Abstract
Robustness against perturbations lies at the heart of topological phenomena. If, however, a perturbation such as disorder becomes dominant, it may cause a topological phase transition between topologically non-trivial and trivial phases. Here we experimentally reveal the competition and interplay between topology and quasi-periodic disorder in a Thouless pump realized with ultracold atoms in an optical lattice, by creating a quasi-periodic potential from weak to strong regimes in a controllable manner. We demonstrate a disorder-induced pumping in which the presence of quasi-periodic disorder can induce a non-trivial pump for a specific pumping sequence, whereas no pump is observed in the clean limit. Our highly controllable system, which can also straightforwardly incorporate interatomic interaction, could be a unique platform for studying various disorder-related effects in a wide range of topological quantum phenomena.
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Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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The codes used for the numerical simulations within this paper are available from the corresponding author upon reasonable request.
References
Lagendijk, A., van Tiggelen, B. & Wiersma, D. S. Fifty years of Anderson localization. Phys. Today 62, 24â29 (2009).
Niu, Q. & Thouless, D. J. Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction. J. Phys. A 17, 2453â2462 (1984).
Niu, Q., Thouless, D. J. & Wu, Y.-S. Quantized Hall conductance as a topological invariant. Phys. Rev. B 31, 3372â3377 (1985).
Kobayashi, K., Ohtsuki, T. & Imura, K.-I. Disordered weak and strong topological insulators. Phys. Rev. Lett. 110, 236803 (2013).
Leung, B. & Prodan, E. Effect of strong disorder in a three-dimensional topological insulator: phase diagram and maps of the \({{\mathbb{Z}}}_{2}\) invariant. Phys. Rev. B 85, 205136 (2012).
Yamakage, A., Nomura, K., Imura, K.-I. & Kuramoto, Y. Disorder-induced multiple transition involving \({{\mathbb{Z}}}_{2}\) topological insulator. J. Phys. Soc. Jpn 80, 053703 (2011).
Li, J., Chu, R.-L., Jain, J. K. & Shen, S.-Q. Topological Anderson insulator. Phys. Rev. Lett. 102, 136806 (2009).
McGinley, M., Knolle, J. & Nunnenkamp, A. Robustness of Majorana edge modes and topological order: exact results for the symmetric interacting Kitaev chain with disorder. Phys. Rev. B 96, 241113 (2017).
Shen, S.-Q. Topological Insulators: Dirac Equation in Condensed Matters 2nd edn (Springer, 2017).
Meier, E. J. et al. Observation of the topological Anderson insulator in disordered atomic wires. Science 362, 929â933 (2018).
Stützer, S. et al. Photonic topological Anderson insulators. Nature 560, 461â465 (2018).
Li, G.-G. et al. Topological Anderson insulator in disordered photonic crystals. Phys. Rev. Lett. 125, 133603 (2020).
Titum, P., Berg, E., Rudner, M. S., Refael, G. & Lindner, N. H. Anomalous FloquetâAnderson insulator as a nonadiabatic quantized charge pump. Phys. Rev. X 6, 021013 (2016).
Sriluckshmy, P. V., Saha, K. & Moessner, R. Interplay between topology and disorder in a two-dimensional semi-Dirac material. Phys. Rev. B 97, 024204 (2018).
Mondragon-Shem, I. & Hughes, T. L. Signatures of metal-insulator and topological phase transitions in the entanglement of one-dimensional disordered fermions. Phys. Rev. B 90, 104204 (2014).
Thouless, D. J. Quantization of particle transport. Phys. Rev. B 27, 6083â6087 (1983).
Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296â300 (2016).
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, 350â354 (2016).
Schweizer, C., Lohse, M., Citro, R. & Bloch, I. Spin pumping and measurement of spin currents in optical superlattices. Phys. Rev. Lett. 117, 170405 (2016).
Lohse, M., Schweizer, C., Price, H. M., Zilberberg, O. & Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55â58 (2018).
Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).
Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59â62 (2018).
Qin, J. & Guo, H. Quantum pumping induced by disorder in one dimension. Phys. Lett. A 380, 2317â2321 (2016).
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).
Imura, K.-I., Yoshimura, Y., Fukui, T. & Hatsugai, Y. Bulk-edge correspondence in topological transport and pumping. J. Phys. Conf. Ser. 969, 012133 (2018).
Kuno, Y. Disorder-induced Chern insulator in the HarperâHofstadterâHatsugai model. Phys. Rev. B 100, 054108 (2019).
Nakagawa, M., Yoshida, T., Peters, R. & Kawakami, N. Breakdown of topological Thouless pumping in the strongly interacting regime. Phys. Rev. B 98, 115147 (2018).
Hayward, A., Schweizer, C., Lohse, M., Aidelsburger, M. & Heidrich-Meisner, F. Topological charge pumping in the interacting bosonic RiceâMele model. Phys. Rev. B 98, 245148 (2018).
