Rigolin, G., Ortiz, G. & Ponce, V. H. Beyond the quantum adiabatic approximation: adiabatic perturbation theory. Phys. Rev. A 78, 052508 (2008).

    Article  ADS  Google Scholar 

  53. Wauters, M. M., Russomanno, A., Citro, R., Santoro, G. E. & Privitera, L. Localization, topology, and quantized transport in disordered Floquet systems. Phys. Rev. Lett. 123, 266601 (2019).

    Article  ADS  Google Scholar 

  54. Hayward, A. L. C., Bertok, E., Schneider, U. & Heidrich-Meisner, F. Effect of disorder on topological charge pumping in the Rice–Mele model. Phys. Rev. A 103, 043310 (2021).

    Article  ADS  Google Scholar 

  55. Hu, S., Ke, Y. & Lee, C. Topological quantum transport and spatial entanglement distribution via a disordered bulk channel. Phys. Rev. A 101, 052323 (2020).

    Article  ADS  Google Scholar 

  56. Ippoliti, M. & Bhatt, R. N. Dimensional crossover of the integer quantum Hall plateau transition and disordered topological pumping. Phys. Rev. Lett. 124, 086602 (2020).

    Article  ADS  Google Scholar 

  57. Marra, P. & Nitta, M. Topologically quantized current in quasiperiodic Thouless pumps. Phys. Rev. Res. 2, 042035 (2020).

    Article  Google Scholar 

  58. Wang, R. & Song, Z. Robustness of the pumping charge to dynamic disorder. Phys. Rev. B 100, 184304 (2019).

    Article  ADS  Google Scholar 

  59. Qin, J. & Guo, H. Quantum pumping induced by disorder in one dimension. Phys. Lett. A 380, 2317–2321 (2016).

    Article  ADS  Google Scholar 

  60. Grifoni, M. & Hänggi, P. Driven quantum tunneling. Phys. Rep. 304, 229–354 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  61. Abrahams, E., Anderson, P. W., Licciardello, D. C. & Ramakrishnan, T. V. Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673–676 (1979).

    Article  ADS  Google Scholar 

  62. Agarwal, K., Ganeshan, S. & Bhatt, R. N. Localization and transport in a strongly driven Anderson insulator. Phys. Rev. B 96, 014201 (2017).

    Article  ADS  Google Scholar 

  63. Hatami, H., Danieli, C., Bodyfelt, J. D. & Flach, S. Quasiperiodic driving of Anderson localized waves in one dimension. Phys. Rev. E 93, 062205 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  64. Nakajima, S. et al. Competition and interplay between topology and quasi-periodic disorder in Thouless pumping of ultracold atoms. Nat. Phys. 17, 844–849 (2021).

    Article  Google Scholar 

  65. Li, J., Chu, R.-L., Jain, J. K. & Shen, S.-Q. Topological Anderson insulator. Phys. Rev. Lett. 102, 136806 (2009).

    Article  ADS  Google Scholar 

  66. Groth, C. W., Wimmer, M., Akhmerov, A. R., Tworzydło, J. & Beenakker, C. W. J. Theory of the topological Anderson insulator. Phys. Rev. Lett. 103, 196805 (2009).

    Article  ADS  Google Scholar 

  67. 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 

  68. Citro, R., Andrei, N. & Niu, Q. Pumping in an interacting quantum wire. Phys. Rev. B 68, 165312 (2003).

    Article  ADS  Google Scholar 

  69. Requist, R. & Gross, E. K. U. Accurate formula for the macroscopic polarization of strongly correlated materials. J. Phys. Chem. Lett. 9, 7045–7051 (2018).

    Article  Google Scholar 

  70. Nakagawa, M., Yoshida, T., Peters, R. & Kawakami, N. Breakdown of topological Thouless pumping in the strongly interacting regime. Phys. Rev. B 98, 115147 (2018).

    Article  ADS  Google Scholar 

  71. Bertok, E., Heidrich-Meisner, F. & Aligia, A. A. Splitting of topological charge pumping in an interacting two-component fermionic Rice–Mele Hubbard model. Phys. Rev. B 106, 045141 (2022).