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).
Mei, F., Chen, G., Goldman, N., Xiao, L. & Jia, S. Topological magnon insulator and quantized pumps from strongly-interacting bosons in optical superlattices. New J. Phys. 21, 095002 (2019).
Rice, M. J. & Mele, E. J. Elementary excitations of a linearly conjugated diatomic polymer. Phys. Rev. Lett. 49, 1455â1459 (1982).
Atala, M. et al. Direct measurement of the Zak phase in topological Bloch bands. Nat. Phys. 9, 795â800 (2013).
Hayward, A. L. C., Bertok, E., Schneider, U. & Heidrich-Meisner, F. Effect of disorder on topological charge pumping in the RiceâMele model. Preprint at http://arxiv.org/abs/2010.15249 (2020).
Asbóth, J. K., Oroszlány, L. & Pályi, A. A Short Course on Topological Insulators Vol. 919 (Springer, 2016).
Zhang, Y.-F. et al. Coupling-matrix approach to the Chern number calculation in disordered systems. Chin. Phys. B 22, 117312 (2013).
Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237â240 (2014).
Das, K. K. & Christ, J. Realizing the Harper model with ultracold atoms in a ring lattice. Phys. Rev. A 99, 013604 (2019).
Marra, P. & Nitta, M. Topologically quantized current in quasiperiodic Thouless pumps. Phys. Rev. Res. 2, 042035 (2020).
Aubry, S. & André, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133â164 (1980).
Huse, D. A., Nandkishore, R., Oganesyan, V., Pal, A. & Sondhi, S. L. Localization-protected quantum order. Phys. Rev. B 88, 014206 (2013).
Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15â38 (2015).
Price, H. M., Zilberberg, O., Ozawa, T., Carusotto, I. & Goldman, N. Four-dimensional quantum Hall effect with ultracold atoms. Phys. Rev. Lett. 115, 195303 (2015).
Ippoliti, M. & Bhatt, R. N. Dimensional crossover of the integer quantum Hall plateau transition and disordered topological pumping. Phys. Rev. Lett. 124, 086602 (2020).
Privitera, L., Russomanno, A., Citro, R. & Santoro, G. E. Nonadiabatic breaking of topological pumping. Phys. Rev. Lett. 120, 106601 (2018).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61â66 (2017).
Xie, B.-Y. et al. Second-order photonic topological insulator with corner states. Phys. Rev. B 98, 205147 (2018).
Li, C.-A., Fu, B., Hu, Z.-A., Li, J. & Shen, S.-Q. Topological phase transitions in disordered electric quadrupole insulators. Phys. Rev. Lett. 125, 166801 (2020).
Araki, H., Mizoguchi, T. & Hatsugai, Y. Phase diagram of a disordered higher-order topological insulator: a machine learning study. Phys. Rev. B 99, 085406 (2019).
Kitagawa, M. et al. Two-color photoassociation spectroscopy of ytterbium atoms and the precise determinations of s-wave scattering lengths. Phys. Rev. A 77, 012719 (2008).
Taie, S. et al. Realization of a SU(2)âÃâSU(6) system of fermions in a cold atomic gas. Phys. Rev. Lett. 105, 190401 (2010).
Acknowledgements
We thank H. Aoki, Y. Hatsugai and K. Imura for valuable discussions and A. Sawada for experimental assistance. This work was supported by Grants-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (JSPS) (grant nos. JP25220711, JP26247064, JP16H00990, JP16H01053, JP17H06138, JP18H05405, JP18H05228 and JP18K13480), the Impulsing Paradigm Change through Disruptive Technologies (ImPACT) programme, the CREST programme of the Japan Science and Technology Agency (grant no. JPMJCR1673) and the Quantum Leap Flagship Program of the Ministry of Education, Culture, Sports, Science and Technology (MEXT Q-LEAP) (grant no. JPMXS0118069021). Y.K. acknowledges the support of a Grant-in-Aid for JSPS Fellows (grant no. 17J00486). P.M. is supported by the JST (CREST grant. no. JPMJCR19T2), by the MEXT-supported Program for the Strategic Research Foundation at Private Universities Topological Science (grant no. S1511006) and by a JSPS Grant-in-Aid for Early-Career Scientists (grant no. 20K14375).
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S.N., N.T. and K.S. carried out experiments and the data analysis. Y.K. and P.M. carried out the theoretical calculation. Y.T. conducted the whole experiment. All the authors contributed to the writing of the manuscript.
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Nakajima, S., Takei, N., Sakuma, K. et al. Competition and interplay between topology and quasi-periodic disorder in Thouless pumping of ultracold atoms. Nat. Phys. 17, 844â849 (2021). https://doi.org/10.1038/s41567-021-01229-9
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DOI: https://doi.org/10.1038/s41567-021-01229-9
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