    Article  ADS  Google Scholar 

  72. Stenzel, L., Hayward, A. L. C., Hubig, C., Schollwöck, U. & Heidrich-Meisner, F. Quantum phases and topological properties of interacting fermions in one-dimensional superlattices. Phys. Rev. A 99, 053614 (2019).

    Article  ADS  Google Scholar 

  73. Berg, E., Levin, M. & Altman, E. Quantized pumping and topology of the phase diagram for a system of interacting bosons. Phys. Rev. Lett. 106, 110405 (2011).

    Article  ADS  Google Scholar 

  74. Qian, Y., Gong, M. & Zhang, C. Quantum transport of bosonic cold atoms in double-well optical lattices. Phys. Rev. A 84, 013608 (2011).

    Article  ADS  Google Scholar 

  75. Grusdt, F. & Höning, M. Realization of fractional Chern insulators in the thin-torus limit with ultracold bosons. Phys. Rev. A 90, 053623 (2014).

    Article  ADS  Google Scholar 

  76. Zeng, T.-S., Zhu, W. & Sheng, D. N. Fractional charge pumping of interacting bosons in one-dimensional superlattice. Phys. Rev. B 94, 235139 (2016).

    Article  ADS  Google Scholar 

  77. Greschner, S., Mondal, S. & Mishra, T. Topological charge pumping of bound bosonic pairs. Phys. Rev. A 101, 053630 (2020).

    Article  ADS  Google Scholar 

  78. Kohn, W. Theory of the insulating state. Phys. Rev. 133, A171 (1964).

    Article  ADS  MathSciNet  Google Scholar 

  79. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    Article  ADS  Google Scholar 

  80. Harper, P. G. Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A 68, 874 (1955).

    Article  ADS  MATH  Google Scholar 

  81. Mostaan, N., Grusdt, F. & Goldman, N. Quantized transport of solitons in nonlinear Thouless pumps: from Wannier drags to ultracold topological mixtures. Preprint at https://arxiv.org/abs/2110.13075 (2022).

  82. Jürgensen, M. & Rechtsman, M. C. Chern number governs soliton motion in nonlinear Thouless pumps. Phys. Rev. Lett. 128, 113901 (2022).

    Article  ADS  Google Scholar 

  83. Fu, Q., Wang, P., Kartashov, Y. V., Konotop, V. V. & Ye, F. Nonlinear Thouless pumping: solitons and transport breakdown. Phys. Rev. Lett. 128, 154101 (2022).

    Article  ADS  Google Scholar 

  84. Jürgensen, M., Mukherjee, S., Jörg, C. & Rechtsman, M. C. Quantized nonlinear Thouless pumping. Nature 596, 63–67 (2021).

    Article  ADS  Google Scholar 

  85. Jung, P. S. et al. Optical Thouless pumping transport and nonlinear switching in a topological low-dimensional discrete nematic liquid crystal array. Phys. Rev. A 105, 013513 (2022).

    Article  ADS  Google Scholar 

  86. Hayward, A., Schweizer, C., Lohse, M., Aidelsburger, M. & Heidrich-Meisner, F. Topological charge pumping in the interacting bosonic Rice–Mele model. Phys. Rev. B 98, 245148 (2018).

    Article  ADS  Google Scholar 

  87. Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  88. Lin, L., Ke, Y. & Lee, C. Interaction-induced topological bound states and Thouless pumping in a one-dimensional optical lattice. Phys. Rev. A 101, 023620 (2020).

    Article  ADS  Google Scholar 

  89. Kuno, Y. & Hatsugai, Y. Interaction-induced topological charge pump. Phys. Rev. Res. 2, 042024 (2020).

    Article  Google Scholar 

  90. Esin, I. et al. Universal transport in periodically driven systems without long-lived quasiparticles. Preprint at https://arXiv.org/abs/2203.01313 (2022).

  91. Ke, Y., Qin, X., Kivshar, Y. S. & Lee, C. Multiparticle Wannier states and Thouless pumping of interacting bosons. Phys. Rev. A 95, 063630 (2017).

    Article  ADS  Google Scholar 

  92. Haug, T., Amico, L., Kwek, L.-C., Munro, W. J. & Bastidas, V. M. Topological pumping of quantum correlations. Phys. Rev. Res. 2, 013135 (2020).

    Article  Google Scholar 

  93. Ando, T., Matsumoto, Y. & Uemura, Y. Theory of Hall effect in a two-dimensional electron system. J. Phys. Soc. Jpn. 39, 279–288 (1975).

    Article  ADS  Google Scholar 

  94. Laughlin, R. B. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 5632–5633 (1981).

    Article  ADS  Google Scholar 

  95. Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).

    Article  ADS  Google Scholar 

  96. Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  97. Stuhl, B. K., Lu, H.-I., Aycock, L. M., Genkina, D. & Spielman, I. B. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1518 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  98. Livi, L. et al. Synthetic dimensions and spin–orbit coupling with an optical clock transition. Phys. Rev. Lett. 117, 220401 (2016).

    Article  ADS  Google Scholar 

  99. Kolkowitz, S. et al. Spin–orbit-coupled fermions in an optical lattice clock. Nature 542, 66–70 (2017).

    Article  ADS  Google Scholar 

  100. An, F. A., Meier, E. J. & Gadway, B. Direct observation of chiral currents and magnetic reflection in atomic flux lattices. Sci. Adv. 3, e1602685 (2017).

    Article  ADS  Google Scholar 

  101. Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356–360 (2019).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  103. Meier, E. J. et al. Observation of the topological Anderson insulator in disordered atomic wires. Science 362, 929–933 (2018).

    Article  ADS  Google Scholar 

  104. Han, J. H., Kang, J. H. & Shin, Y. Band gap closing in a synthetic Hall tube of neutral fermions. Phys. Rev. Lett. 122, 065303 (2019).

    Article  ADS  Google Scholar 

  105. Liang, Q.-Y. et al. Coherence and decoherence in the Harper–Hofstadter model. Phys. Rev. Res. 3, 023058 (2021).

    Article  Google Scholar 

  106. Li, C.-H. et al. Bose–Einstein condensate on a synthetic topological Hall cylinder. PRX Quantum 3, 010316 (2022).

    Article  ADS  Google Scholar 

  107. Fabre, A., Bouhiron, J.-B., Satoor, T., Lopes, R. & Nascimbene, S. Laughlin’s topological charge pump in an atomic Hall cylinder. Phys. Rev. Lett. 128, 173202 (2022).

    Article  ADS  Google Scholar 

  108. Price, H., Zilberberg, O., Ozawa, T., Carusotto, I. & Goldman, N. Four-dimensional quantum Hall effect with ultracold atoms. Phys. Rev. Lett. 115, 195303 (2015).

    Article  ADS  Google Scholar 

  109. Ozawa, T., Price, H. M., Goldman, N., Zilberberg, O. & Carusotto, I. Synthetic dimensions in integrated photonics: from optical isolation to four-dimensional quantum Hall physics. Phys. Rev. A 93, 043827 (2016).

    Article  ADS  Google Scholar 

  110. Lu, L., Gao, H. & Wang, Z. Topological one-way fiber of second Chern number. Nat. Commun. 9, 5384 (2018).

    Article  ADS  Google Scholar 

  111. Kolodrubetz, M. Measuring the second Chern number from nonadiabatic effects. Phys. Rev. Lett. 117, 015301 (2016).

    Article  ADS  Google Scholar 

  112. Kraus, Y. E., Ringel, Z. & Zilberberg, O. Four-dimensional quantum Hall effect in a two-dimensional quasicrystal. Phys. Rev. Lett. 111, 226401 (2013).

    Article  ADS  Google Scholar 

  113. Lohse, M., Schweizer, C., Price, H. M., Zilberberg, O. & Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55–58 (2018).

    Article  ADS  Google Scholar 

  114. Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59–62 (2018).

    Article  ADS  Google Scholar 

  115. 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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  116. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  117. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).

    Article  ADS  Google Scholar 

  118. Benalcazar, W. A. et al. Higher-order topological pumping. Phys. Rev. B 105, 195129 (2022).

    Article  ADS  Google Scholar 

  119. Noguchi, R. et al. Evidence for a higher-order topological insulator in a three-dimensional material built from van der Waals stacking of bismuth–halide chains. Nat. Mater. 20, 473–479 (2021).

    Article  ADS  Google Scholar 

  120. Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018).

    Article  ADS  Google Scholar 

  121. Imhof, S. et al. Topoelectrical-circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).

    Article  Google Scholar 

  122. Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).

    Article  ADS  Google Scholar 

  123. Bao, J. et al. Topoelectrical circuit octupole insulator with topologically protected corner states. Phys. Rev. B 100, 201406 (2019).

    Article  ADS  Google Scholar 

  124. Mittal, S. et al. Photonic quadrupole topological phases. Nat. Photon. 13, 692–696 (2019).

    Article  ADS  Google Scholar 

  125. Ni, X., Weiner, M., Alú, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).

    Article  Google Scholar 

  126. Ni, X., Li, M., Weiner, M., Alú, A. & Khanikaev, A. B. Demonstration of a quantized acoustic octupole topological insulator. Nat. Commun. 11, 2108 (2020).

    Article  ADS  Google Scholar 

  127. Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higher-order topological insulator on a Kagome lattice. Nat. Mater. 18, 108–112 (2018).

    Article  Google Scholar 

  128. Dutt, A., Minkov, M., Williamson, I. A. D. & Fan, S. Higher-order topological insulators in synthetic dimensions. Light Sci. Appl. https://doi.org/10.1038/s41377-020-0334-8 (2020).

  129. Kang, B., Shiozaki, K. & Cho, G. Y. Many-body order parameters for multipoles in solids. Phys. Rev. B 100, 245134 (2019).

    Article  ADS  Google Scholar 

  130. Petrides, I. & Zilberberg, O. Higher-order topological insulators, topological pumps and the quantum Hall effect in high dimensions. Phys. Rev. Res. 2, 022049 (2020).

    Article  Google Scholar 

  131. Kang, B., Lee, W. & Cho, G. Y. Many-body invariants for Chern and chiral hinge insulators. Phys. Rev. Lett. 126, 016402 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  132. Wienand, J. F., Horn, F., Aidelsburger, M., Bibo, J. & Grusdt, F. Thouless pumps and bulk-boundary correspondence in higher-order symmetry-protected topological phases. Phys. Rev. Lett. 128, 246602 (2022).

    Article  ADS  Google Scholar 

  133. Resta, R. Quantum-mechanical position operator in extended systems. Phys. Rev. Lett. 80, 1800–1803 (1998).

    Article  ADS  Google Scholar 

  134. Araki, H., Mizoguchi, T. & Hatsugai, Y. \({{\mathbb{Z}}}_{Q}\) Berry phase for higher-order symmetry-protected topological phases. Phys. Rev. Res. 2, 012009 (2020).

  135. Berry, B. M. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. Ser. A 392, 45 (1984).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  136. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).

    Article  ADS  Google Scholar 

  137. Brosco, V., Pilozzi, L., Fazio, R. & Conti, C. Non-Abelian Thouless pumping in a photonic lattice. Phys. Rev. A 103, 063518 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  138. Resta, R. Macroscopic polarization in crystalline dielectrics: the geometric phase approach. Rev. Mod. Phys. 66, 899–915 (1994).

    Article  ADS  Google Scholar 

  139. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

    Article  ADS  Google Scholar 

  140. Harper, P. G. The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals. Proc. Phys. Soc. A 68, 879–892 (1955).

    Article  ADS  MATH  Google Scholar 

  141. Azbel, M. Y. Energy spectrum of a conduction electron in a magnetic field. Zh. Eksp. Teor. Fiz. 46, 929–946 (1964).

    Google Scholar 

  142. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    Article  ADS  Google Scholar 

  143. Aubry, S. & André, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133–140 (1980).

    MathSciNet  MATH  Google Scholar 

  144. Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).

    Article  ADS  Google Scholar 

  145. Marra, P., Citro, R. & Ortix, C. Fractional quantization of the topological charge pumping in a one-dimensional superlattice. Phys. Rev. B 91, 125411 (2015).

    Article  ADS  Google Scholar 

  146. Wei, R. & Mueller, E. J. Anomalous charge pumping in a one-dimensional optical superlattice. Phys. Rev. A 92, 013609 (2015).

    Article  ADS  Google Scholar 

  147. Resta, R. Theory of the electric polarization in crystals. Ferroelectrics 136, 51–55 (1992).

    Article  ADS  Google Scholar 

  148. King-Smith, R. D. & Vanderbilt, D. Theory of polarization of crystalline solids. Phys. Rev. B 47, 1651–1654 (1993).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors thank all of their collaborators on the topic of topological pumping and related work, with whom they have had many stimulating interactions and discussions. M.A. acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Research Unit FOR 2414 under project number 277974659 and under Germany’s Excellence Strategy — EXC-2111 — 390814868. R.C. acknowledges the project Quantox of QuantERA ERA-NET Cofund in Quantum Technologies (grant agreement number 731473).

Author information

Authors and Affiliations

  1. Physics Department ‘E. R. Caianiello’ and CNR-SPIN, Università degli Studi di Salerno, Fisciano, (SA), Italy

    Roberta Citro

  2. INFN — Gruppo Collegato di Salerno, Salerno, Italy

    Roberta Citro

  3. Faculty of Physics, Ludwig-Maximilians-Universität München, Munich, Germany

    Monika Aidelsburger

  4. Munich Center for Quantum Science and Technology (MCQST), Munich, Germany

    Monika Aidelsburger

Authors
  1. Roberta Citro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monika Aidelsburger
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed equally to all aspects of the article.

Corresponding authors

Correspondence to Roberta Citro or Monika Aidelsburger.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Reviews Physics thanks Sebabrata Mukherjee, Shuta Nakajima, Chaohong Lee and Yongguan Ke for their contribution to the peer review of this work.

Additional information

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

Glossary

Anderson-localized

The absence of diffusion in a disordered medium, which enables electron localization in a lattice potential, provided that the degree of disorder is sufficiently large.

Bare tunnelling

Probability of a particle jumping from one lattice site to another in the absence of interaction among the particles.

Glide symmetry

In 2D, a glide reflection symmetry is a symmetry operation that consists in a reflection with respect to a line and then a translation along that line, combined into a single operation.

Hardcore interaction

An infinite interaction between particles, so that particles cannot overlap in space; represented by a Dirac delta potential that diverges, when the positions of the particles coincide.

Hardcore bosons

Bosons that cannot occupy the same quantum state, like fermions; they interact with a Dirac delta potential,  but without antisymmetric exchange statistics.

Kubo formula

A formula expressing the linear response of a system quantity to an external time-dependent perturbation, which can be used to calculate the susceptibility of particles systems in response to a field.

Landau–Zener transition

The probability of finding the system in the upper energy branch, when changing the system parameters through an avoided crossing. In the adiabatic limit, there will be no excitation to the upper branch.

Slater determinant

An expression that describes the fermionic many-body wavefunction, satisfying the Pauli principle and the antisymmetric condition under the exchange of two particles.

Solitons and antisolitons

The soliton is an excitation (like a kink) that propagates at a constant velocity in a nonlinear medium. The antisoliton is a kink that propagates in the opposite direction.

Wannier orbitals

Localized molecular orbitals of crystalline systems that belong to a complete set of orthogonal functions used in solid-state physics.

Wannier-state formalism

A solid-state physics formalism that uses Wannier functions as a complete set of orthogonal functions.

Winding number

An integer representing the number of times that the curve travels anticlockwise around a singularity. The winding number depends on the orientation of the curve.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Citro, R., Aidelsburger, M. Thouless pumping and topology. Nat Rev Phys 5, 87–101 (2023). https://doi.org/10.1038/s42254-022-00545-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-022-00545-0

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